Steven Schaefer

Monoids for Day Convolution

2026-10-02T00:00:00Z

If $𝒞︀$ is a monoidal, $𝒱︀$ -enriched category, then monoid objects for the Day Convolution over $𝒞︀$ are given by lax monoidal functors from $𝒞︀$ to $𝒱︀$ .

Getting Organized Again: My First Weeknote

2026-10-01T00:00:00Z

Notetaking in Helianthus

The running theme this week is that I ought to get organized again. Over the last few weeks, I have experimented a bit writing in Typst rather than LaTeX, and it has been a delight. A document builds in an instant! This gives a much faster feedback loop than TeX, and this makes writing much more enjoyable.

I have been using Forester, which builds atop LaTeX, for a couple of years now. In that time, I have dabbled with some extensions to Forester that met my needs, but none

Are maintainable enough to merge upstream, or
that are large enough for me to feel equipped for a Typst rewrite

of the tool

So this last week, I am experimenting with the reality that I might not even need to write the tool myself.

This resulted in the creation of Helianthus, which I am now using to keep my notes and build my personal website.

I am currently in the process of porting over my old notes, my recent Typst notes, and writing up more on the dependent parser combinators in Lambek ${\begin{matrix} \end{matrix}}^{𝙳}$ .

Sharing Notes

I was reading Jon Sterling’s Research Group Manual. One thing he suggest is that everyone in his students write weeknotes, published in an RSS feed, which he reads before any meetings. He further suggests the everyone should subscribe to each other’s RSS feeds. Should anyone be interested in participating in this process, here is my feed. I would also love to read any other RSS-published notes that anyone else may take, with the selfish goal of being able to intergrate them throughout the rest of my notetaking system.

This kind of collaborative knowledge sharing is what I’d hoped to get off of the ground with the lab’s Forest, but I think there may have been too much friction for that to get off of the ground. Even using Forester myself, I often wouldn’t want to go through the extra effort of polishing my notes for the lab’s Forest, nor did I have much faith that other’s would be reading it. Rather than an external system that needs much buy-in from the rest of the group, I think that a passive system like this has a higher likelihood of benefitting us.

I am not aiming to ask anyone to change whatever method might work best for them, this is soley an open request for anyone who may be interested. I also am not asking anyone to necessarily ever read them.

Max seems to prefer Zulip, I could also wire up a hacky Zulip bot (for which people should probably turn notifications off) for relaying notes periodically.

Learning about monoids and modules

I spent some time this week learning about monoid object and their modules. After seeing Max and Eric talk about modules for monads, I wanted to figure out if similar ideas had any connection to my parser combinators.

The first question I wanted to figure out was: what are the monoid objects for the Day monoidal structure? And secondly: what are the modules of these monoid objects?

(For my own sake, I still need to write these observations up in their own notes)

I was surprised to learn that the monoid objects in Lambek ${\begin{matrix} \end{matrix}}^{𝙳}$ are just monoids with a monoid homomorphism into the alphabet, which is underwhelming.

The Schanuel Topos

2026-10-01T00:00:00Z

Let $𝕀$ be the category of finite sets and injections. The Schanuel topos is the category of pullback-preserving functors $𝕀^{op} \to 𝐒𝐞𝐭$ .

Equivalently, it is the category of nominal sets, which are sets equipped with an action by the group of permutations on a countable set of names $𝔸$ , such that every element has finite support.

Nominal Sets as Day Quotients

2026-10-01T00:00:00Z

In the Schanuel topos, the underlying category for Day convolution is $𝐶 = 𝕀^{op}$ , where $𝕀$ is the category of finite sets and injections.

Given a nominal set $𝑋$ , its presheaf action describes elements supported by $𝑡$ :

𝑋 (𝑡) = {𝑥 \in 𝑋 | 𝗌𝗎𝗉𝗉 (𝑥) \subseteq 𝑡}

Instead of taking $𝑋$ a priori as a presheaf, we can view it as a finitely- supported $𝖯𝖾𝗋𝗆 (𝔸)$ -set. We can restrict our attention to the groupoid core $𝐺 = 𝖼𝗈𝗋𝖾 (𝕀)$ , asking for the support to be exactly the input:

𝑋_{𝑠} = {𝑥 \in 𝑋 | 𝗌𝗎𝗉𝗉 (𝑥) = 𝑠}

This family $𝑋_{•} : 𝐺 \to 𝐒𝐞𝐭$ is functorial on finite sets and bijections.

By extending this functor along the inclusions described in Quotient Coincidence, we can extend $𝑋_{•}$ to both a presheaf $𝗆𝗄𝖯𝗌𝗁 (𝑋_{•})$ and a copresheaf $𝗆𝗄𝖢𝗈𝗉𝗌𝗁 (𝑋_{•})$ on $𝐶$ . This suggests the equivalence:

𝗆𝗄𝖯𝗌𝗁 (𝑋_{•}) ⊸ 𝑌 ≅ 𝐷_{𝗆𝗄𝖢𝗈𝗉𝗌𝗁 (𝑋_{•})} 𝑌

I suspect this allows us to describe name abstraction $[𝑋] 𝑌$ [1] —which ordinarily looks like an operation on two presheaves of the same variance—as the quotient of $𝑌$ by the induced copresheaf $𝗆𝗄𝖢𝗈𝗉𝗌𝗁 (𝑋_{•})$ .

Concretely, $[𝑋] 𝑌$ is usually given by the quotient of the product by an equivalence relation:

[𝑋] 𝑌 ≅ \frac{𝑋 \times 𝑌}{\sim}

where $(𝑥, 𝑦) \sim (𝜋 𝑥, 𝜋 𝑦)$ for permutations $𝜋$ fixing $𝗌𝗎𝗉𝗉 (𝑦) - 𝗌𝗎𝗉𝗉 (𝑥)$ .

On the other hand, the Day quotient computes to a coend:

(𝐷_{𝗆𝗄𝖢𝗈𝗉𝗌𝗁 (𝑋_{•})} 𝑌) (𝑐) = \int^{𝑠} 𝑋_{𝑠} \times 𝑌 (𝑠 ⊎ 𝑐)

A priori, a coend over $𝑠$ of the product $𝑋_{𝑠} \times 𝑌 (𝑠 ⊎ 𝑐)$ is expressed as the quotient of a set of triples by an equivalence relation $\approx$ :

{(𝑠, 𝑥, 𝑦) | 𝑠 = 𝗌𝗎𝗉𝗉 (𝑥), 𝗌𝗎𝗉𝗉 (𝑦) \subseteq 𝑠 ⊎ 𝑐} / \approx

However, because the comprehension formula fixes $𝑠 = 𝗌𝗎𝗉𝗉 (𝑥)$ , we reduce to a quotient of pairs:

{(𝑥, 𝑦) | 𝗌𝗎𝗉𝗉 (𝑦) \subseteq 𝗌𝗎𝗉𝗉 (𝑥) ⊎ 𝑐} / \approx

I suspect that this quotient will equate to the one given by $[𝑋] 𝑌$ , thus resolving the apparent issues with variance. I further suspect that one will need the sheaf condition (pullback-preservation) of nominal sets to establish this equivalence.

Nominal Sets

2026-10-01T00:00:00Z

Fix a countably infinite set of names $𝔸$ . A nominal set [1] is a set $𝑋$ equipped with an action by the group of finite permutations $𝖯𝖾𝗋𝗆 (𝔸)$ , such that every element $𝑥 \in 𝑋$ has a finite support.

A finite set of names $𝑆 \subseteq 𝔸$ supports $𝑥$ if any permutation fixing $𝑆$ pointwise also fixes $𝑥$ .

The category of nominal sets is equivalent to the Schanuel topos. Under this equivalence, a nominal set $𝑋$ corresponds to a functor $𝕀 \to 𝐒𝐞𝐭$ , where $𝕀$ is the category of finite sets and injections, given by mapping a finite set of names $𝑡$ to the set of elements supported by $𝑡$ :

𝑋 (𝑡) = {𝑥 \in 𝑋 | 𝑡 supports 𝑥}

Navigating This Site

2026-10-01T00:00:00Z

This site is a gross amalgamation of a research notebook, reference manager, and personal website, among other things. It holds 112 notes, 815 references, 2,179 people, 4 talks and 3 recipes.

Click a heading to fold or unfold its section. Press Ctrl-K/⌘K to search. To follow new entries, subscribe to the Atom feed.

Mechanized Category Theory

2026-10-01T00:00:00Z

Our research lab has a library of mechanized category theory in Agda that I help maintain, cubical-categorical-logic.

Cubical categorical logic

Lessons from Mechanizing Categorical Logic in Cubical Agda onpls-2026-talkSteven Schaefer, Johnson He, Max S. New · ONPLS · June 1, 2026

We present cubical-categorical-logic, a library of formalized category theory in Cubical Agda. The library’s core idea is to treat syntax as a free categorical structure whose dependent eliminator is stated via (displayed) universal properties. From the same reusable components we have proven canonicity and conservativity results across several type theories, as well as the coherence theorem for monoidal categories. In this talk, we discuss these applications and reflect on Cubical Agda as a host for mechanized metatheory, where it is sometimes a boon and sometimes a bane.

Slides

Further, there are plenty of notes and thoughts related to category theory more broadly. Although, these thoughts are often formalized, the formalization is currently not much represented in these notes. Instead, they just talk about math.

Category theory

Definition. Day Convolution day-convolution

Let $𝒱︀$ be a symmetric monoidal closed category that is complete and cocomplete. Let $(𝒞︀, \otimes_{𝒞︀}, 𝐼)$ be a small monoidal $𝒱︀$ -enriched category. The Day convolution of $𝒱︀$ -enriched presheaves $𝐴, 𝐵 : {𝒞︀}^{op} \to 𝒱︀$ is the enriched presheaf

(𝐴 \otimes_{Day} 𝐵) 𝑐 = \int^{𝑢, 𝑣} 𝒞︀ (𝑐, 𝑢 \otimes_{𝒞︀} 𝑣) \otimes_{𝒱︀} 𝐴 𝑢 \otimes_{𝒱︀} 𝐵 𝑣,

with unit the representable $𝒞︀ (-, 𝐼)$ .

Day convolution makes the enriched presheaf category $[{𝒞︀}^{op}, 𝒱︀]$ a monoidal category, symmetric when $𝒞︀$ is [1]. It is moreover closed.

Under the enriched Yoneda embedding the convolution of representables is representable, $𝒞︀ (-, 𝑥) \otimes_{Day} 𝒞︀ (-, 𝑦) ≅ 𝒞︀ (-, 𝑥 \otimes_{𝒞︀} 𝑦)$ , so Day convolution is the cocontinuous extension of the tensor of $𝒞︀$ .

As a Kan Extension

Equivalently, $𝐴 \otimes_{Day} 𝐵$ is the left Kan extension of $(𝑢, 𝑣) \mapsto 𝐴 𝑢 \otimes_{𝒱︀} 𝐵 𝑣$ along $\otimes_{𝒞︀}^{op} : {𝒞︀}^{op} \times {𝒞︀}^{op} \to {𝒞︀}^{op}$ .

In $𝐒𝐞𝐭$

When $𝒱︀ = 𝐒𝐞𝐭$ , we recover the ordinary Day convolution of presheaves $𝐴, 𝐵 : {𝒞︀}^{op} \to 𝐒𝐞𝐭$ , where the formula simplifies to:

(𝐴 \otimes_{Day} 𝐵) 𝑐 = \int^{𝑢, 𝑣} 𝒞︀ [𝑐, 𝑢 \otimes_{𝒞︀} 𝑣] \times 𝐴 𝑢 \times 𝐵 𝑣 .

Definition. The Quotient and its Right Adjoint in Day Convolution day-quotient

Let $𝐶$ be a small monoidal category. Recall that for presheaves $𝐴, 𝐵 : 𝐶^{op} \to 𝐒𝐞𝐭$ , the Day convolution provides a closed monoidal structure:

(𝐴 \otimes 𝐵) (𝑐) = \int^{𝑢, 𝑣} 𝐶 [𝑐, 𝑢 \otimes 𝑣] \times 𝐴 (𝑢) \times 𝐵 (𝑣)

(𝐴 ⊸ 𝐵) (𝑐) = \int_{𝑢} 𝐴 (𝑢) \to 𝐵 (𝑢 \otimes 𝑐)

which forms an adjunction $𝐴 \otimes - ⊣ 𝐴 ⊸ -$ .

For covariant functors $𝐴 : 𝐶 \to 𝐒𝐞𝐭$ and presheaves $𝐵 : 𝐶^{op} \to 𝐒𝐞𝐭$ , we can define the quotient and its right adjoint, each of which is a presheaf on $𝐶$ :

(𝐷_{𝐴} 𝐵) (𝑐) = \int^{𝑢} 𝐴 (𝑢) \times 𝐵 (𝑢 \otimes 𝑐)

(\sqrt[𝐴]{𝐵}) (𝑐) = \int_{𝑢, 𝑣} 𝐶 [𝑢 \otimes 𝑣, 𝑐] \to (𝐴 (𝑢) \to 𝐵 (𝑣))

These form the adjunction $𝐷_{𝐴} ⊣ \sqrt[𝐴]{-}$ .

Definition. The Quotient and Residual Coincidence in Day Convolution day-quotient-coincidence

The Day quotient, $𝐷_{𝐴} 𝐵$ , takes $𝐴$ to be a covariant functor and $𝐵$ to be a contravariant one. While the residual $𝐴 ⊸ 𝐵$ takes both to be contravariant.

This apparent variance mismatch disappears when we restrict to the groupoid core $𝐺 = 𝖼𝗈𝗋𝖾 (𝐶)$ , where variance is trivialized since $𝐺 ≅ 𝐺^{op}$ .

For any functor $𝐹 : 𝐺 \to 𝐒𝐞𝐭$ on the core, we can freely extend it to both a presheaf and a covariant copresheaf on $𝐶$ by left Kan extension along the respective inclusions $𝐺 \to 𝐶^{op}$ and $𝐺 \to 𝐶$ :

𝗆𝗄𝖯𝗌𝗁 (𝐹) (𝑐) = \int^{𝑠 \in 𝐺} 𝐶 [𝑐, 𝑠] \times 𝐹 (𝑠)

𝗆𝗄𝖢𝗈𝗉𝗌𝗁 (𝐹) (𝑐) = \int^{𝑠 \in 𝐺} 𝐶 [𝑠, 𝑐] \times 𝐹 (𝑠)

By substituting these extensions into the definitions of the residual and the quotient, we obtain a general coincidence for any $𝐹 : 𝐺 \to 𝐒𝐞𝐭$ and $𝐵 : 𝐶^{op} \to 𝐒𝐞𝐭$ :

𝗆𝗄𝖯𝗌𝗁 (𝐹) ⊸ 𝐵 ≅ 𝐷_{𝗆𝗄𝖢𝗈𝗉𝗌𝗁 (𝐹)} 𝐵

This equivalence states that computing the residual against the presheaf extension of $𝐹$ is perfectly isomorphic to taking the quotient by its covariant extension.

The Representable Case

Instantiating the above theorem at a representable functor on the core, $𝐹 = 𝐺 [𝑥, -]$ : the co-Yoneda lemma says that the left Kan extensions compute to the representables on $𝐶$ :

𝗆𝗄𝖯𝗌𝗁 (𝐺 [𝑥, -]) ≅ 𝐶 [-, 𝑥]

𝗆𝗄𝖢𝗈𝗉𝗌𝗁 (𝐺 [𝑥, -]) ≅ 𝐶 [𝑥, -]

Applying the general coincidence theorem, the left and right adjoints coincide precisely on (opposite-variance) representables:

(𝐶 [-, 𝑥] ⊸ 𝐵) (𝑐) ≅ 𝐵 (𝑥 \otimes 𝑐) ≅ (𝐷_{𝐶 [𝑥, -]} 𝐵) (𝑐)

When $𝐶$ is the discrete monoidal category of strings, this recovers the derivative of formal grammars.

In nominal sets, I suspect that this construction also describes name abstraction and the freshness quantifier, although I have not check all of the details of the proof.

Definition. Nominal Sets nominal-set

A finite set of names $𝑆 \subseteq 𝔸$ supports $𝑥$ if any permutation fixing $𝑆$ pointwise also fixes $𝑥$ .

𝑋 (𝑡) = {𝑥 \in 𝑋 | 𝑡 supports 𝑥}

Definition. Nominal Sets as Day Quotients nominal-sets-quotient

In the Schanuel topos, the underlying category for Day convolution is $𝐶 = 𝕀^{op}$ , where $𝕀$ is the category of finite sets and injections.

Given a nominal set $𝑋$ , its presheaf action describes elements supported by $𝑡$ :

𝑋 (𝑡) = {𝑥 \in 𝑋 | 𝗌𝗎𝗉𝗉 (𝑥) \subseteq 𝑡}

𝑋_{𝑠} = {𝑥 \in 𝑋 | 𝗌𝗎𝗉𝗉 (𝑥) = 𝑠}

This family $𝑋_{•} : 𝐺 \to 𝐒𝐞𝐭$ is functorial on finite sets and bijections.

𝗆𝗄𝖯𝗌𝗁 (𝑋_{•}) ⊸ 𝑌 ≅ 𝐷_{𝗆𝗄𝖢𝗈𝗉𝗌𝗁 (𝑋_{•})} 𝑌

Concretely, $[𝑋] 𝑌$ is usually given by the quotient of the product by an equivalence relation:

[𝑋] 𝑌 ≅ \frac{𝑋 \times 𝑌}{\sim}

where $(𝑥, 𝑦) \sim (𝜋 𝑥, 𝜋 𝑦)$ for permutations $𝜋$ fixing $𝗌𝗎𝗉𝗉 (𝑦) - 𝗌𝗎𝗉𝗉 (𝑥)$ .

On the other hand, the Day quotient computes to a coend:

(𝐷_{𝗆𝗄𝖢𝗈𝗉𝗌𝗁 (𝑋_{•})} 𝑌) (𝑐) = \int^{𝑠} 𝑋_{𝑠} \times 𝑌 (𝑠 ⊎ 𝑐)

A priori, a coend over $𝑠$ of the product $𝑋_{𝑠} \times 𝑌 (𝑠 ⊎ 𝑐)$ is expressed as the quotient of a set of triples by an equivalence relation $\approx$ :

{(𝑠, 𝑥, 𝑦) | 𝑠 = 𝗌𝗎𝗉𝗉 (𝑥), 𝗌𝗎𝗉𝗉 (𝑦) \subseteq 𝑠 ⊎ 𝑐} / \approx

However, because the comprehension formula fixes $𝑠 = 𝗌𝗎𝗉𝗉 (𝑥)$ , we reduce to a quotient of pairs:

{(𝑥, 𝑦) | 𝗌𝗎𝗉𝗉 (𝑦) \subseteq 𝗌𝗎𝗉𝗉 (𝑥) ⊎ 𝑐} / \approx

Definition. The Schanuel Topos schanuel-topos

Let $𝕀$ be the category of finite sets and injections. The Schanuel topos is the category of pullback-preserving functors $𝕀^{op} \to 𝐒𝐞𝐭$ .

Equivalently, it is the category of nominal sets, which are sets equipped with an action by the group of permutations on a countable set of names $𝔸$ , such that every element has finite support.

Theorem. Day Convolution is Closed day-closed-structure

Let $𝒱︀$ be a symmetric monoidal closed category that is complete and cocomplete. Let $(𝒞︀, \otimes_{𝒞︀}, 𝐼)$ be a small monoidal $𝒱︀$ -enriched category and $𝐴, 𝐵$ be $𝒱︀$ -enriched presheaves on $𝒞︀$ . Define

(𝐴 ⊸ 𝐵) 𝑐 = \int_{𝑢} 𝐴 𝑢 ⊸_{𝒱︀} 𝐵 (𝑢 \otimes_{𝒞︀} 𝑐) .

Then, for Day convolution $\otimes_{Day}$ ,

𝐴 \otimes_{Day} - ⊣ 𝐴 ⊸_{Day} - .

Proof. Proof that Day Convolution is Closed day-closed-structure-proof

For $𝑋, 𝐵 : {𝒞︀}^{op} \to 𝒱︀$ , the enriched hom in $[{𝒞︀}^{op}, 𝒱︀]$ is given by the end:

\begin{aligned} \int_{𝑐} ((𝐴 \otimes_{Day} 𝑋) 𝑐 ⊸_{𝒱︀} 𝐵 𝑐) & ≅ \int_{𝑐} ((\int^{𝑢, 𝑣} 𝒞︀ (𝑐, 𝑢 \otimes_{𝒞︀} 𝑣) \otimes_{𝒱︀} 𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} 𝐵 𝑐) \\ ≅ \int_{𝑐} \int_{𝑢, 𝑣} ((𝒞︀ (𝑐, 𝑢 \otimes_{𝒞︀} 𝑣) \otimes_{𝒱︀} 𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} 𝐵 𝑐) \\ ≅ \int_{𝑢, 𝑣} ((𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} \int_{𝑐} (𝒞︀ (𝑐, 𝑢 \otimes_{𝒞︀} 𝑣) ⊸_{𝒱︀} 𝐵 𝑐)) \\ ≅ \int_{𝑢, 𝑣} ((𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} 𝐵 (𝑢 \otimes_{𝒞︀} 𝑣)) \\ ≅ \int_{𝑣} (𝑋 𝑣 ⊸_{𝒱︀} \int_{𝑢} (𝐴 𝑢 ⊸_{𝒱︀} 𝐵 (𝑢 \otimes_{𝒞︀} 𝑣))) \\ ≅ \int_{𝑣} (𝑋 𝑣 ⊸_{𝒱︀} (𝐴 ⊸ 𝐵) 𝑣) . \end{aligned}

∎

Symmetrically, $(𝐵 ⟜ 𝐴) 𝑐 = \int_{𝑣} 𝐴 𝑣 ⊸_{𝒱︀} 𝐵 (𝑐 \otimes_{𝒞︀} 𝑣)$ and $- \otimes_{Day} 𝐴 ⊣ - ⟜ 𝐴$ , so the enriched presheaf category $[{𝒞︀}^{op}, 𝒱︀]$ is biclosed [1].

Definition. The Bicategory of Categories bicategory-of-categories

The bicategory of categories $𝖢𝖠𝖳$ has

as 0-cells, categories (at a fixed pair of universe levels, for objects and for morphisms);
as hom-category $𝖢𝖠𝖳 (𝒞︀, 𝒟︀)$ , the functor category $[𝒞︀, 𝒟︀]$ , so 1-cells are functors and 2-cells are natural transformations;
as identity 1-cell, the identity functor;
as composition, $𝐹 ⋆ 𝐺 = 𝐺 \circ 𝐹$ on functors. On natural transformations $𝛽 : 𝐹 \Rightarrow 𝐹'$ and $𝛾 : 𝐺 \Rightarrow 𝐺'$ the horizontal composite is given directly by its components
${(𝛽 ⋆ 𝛾)}_{𝑐} = 𝐺 (𝛽_{𝑐}) ⋆ 𝛾_{𝐹' 𝑐} .$

Every component of the left unitor, the right unitor and the associator is an identity morphism, and their inverses are identities too. So the only content of the triangle and pentagon is that composites of identities are identities.

$𝖢𝖠𝖳$ is still a bicategory and not a strict 2-category: $Id \circ 𝐹$ and $𝐹$ agree on objects and on morphisms, but in the formalization they are not the same functor definitionally. The structure cells are there to name that agreement.

A monad in $𝖢𝖠𝖳$ is an ordinary monad on a category, and a prestack is a pseudofunctor into $𝖢𝖠𝖳$ .

Definition. Corecursive algebra corecursive-algebra

Fix an endofunctor $𝐹 : 𝒞︀ \to 𝒞︀$ . An algebra $𝛼 : 𝐹 𝐵 \to 𝐵$ is corecursive when for every coalgebra $𝛾 : 𝑋 \to 𝐹 𝑋$ there is exactly one hylomorphism from $𝛾$ to $𝛼$ . Equivalently, the functor $𝖧𝗒𝗅𝗈 (-, 𝛼)$ of the hylomorphism profunctor is constantly a singleton. It is the dual of a recursive coalgebra: an $𝐹$ -algebra in $𝒞︀$ is corecursive exactly when it is recursive as an $𝐹^{op}$ -coalgebra in ${𝒞︀}^{op}$ .

Example. If $(𝜈 𝐹, 𝗈𝗎𝗍)$ is a terminal coalgebra, then $𝗈𝗎𝗍$ is invertible and ${𝗈𝗎𝗍}^{- 1} : 𝐹 (𝜈 𝐹) \to 𝜈 𝐹$ is a corecursive algebra: a solution of $ℎ = 𝛾 ⋆ 𝐹 ℎ ⋆ {𝗈𝗎𝗍}^{- 1}$ is the same thing as a solution of $ℎ ⋆ 𝗈𝗎𝗍 = 𝛾 ⋆ 𝐹 ℎ$ , that is, a coalgebra map into the terminal coalgebra, and there is exactly one, $𝗎𝗇𝖿𝗈𝗅𝖽 𝛾$ .

Definition. Hylomorphism hylomorphism

Fix an endofunctor $𝐹 : 𝒞︀ \to 𝒞︀$ , a coalgebra $𝛾 : 𝑋 \to 𝐹 𝑋$ and an algebra $𝛼 : 𝐹 𝐵 \to 𝐵$ . A hylomorphism (or coalgebra-to-algebra morphism) from $𝛾$ to $𝛼$ is a morphism $ℎ : 𝑋 \to 𝐵$ satisfying

ℎ = 𝛾 ⋆ 𝐹 ℎ ⋆ 𝛼 .

Definition. The hylomorphism profunctor hylomorphism-profunctor

Fix an endofunctor $𝐹 : 𝒞︀ \to 𝒞︀$ . Hylomorphisms form a profunctor from coalgebras to algebras,

𝖧𝗒𝗅𝗈 : 𝖢𝗈𝖺𝗅𝗀 {(𝐹)}^{op} \times 𝖠𝗅𝗀 (𝐹) \to 𝐒𝐞𝐭,

where $𝖧𝗒𝗅𝗈 ((𝑋, 𝛾), (𝐵, 𝛼))$ is the set of $ℎ : 𝑋 \to 𝐵$ with $ℎ = 𝛾 ⋆ 𝐹 ℎ ⋆ 𝛼$ .

The action is by composition. If $𝑔 : (𝑌, 𝛿) \to (𝑋, 𝛾)$ is a coalgebra morphism ( $𝑔 ⋆ 𝛾 = 𝛿 ⋆ 𝐹 𝑔$ ) and $𝑘 : (𝐵, 𝛼) \to (𝐵', 𝛼')$ is an algebra morphism ( $𝛼 ⋆ 𝑘 = 𝐹 𝑘 ⋆ 𝛼'$ ), then $𝑔 ⋆ ℎ ⋆ 𝑘$ is again a hylomorphism:

𝛿 ⋆ 𝐹 (𝑔 ⋆ ℎ ⋆ 𝑘) ⋆ 𝛼' = 𝛿 ⋆ 𝐹 𝑔 ⋆ 𝐹 ℎ ⋆ 𝛼 ⋆ 𝑘 = 𝑔 ⋆ 𝛾 ⋆ 𝐹 ℎ ⋆ 𝛼 ⋆ 𝑘 = 𝑔 ⋆ ℎ ⋆ 𝑘 .

In these terms, a coalgebra $𝛾$ is recursive when $𝖧𝗒𝗅𝗈 (𝛾, -)$ is the terminal functor, and an algebra $𝛼$ is corecursive when $𝖧𝗒𝗅𝗈 (-, 𝛼)$ is. For the inverse of an initial algebra the profunctor is representable, $𝖧𝗒𝗅𝗈 ({𝗂𝗇}^{- 1}, -) ≅ 𝖠𝗅𝗀 (𝐹) (𝗂𝗇, -)$ , and dually $𝖧𝗒𝗅𝗈 (-, {𝗈𝗎𝗍}^{- 1}) ≅ 𝖢𝗈𝖺𝗅𝗀 (𝐹) (-, 𝗈𝗎𝗍)$ for a terminal coalgebra.

When every value of $𝖧𝗒𝗅𝗈$ is a singleton, every divide-and-conquer specification over $𝐹$ has exactly one solution. This is what local contractivity guarantees.

Definition. Lax Functor lax-functor

A lax functor $𝐹 : ℬ︀ \to 𝒞︀$ between bicategories consists of

a map on 0-cells, $𝑥 \mapsto 𝐹 𝑥$ ;
for all $𝑥, 𝑦$ , a functor $𝐹_{𝑥, 𝑦} : ℬ︀ (𝑥, 𝑦) \to 𝒞︀ (𝐹 𝑥, 𝐹 𝑦)$ , acting on 1-cells and 2-cells;
a unit comparison, natural 2-cells $𝐹_{𝑥}^{0} : 1_{𝐹 𝑥} \Rightarrow 𝐹 (1_{𝑥})$ ;
a composition comparison, 2-cells $𝐹_{𝑓, 𝑔}^{2} : 𝐹 𝑓 ⋆ 𝐹 𝑔 \Rightarrow 𝐹 (𝑓 ⋆ 𝑔)$ natural in $𝑓$ and $𝑔$ ;
such that three coherence laws hold, one for each structure cell of $ℬ︀$ :
- left unit: $(𝐹^{0} ▷ 𝐹 𝑓) ⋆ 𝐹_{1, 𝑓}^{2} ⋆ 𝐹 (𝜆_{𝑓}) = 𝜆_{𝐹 𝑓}$ ;
- right unit: $(𝐹 𝑓 ◁ 𝐹^{0}) ⋆ 𝐹_{𝑓, 1}^{2} ⋆ 𝐹 (𝜌_{𝑓}) = 𝜌_{𝐹 𝑓}$ ;
- associativity: $(𝐹_{𝑓, 𝑔}^{2} ▷ 𝐹 ℎ) ⋆ 𝐹_{𝑓 ⋆ 𝑔, ℎ}^{2} ⋆ 𝐹 (𝛼_{𝑓, 𝑔, ℎ}) = 𝛼_{𝐹 𝑓, 𝐹 𝑔, 𝐹 ℎ} ⋆ (𝐹 𝑓 ◁ 𝐹_{𝑔, ℎ}^{2}) ⋆ 𝐹_{𝑓, 𝑔 ⋆ ℎ}^{2}$ .

Here $⋆$ between 2-cells is vertical composition, and $𝜃 ▷ ℎ$ and $ℎ ◁ 𝜃$ are whiskerings.

Lax functors compose: ${(𝐺 \circ 𝐹)}^{0} = 𝐺^{0} ⋆ 𝐺 (𝐹^{0})$ and ${(𝐺 \circ 𝐹)}_{𝑓, 𝑔}^{2} = 𝐺_{𝐹 𝑓, 𝐹 𝑔}^{2} ⋆ 𝐺 (𝐹_{𝑓, 𝑔}^{2})$ . The coherence laws of the composite follow from those of $𝐹$ and $𝐺$ and naturality of $𝐺^{2}$ , without using the triangle or pentagon of any of the bicategories involved.

A lax functor whose comparison cells are invertible is a pseudofunctor.

Definition. Lax and Pseudonatural Transformations lax-natural-transformation

Let $𝐹, 𝐺 : ℬ︀ \to 𝒞︀$ be lax functors. A lax natural transformation $𝜎 : 𝐹 \Rightarrow 𝐺$ consists of

for each 0-cell $𝑥$ , a 1-cell $𝜎_{𝑥} : 𝐹 𝑥 \to 𝐺 𝑥$ ;
for each 1-cell $𝑓 : 𝑥 \to 𝑦$ , a 2-cell filling the naturality square,
$𝜎_{𝑓} : 𝐹 𝑓 ⋆ 𝜎_{𝑦} \Rightarrow 𝜎_{𝑥} ⋆ 𝐺 𝑓;$
such that $𝜎_{𝑓}$ is natural in $𝑓$ : for a 2-cell $𝜃 : 𝑓 \Rightarrow 𝑔$ , $(𝐹 𝜃 ▷ 𝜎_{𝑦}) ⋆ 𝜎_{𝑔} = 𝜎_{𝑓} ⋆ (𝜎_{𝑥} ◁ 𝐺 𝜃)$ ;
and such that $𝜎$ respects the comparison cells of $𝐹$ and $𝐺$ : one law relating $𝜎_{1_{𝑥}}$ to $𝐹^{0}$ , $𝐺^{0}$ and the unitors, and one relating $𝜎_{𝑓 ⋆ 𝑔}$ to $𝜎_{𝑓}$ , $𝜎_{𝑔}$ , $𝐹^{2}$ , $𝐺^{2}$ and the associators.

A lax natural transformation is pseudonatural when every $𝜎_{𝑓}$ is invertible. As with pseudofunctors, this is a property, so the pseudonatural transformations $𝐹 \Rightarrow 𝐺$ are a full subcategory of the category of lax transformations and modifications.

Between prestacks the 1-cells are taken to be pseudonatural. The reason is biuniversality: a transformation whose components $𝜎_{𝑥}$ are all equivalences of categories is an equivalence of prestacks only if its naturality cells are invertible, and a biuniversal element should be exactly a representation of a prestack up to such an equivalence.

Definition. Locally Discrete Bicategory locally-discrete-bicategory

Every category $𝒞︀$ is a bicategory $𝖫𝖣 (𝒞︀)$ with only identity 2-cells. Its 0-cells are the objects of $𝒞︀$ , and the hom-category $𝖫𝖣 (𝒞︀) (𝑥, 𝑦)$ is the discrete category on the set $𝒞︀ (𝑥, 𝑦)$ : a 2-cell $𝑓 \Rightarrow 𝑔$ is a proof that $𝑓 = 𝑔$ . Composition and identities are those of $𝒞︀$ ; the unitors and associator are the unit and associativity laws of $𝒞︀$ . Since the homs of $𝒞︀$ are sets, any two parallel 2-cells are equal, so the triangle and pentagon hold trivially.

A functor $𝐹 : 𝒞︀ \to 𝒟︀$ gives a pseudofunctor $𝖫𝖣 (𝐹) : 𝖫𝖣 (𝒞︀) \to 𝖫𝖣 (𝒟︀)$ , whose comparison 2-cells are the functor laws of $𝐹$ .

The locally discrete bicategory is how ordinary indexed categories enter bicategorical language: a prestack on $𝖫𝖣 (𝒞︀)$ is a pseudofunctor ${𝒞︀}^{op} \to 𝖢𝖠𝖳$ , and its Grothendieck construction is a displayed category over $𝒞︀$ .

Definition. Opposite Bicategory opposite-bicategory

The opposite ${ℬ︀}^{op}$ of a bicategory $ℬ︀$ has the same 0-cells and reverses the 1-cells but not the 2-cells:

{ℬ︀}^{op} (𝑥, 𝑦) = ℬ︀ (𝑦, 𝑥) .

Composition swaps its arguments, $𝑓 ⋆^{op} 𝑔 = 𝑔 ⋆ 𝑓$ . The left unitor of ${ℬ︀}^{op}$ is the right unitor of $ℬ︀$ and vice versa, and the associator of ${ℬ︀}^{op}$ is the inverse of the associator of $ℬ︀$ , with its arguments reversed.

Since the 2-cells keep their direction, a lax functor $𝐹 : ℬ︀ \to 𝒞︀$ induces a lax (not oplax) functor ${ℬ︀}^{op} \to {𝒞︀}^{op}$ with the same action on cells. Reversing the 2-cells instead gives the bicategory ${ℬ︀}^{co}$ , whose hom-categories are the opposites ${(ℬ︀ (𝑥, 𝑦))}^{op}$ .

Duality saves work: a coherence lemma about $ℬ︀$ can often be obtained by instantiating a companion lemma at ${ℬ︀}^{op}$ , which swaps left and right.

Definition. Pseudofunctor pseudofunctor

A pseudofunctor $𝐹 : ℬ︀ \to 𝒞︀$ is a lax functor whose unit and composition comparisons

𝐹_{𝑥}^{0} : 1_{𝐹 𝑥} \Rightarrow 𝐹 (1_{𝑥}) 𝐹_{𝑓, 𝑔}^{2} : 𝐹 𝑓 ⋆ 𝐹 𝑔 \Rightarrow 𝐹 (𝑓 ⋆ 𝑔)

are invertible 2-cells. So $𝐹$ preserves identities and composition up to coherent isomorphism.

Being pseudo is a property of a lax functor: invertibility of a 2-cell is a proposition, since inverses are unique. The data of a pseudofunctor is exactly the data of a lax functor, and everything proved about lax functors applies to pseudofunctors unchanged. Pseudofunctors are closed under composition and identities, as lax functors are, because invertible 2-cells are closed under composition and under the action of a functor on hom-categories.

The main examples here are prestacks, pseudofunctors ${ℬ︀}^{op} \to 𝖢𝖠𝖳$ . When $ℬ︀$ is locally discrete on a category $𝒞︀$ , these are the pseudofunctors ${𝒞︀}^{op} \to 𝖢𝖠𝖳$ of fibred category theory.

Definition. Recursive coalgebra recursive-coalgebra

Fix an endofunctor $𝐹 : 𝒞︀ \to 𝒞︀$ . A coalgebra $𝛾 : 𝑋 \to 𝐹 𝑋$ is recursive when for every algebra $𝛼 : 𝐹 𝐵 \to 𝐵$ there is exactly one hylomorphism $ℎ : 𝑋 \to 𝐵$ from $𝛾$ to $𝛼$ , that is, exactly one solution of

ℎ = 𝛾 ⋆ 𝐹 ℎ ⋆ 𝛼 .

Equivalently, the functor $𝖧𝗒𝗅𝗈 (𝛾, -)$ of the hylomorphism profunctor is constantly a singleton.

Recursiveness is a coalgebraic form of well-foundedness: $𝛾$ decomposes each input into subproblems, and recursiveness says that every divide-and-conquer program built on this decomposition has a unique meaning, without mentioning an order on inputs. [1] use recursive coalgebras on categories of indexed families to obtain algorithms that are correct by the type of the map they compute.

Example. If $(𝜇 𝐹, 𝗂𝗇)$ is an initial algebra, then $𝗂𝗇$ is invertible (Lambek’s lemma) and ${𝗂𝗇}^{- 1} : 𝜇 𝐹 \to 𝐹 (𝜇 𝐹)$ is a recursive coalgebra. Precomposing with the isomorphism $𝗂𝗇$ , the equation $ℎ = {𝗂𝗇}^{- 1} ⋆ 𝐹 ℎ ⋆ 𝛼$ is equivalent to $𝗂𝗇 ⋆ ℎ = 𝐹 ℎ ⋆ 𝛼$ , which says $ℎ$ is an algebra map out of the initial algebra; there is exactly one, $𝖿𝗈𝗅𝖽 𝛼$ .

The dual notion is a corecursive algebra.

Definition. Total Bicategory total-bicategory

The total bicategory of a displayed bicategory over $𝒦︀$ packages the base and the displayed data together, one dimension up from the total category of a displayed category.

Its 0-cells are pairs of a 0-cell of $𝒦︀$ and a displayed 0-cell over it.
Its hom-category from to is the total category of the displayed hom-category . So a 1-cell is a pair and a 2-cell is a pair .
Identities, composition, unitors and associator are pairs of the base structure and the displayed structure over it, and the triangle and pentagon hold because they hold in the base and, over that, in the displayed bicategory.

Projecting to first components is a pseudofunctor whose unit and composition comparisons are identity 2-cells.

Predecessors simplify later predecessors-simplify-later

A predecessor of $𝑥$ is a top element of its strict downset: a strict morphism $𝜌 : 𝑝 \to 𝑥$ through which every strict morphism into $𝑥$ factors uniquely. Equivalently, $𝖲𝗍𝗋𝗂𝖼𝗍𝖣𝗈𝗐𝗇 (𝑥) ≅ 𝗒 𝑝$ — the downset is representable.

The Yoneda lemma then collapses later to evaluation:

⊳ 𝑃 (𝑥) = 𝖯𝗌𝗁 𝒞︀ (𝖲𝗍𝗋𝗂𝖼𝗍𝖣𝗈𝗐𝗇 (𝑥), 𝑃) ≅ 𝖯𝗌𝗁 𝒞︀ (𝗒 𝑝, 𝑃) ≅ 𝑃 (𝑝),

with $𝗇𝖾𝗑𝗍$ becoming restriction along $𝜌$ . The name is from the naturals: every strict map into $𝑛 + 1$ factors through $𝑛 \to 𝑛 + 1$ , so on $𝜔$ — in the topos of trees — later is just the shift

⊳ 𝑃 (0) ≅ ⊤, ⊳ 𝑃 (𝑛 + 1) ≅ 𝑃 (𝑛),

and a Löb step is a base value together with a rule producing the value at $𝑛 + 1$ from the value at $𝑛$ .

When this predecessor exists, we can give a simpler description of later, as in the topos of trees, but this may not be possible in all direct categories.

Algebras as a displayed category algebras-displayed

Fix an endofunctor $𝐹 : 𝒞︀ \to 𝒞︀$ . The $𝐹$ -algebras form a displayed category $AlgStr (𝐹)$ over $𝒞︀$ .

Over an object $𝑥$ , a displayed object of $AlgStr (𝐹)$ is a structure map

𝛼 \in 𝒞︀ (𝐹 𝑥, 𝑥) .

Over $𝑓 : 𝑥 \to 𝑦$ , a displayed morphism from $𝛼$ to $𝛽$ is the proposition that $𝑓$ is an algebra homomorphism:

𝛼 ⋆ 𝑓 = 𝐹 𝑓 ⋆ 𝛽 .

The total category $Alg (𝐹) = \int AlgStr (𝐹)$ is the category of $𝐹$ -algebras.

The co-Eilenberg–Moore category as a displayed category co-eilenberg-moore-displayed

A comonad on $𝒞︀$ is a monad $𝑊$ on ${𝒞︀}^{op}$ . Everything about its coalgebras is then inherited from the Eilenberg–Moore construction, instantiated at the opposite category — nothing is defined twice.

Algebras of $𝑊$ over ${𝒞︀}^{op}$ are coalgebras $𝛾 \in 𝒞︀ (𝑥, 𝑊 𝑥)$ of the underlying endofunctor, and the monad algebra laws, read in ${𝒞︀}^{op}$ , are the comonad coalgebra laws — the unit and multiplication of $𝑊$ , viewed in $𝒞︀$ , are the counit $𝜀$ and comultiplication $𝛿$ :

𝛾 ⋆ 𝜀_{𝑥} = {𝗂𝖽}_{𝑥} 𝛾 ⋆ 𝛿_{𝑥} = 𝛾 ⋆ 𝑊 𝛾 .

The co-Eilenberg–Moore category is the opposite of the total category:

coEM (𝑊) = {(EM (𝑊))}^{op} .

Coalgebras as a displayed category coalgebras-displayed

Fix an endofunctor $𝐹 : 𝒞︀ \to 𝒞︀$ . Coalgebras require no new construction: a coalgebra is an algebra in the opposite category. Define

CoalgStr (𝐹) = AlgStr (𝐹^{op}),

a displayed category over ${𝒞︀}^{op}$ , where $𝐹^{op} : {𝒞︀}^{op} \to {𝒞︀}^{op}$ is $𝐹$ acting on the opposite category.

Concretely, over an object $𝑥$ a displayed object is a structure map

𝛾 \in 𝒞︀ (𝑥, 𝐹 𝑥) .

The category of coalgebras is the opposite of the total category:

Coalg (𝐹) = {(\int CoalgStr (𝐹))}^{op} .

The outer opposite returns morphisms to the direction of $𝒞︀$ : a morphism $(𝑥, 𝛾) \to (𝑦, 𝛿)$ is a map $𝑓 : 𝑥 \to 𝑦$ with

𝑓 ⋆ 𝛿 = 𝛾 ⋆ 𝐹 𝑓 .

Definition. The comparison functor of an adjunction comparison-functor

An adjunction $𝐹 ⊣ 𝑈$ with $𝐹 : 𝒞︀ \to 𝒟︀$ and $𝑈 : 𝒟︀ \to 𝒞︀$ induces a monad $𝑇 = 𝑈 \circ 𝐹$ on $𝒞︀$ . Write $𝜀 : 𝐹 \circ 𝑈 \Rightarrow 𝖨𝖽$ for the counit of the adjunction. Every object $𝑑$ of $𝒟︀$ then induces a $𝑇$ -algebra carried by the object $𝑈 𝑑$ , witnessed by the map

𝑈 𝜀_{𝑑} : 𝑈 𝐹 𝑈 𝑑 \to 𝑈 𝑑 .

This assignment extends to a functor into the Eilenberg–Moore category,

𝐾 : 𝒟︀ \to EM (𝑇),

the comparison functor of the adjunction.

Dually, an adjunction induces a comonad on the other side and a comparison into the co-Eilenberg–Moore category. When these comparisons are equivalences we say that the adjunction $𝐹 ⊣ 𝑈$ is (co)monadic.

Definition. Direct categories direct-category

A well-founded order is a set $𝐷$ with a proposition-valued transitive relation $<$ admitting no infinite descent: every element is accessible. Write $𝑎 \leq 𝑏$ for $(𝑎 < 𝑏) \lor (𝑎 = 𝑏)$ .

A direct structure on a category $𝒞︀$ over $(𝐷, <)$ is a functor

deg : 𝒞︀ \to (𝐷, \leq)

into the well-founded order viewed as a poset category. The functor organizes two pieces of data at once: an ordering on the objects, and the invariant that morphisms respect it — $𝑓 : 𝑥 \to 𝑦$ forces $deg 𝑥 \leq deg 𝑦$ .

A direct structure equips the objects with a well-founded strict relation

𝑥 ≺ 𝑦 ⟺ deg 𝑥 < deg 𝑦 .

Intuitively, direct categories are the right generalization of well-foundedness to the categorical setting: a direct category is essentially one whose underlying graph is a directed acyclic graph, layered by degree, so that data at an object may be defined by recursion from data at all objects strictly below it.

The degrees order the objects, while the morphisms of $𝒞︀$ say how an object sits over its predecessors.

Definition. Earlier on presheaves earlier-presheaf

Later takes a limit over smaller indices. Dually, the earlier modality takes a colimit over larger indices: an element of $⊲ 𝑃$ at $𝑥$ is a $𝑃$ -element sitting at some object $𝑦$ strictly above $𝑥$ , carried down along a chosen morphism $𝑥 \to 𝑦$ .

Earlier is left adjoint to later:

⊲ ⊣ ⊳

Under this adjunction, $𝗇𝖾𝗑𝗍 : 𝑃 \to ⊳ 𝑃$ corresponds to

𝗉𝗋𝖾𝗏 : ⊲ 𝑃 \to 𝑃,

The Eilenberg–Moore category as a displayed category eilenberg-moore-displayed

Fix a monad $(𝑇, 𝜂, 𝜇)$ on $𝒞︀$ . Its Eilenberg–Moore category arises in two displayed layers. The first layer is the displayed category of algebras $AlgStr (𝑇)$ of the underlying endofunctor.

The second layer, $EMStr (𝑇)$ , is displayed over the total category $Alg (𝑇)$ . Over an algebra $(𝑥, 𝛼)$ the displayed objects are the propositions that $𝛼$ satisfies the monad algebra laws:

𝜂_{𝑥} ⋆ 𝛼 = {𝗂𝖽}_{𝑥} 𝜇_{𝑥} ⋆ 𝛼 = 𝑇 𝛼 ⋆ 𝛼 .

The Eilenberg–Moore category is the total category of the tower:

EM (𝑇) = \int EMStr (𝑇) .

Definition. Initial algebra initial-algebra

Fix an endofunctor $𝐹 : 𝒞︀ \to 𝒞︀$ . An initial $𝐹$ -algebra, written $𝜇 𝐹$ , is an initial object of the category of algebras $Alg (𝐹)$ .

Unfolding the universal property: an initial algebra is an algebra $(𝜇 𝐹, in)$ such that every algebra $(𝑥, 𝛼)$ admits a unique morphism $fold 𝛼 : 𝜇 𝐹 \to 𝑥$ satisfying

in ⋆ (fold 𝛼) = 𝐹 (fold 𝛼) ⋆ 𝛼 .

Definition. Later on families later-family

Conjugation with the adjunction between presheaves and families $𝑈 ⊣ Cofree$ lets us induce a later construction on families from the one on presheaves,

⊳_{Fam} = 𝑈 \circ ⊳ \circ Cofree : Fam (𝒞︀) \to Fam (𝒞︀) .

Concretely, later on families evaluates to

⊳_{Fam} 𝐴 (𝑥) ≅ \prod_{𝑦 ≺ 𝑥} \prod_{𝑓 : 𝑦 \to 𝑥} 𝐴 (𝑦)

Definition. Later on presheaves later-presheaf

The later modality for presheaves on a direct category is given by the presheaf of natural transformations

out of the strict downset.

(⊳ 𝑃) (𝑥) = 𝖯𝗌𝗁 𝒞︀ (𝖲𝗍𝗋𝗂𝖼𝗍𝖣𝗈𝗐𝗇 (𝑥), 𝑃) .

An element of $⊳ 𝑃$ at $𝑥$ is a coherent choice of $𝑃$ -elements at all objects strictly smaller than $𝑥$ .

Restriction in $⊳ 𝑃$ along $𝑓 : 𝑦 \to 𝑥$ precomposes with the induced map $𝖲𝗍𝗋𝗂𝖼𝗍𝖣𝗈𝗐𝗇 (𝑦) \to 𝖲𝗍𝗋𝗂𝖼𝗍𝖣𝗈𝗐𝗇 (𝑥)$ . At an object of minimal degree the strict downset is empty, so $⊳ 𝑃$ is trivial there.

Via functoriality, every presheaf restricts to smaller indices. Thus we may define the map

𝗇𝖾𝗑𝗍 : 𝑃 \to ⊳ 𝑃

that sends an element $𝑝$ over $𝑥$ to the family of all its restrictions along morphisms from strictly lower objects.

Theorem. Löb induction on families lob-family

Like later on families, the recursion principle for families is inherited from that on presheaves. Given a family $𝐴$ and a step

𝜑_{𝑥} : ⊳_{Fam} 𝐴 (𝑥) \to 𝐴 (𝑥) for each 𝑥,

the construction is a chain of transpositions with Löb for presheaves used in the middle:

Just as for presheaves, the fixed point constructed above is unique: the two transpositions are bijections, and the presheaf-level fixed point is already unique.

Theorem. Löb induction for presheaves on a direct category lob-presheaf

Let $𝒞︀$ be a direct category and $𝑃$ a presheaf on it. Every map

𝜑 : ⊳ 𝑃 \to 𝑃

has a fixed point: a global element $𝗅ö𝖻 𝜑 : ⊤ \to 𝑃$ with

𝗅ö𝖻 𝜑 = 𝗅ö𝖻 𝜑 ⋆ 𝗇𝖾𝗑𝗍 ⋆ 𝜑,

and this fixed point is unique.

The hypothesis says: the value of $𝑃$ at any object is determined by its values over the strict past — $𝜑$ turns a coherent family over the strict downset of $𝑥$ into a value at $𝑥$ . The proof is recursion along the well-founded $≺$ : at each $𝑥$ , the section already constructed over the past assembles into an element of $(⊳ 𝑃) (𝑥)$ , and $𝜑$ extends it to $𝑥$ .

Definition. Locally contractive endofunctors locally-contractive-functor

Write $𝑋 \Rightarrow 𝑌$ for the presheaf of morphisms $𝑋 \to 𝑌$ . An endofunctor $𝐹$ on presheaves is locally contractive when its action on morphisms factors through later: there is a map

𝐹_{𝛿} : ⊳ (𝑋 \Rightarrow 𝑌) \to (𝐹 𝑋 \Rightarrow 𝐹 𝑌)

Definition. Monadicity and comonadicity monadicity-comonadicity

An adjunction $𝐹 ⊣ 𝑈$ with $𝐹 : 𝒞︀ \to 𝒟︀$ and $𝑈 : 𝒟︀ \to 𝒞︀$ induces a monad $𝑇 = 𝑈 \circ 𝐹$ on $𝒞︀$ , and a comparison functor

𝐾 : 𝒟︀ \to EM (𝑇)

sending each object of $𝒟︀$ to the $𝑇$ -algebra it carries.

The functor $𝑈$ is monadic when $𝐾$ is an equivalence: the adjunction exhibits $𝒟︀$ as objects of $𝒞︀$ equipped with algebraic structure for $𝑇$ , the Eilenberg–Moore category.

Comonadicity is monadicity in the opposite category: a left adjoint $𝐿 : 𝒟︀ \to 𝒞︀$ with right adjoint $𝑅$ induces a comonad $𝑊 = 𝐿 \circ 𝑅$ on $𝒞︀$ , a comparison $𝒟︀ \to coEM (𝑊)$ into the co-Eilenberg–Moore category, and $𝐿$ is comonadic when this comparison is an equivalence.

The adjoint triple between presheaves and families presheaf-family-adjoint-triple

A family over $𝒞︀$ is a set $𝐴 (𝑥)$ for each object $𝑥$ , with no action of morphisms. Families form a category $Fam (𝒞︀)$ : a morphism $𝐴 \to 𝐵$ is a function $𝐴 (𝑥) \to 𝐵 (𝑥)$ for each $𝑥$ .

Forgetting the restriction maps of a presheaf gives a functor

𝑈 : 𝖯𝗌𝗁 𝒞︀ \to Fam (𝒞︀) .

It has both a left and a right adjoint,

Free ⊣ 𝑈 ⊣ Cofree .

The two adjoints demonstrate different means of forcing a family to be functorial. The right adjoint universally quantifies over morphisms in,

Cofree (𝐴) (𝑥) = \prod_{𝑦} 𝒞︀ (𝑦, 𝑥) \to 𝐴 (𝑦),

with restriction along $𝑓$ given by precomposition. The left adjoint instead existentially quantifiers over morphisms out:

Free (𝐴) (𝑥) = \sum_{𝑦} 𝒞︀ (𝑥, 𝑦) \times 𝐴 (𝑦),

with restriction acting on the first component. (For $Free$ we ask that $𝒞︀$ have a set of objects, so that this sum is a set and thus $Free$ defines a presheaf.)

Theorem. Presheaves are monadic and comonadic over families presheaves-monadic-comonadic-over-families

The adjoint triple $Free ⊣ 𝑈 ⊣ Cofree$ induces a monad $𝑇 = 𝑈 \circ Free$ and a comonad $𝑊 = 𝑈 \circ Cofree$ on $Fam (𝒞︀)$ .

Both comparison functors are equivalences: presheaves are the Eilenberg–Moore algebras of $𝑇$ and the co-Eilenberg–Moore coalgebras of $𝑊$ ,

𝖯𝗌𝗁 𝒞︀ ≃ EM (𝑇) 𝖯𝗌𝗁 𝒞︀ ≃ coEM (𝑊) .

So presheaves are both monadic and comonadic over families.

Reading the algebra structure concretely: a $𝑇$ -algebra on a family $𝐴$ is a map $\sum_{𝑦} 𝒞︀ (𝑥, 𝑦) \times 𝐴 (𝑦) \to 𝐴 (𝑥)$ for each $𝑥$ , subject to the monad algebra laws — that is, exactly a functorial action of restriction.

The comonadic reading is the same structure seen from the element’s side: a $𝑊$ -coalgebra is a map $𝐴 (𝑥) \to \prod_{𝑦} 𝒞︀ (𝑦, 𝑥) \to 𝐴 (𝑦)$ , giving each value its restriction along every morphism into $𝑥$ . Where the monad says restriction acts on values, the comonad says a value already carries all of its restrictions — and the coalgebra laws say it does so coherently.

Definition. Proper and maximal sieves proper-maximal-sieve

The representable $𝗒 𝑥$ is itself a sieve on $𝑥$ . A sieve on $𝑥$ is proper when it is not equal to the representable.

Say that a proper sieve is maximal when it contains all other proper sieves as a sub-sieve.

Definition. Sieves sieve

A sieve on an object $𝑥$ of $𝒞︀$ is a subobject of the representable presheaf $𝗒 𝑥$ : a presheaf $𝑆$ with a monic morphism $𝑆 ↣ 𝗒 𝑥$ . Sieves are a generalization from the notion of ideal found in ring theory to category theory.

A morphism $𝑓 : 𝑦 \to 𝑥$ belongs to $𝑆$ , written $𝑆 ∋ 𝑓$ , when $𝑓$ lies in the image of the inclusion at $𝑦$ . Because $𝑆$ is a presheaf and the inclusion is natural, membership is closed under precomposition:

𝑆 ∋ 𝑓 \Rightarrow 𝑆 ∋ 𝑔 ⋆ 𝑓 for every 𝑔 : 𝑧 \to 𝑦 .

A sieve is thus a “downward closed” collection of morphisms into $𝑥$ .

Sieves on $𝑥$ are ordered by refinement: $𝑆 \subseteq 𝑇$ when every morphism belonging to $𝑆$ belongs to $𝑇$ .

Theorem. Maximality of the strict downset among proper sieves strict-downset-maximal

Call a direct structure reflecting when every morphism between objects of equal degree is a split epimorphism. In a reflecting direct category, every non-invertible-in-degree morphism strictly raises degree, and the strict downset is as large as a proper sieve can be:

If the direct structure is reflecting, then every proper sieve $𝑆$ on $𝑥$ refines into the strict downset:

𝑆 \subseteq 𝖲𝗍𝗋𝗂𝖼𝗍𝖣𝗈𝗐𝗇 (𝑥) .

Suppose $𝑓 \in 𝑆$ with $𝑓 : 𝑦 \to 𝑥$ of equal degree. By reflection $𝑓$ has a section $𝑠$ , and closure under precomposition gives $𝑠 ⋆ 𝑓 = {𝗂𝖽}_{𝑥} \in 𝑆$ , contradicting properness.

So every morphism in $𝑆$ strictly raises degree. That is, every morphism in $𝑆$ is also a member of $𝖲𝗍𝗋𝗂𝖼𝗍𝖣𝗈𝗐𝗇 (𝑥)$ .

Definition. The strict downset sieve of a direct category strict-downset-sieve

Let $𝒞︀$ carry a direct structure. The strict downset $𝖲𝗍𝗋𝗂𝖼𝗍𝖣𝗈𝗐𝗇 (𝑥)$ of an object $𝑥$ is the presheaf of morphisms into $𝑥$ from strictly lower objects:

𝖲𝗍𝗋𝗂𝖼𝗍𝖣𝗈𝗐𝗇 (𝑥) (𝑦) = {𝑓 : 𝑦 \to 𝑥 ∣ 𝑦 ≺ 𝑥},

with restriction by precomposition — well defined since degrees are non-decreasing, so precomposing can only stay strictly below.

The evident inclusion $𝖲𝗍𝗋𝗂𝖼𝗍𝖣𝗈𝗐𝗇 (𝑥) ↣ よ 𝑥$ makes $𝖲𝗍𝗋𝗂𝖼𝗍𝖣𝗈𝗐𝗇 (𝑥)$ a sieve on $𝑥$ . It is moreover a proper sieve, as it exlcudes the identity.

Definition. Terminal coalgebra terminal-coalgebra

Fix an endofunctor $𝐹 : 𝒞︀ \to 𝒞︀$ . A terminal $𝐹$ -coalgebra, written $𝜈 𝐹$ , is a terminal object of the category of coalgebras $Coalg (𝐹)$ — equivalently, an initial algebra for $𝐹^{op}$ .

Unfolding the universal property: a terminal coalgebra is a coalgebra $(𝜈 𝐹, out)$ such that every coalgebra $(𝑥, 𝛾)$ admits a unique morphism $unfold 𝛾 : 𝑥 \to 𝜈 𝐹$ satisfying

(unfold 𝛾) ⋆ out = 𝛾 ⋆ 𝐹 (unfold 𝛾) .

Well-founded posets are thin direct categories well-founded-poset-as-thin

Every poset forms a thin category. Similarly, if the poset is well-founded then it induces a thin direct category.

Definition. Category category

A category $𝒞︀$ consists of

A type of objects ${𝒞︀}_{0}$
For each pair of objects $𝑥, 𝑦 : {𝒞︀}_{0}$ a set of morphisms $𝒞︀ (𝑥, 𝑦)$ . We may simply write a morphism with an arrow, denote $𝑓 : 𝒞︀ (𝑥, 𝑦)$ as $𝑓 : 𝑥 \to 𝑦$ or $𝑥 \overset{𝑓}{\to} 𝑦$ or similar
A composition operation on morphisms. For $𝑓 : 𝑥 \to 𝑦$ and $𝑔 : 𝑦 \to 𝑧$ , there is a morphism $𝑓 ⋆ 𝑔 : 𝑥 \to 𝑧$
For each $𝑥 : {𝒞︀}_{0}$ , an identity morphism ${𝗂𝖽}_{𝑥} : 𝑥 \to 𝑥$
Left-unitality of composition: for all $𝑓 : 𝑥 \to 𝑦$ , an equality
${𝗂𝖽𝖫}_{𝑓} : {𝗂𝖽}_{𝑥} ⋆ 𝑓 = 𝑓$
Right-unitality of composition: for all $𝑓 : 𝑥 \to 𝑦$ , an equality
${𝗂𝖽𝖱}_{𝑓} : 𝑓 ⋆ {𝗂𝖽}_{𝑦} = 𝑓$
Associativity of composition: for all $𝑓 : 𝑥 \to 𝑦$ , $𝑔 : 𝑦 \to 𝑧$ , $ℎ : 𝑧 \to 𝑤$ , an equality
${𝖺𝗌𝗌𝗈𝖼}_{𝑓, 𝑔, ℎ} : (𝑓 ⋆ 𝑔) ⋆ ℎ = 𝑓 ⋆ (𝑔 ⋆ ℎ)$

Concretely, the definition above is meant to model the one used in the Cubical standard library [1].

However, the notion of category is flexible. Depending on the context, we may be talking of small, locally small, wild, or any other kind of category that may augment which things we require to be (homotopy) sets, which things we require to be small types, etc. For the most part, the same idea of a category will apply across all of these settings.

Definition. Bicategory bicategory

A bicategory is a notion of weak 2-category that arises as a category weakly enriched in categories. That is, instead of having hom sets, between any two objects a bicategory has hom categories such that the enriched category laws hold up to invertible 2-cell rather than strictly.

A bicategory $𝒦︀$ consists of

A type of objects ${𝒦︀}_{0}$ , or 0-cells
For all $𝑥, 𝑦 : {𝒦︀}_{0}$ , a category ${𝒦︀}_{1} (𝑥, 𝑦)$ . We may elide the subscript and simply write this as $𝒦︀ (𝑥, 𝑦)$ . Refer to the objects of $𝒦︀ (𝑥, 𝑦)$ as 1-cells between $𝑥$ and $𝑦$ , and we may write $𝑓 : 𝒦︀ (𝑥, 𝑦)$ as $𝑓 : 𝑥 \to 𝑦$ or $𝑥 \overset{𝑓}{\to} 𝑦$ . For $𝑓, 𝑔 : 𝒦︀ (𝑥, 𝑦)$ , refer to the morphisms in $𝒦︀ (𝑥, 𝑦)$ between $𝑓$ and $𝑔$ as 2-cells and write the morphism $𝛼 : (𝒦︀ (𝑥, 𝑦)) (𝑓, 𝑔)$ as $𝛼 : 𝑓 \Rightarrow 𝑔$ or $𝑓 \overset{𝛼}{\Rightarrow} 𝑔$
For each $𝑥 : {𝒦︀}_{0}$ , an identity 1-cell $1_{𝑥} : 𝒦︀ (𝑥, 𝑥)$
For all $𝑥, 𝑦, 𝑧 : {𝒦︀}_{0}$ , a composition functor ${𝒦︀}_{⋆}^{𝑥, 𝑦, 𝑧} : 𝒦︀ (𝑥, 𝑦) \times 𝒦︀ (𝑦, 𝑧) \to 𝒦︀ (𝑥, 𝑧)$ . For 1-cells $𝑓 : 𝑥 \to 𝑦$ and $𝑔 : 𝑦 \to 𝑧$ , write their composite as $𝑓 ⋆ 𝑔 : 𝑥 \to 𝑧$
For all $𝑤, 𝑥, 𝑦, 𝑧 : {𝒦︀}_{0}$ , a natural isomorphism, the associator $𝛼$ between the two composite functors $𝒦︀ (𝑤, 𝑥) \times 𝒦︀ (𝑥, 𝑦) \times 𝒦︀ (𝑦, 𝑧) \to 𝒦︀ (𝑤, 𝑧)$ that compose the leftmost, respectively rightmost, pair first:
${𝒦︀}_{⋆}^{𝑤, 𝑦, 𝑧} \circ ({𝒦︀}_{⋆}^{𝑤, 𝑥, 𝑦} \times id) \Rightarrow {𝒦︀}_{⋆}^{𝑤, 𝑥, 𝑧} \circ (id \times {𝒦︀}_{⋆}^{𝑥, 𝑦, 𝑧})$
Its component at 1-cells $𝑓, 𝑔, ℎ$ is the invertible 2-cell
$𝛼_{𝑓, 𝑔, ℎ} : (𝑓 ⋆ 𝑔) ⋆ ℎ \Rightarrow 𝑓 ⋆ (𝑔 ⋆ ℎ)$
For all $𝑥, 𝑦 : {𝒦︀}_{0}$ , natural isomorphisms, the left unitor $𝜆$ and right unitor $𝜌$ , each between an endofunctor of $𝒦︀ (𝑥, 𝑦)$ and the identity functor:
${𝒦︀}_{⋆}^{𝑥, 𝑥, 𝑦} \circ ⟨ 1_{𝑥}, id ⟩ \Rightarrow id {𝒦︀}_{⋆}^{𝑥, 𝑦, 𝑦} \circ ⟨ id, 1_{𝑦} ⟩ \Rightarrow id$
where $1_{𝑥}$ and $1_{𝑦}$ in the pairings $⟨ -, - ⟩$ denote the constant functors at the identity 1-cells. The components at a 1-cell $𝑓 : 𝒦︀ (𝑥, 𝑦)$ are the invertible 2-cells
$𝜆_{𝑓} : 1_{𝑥} ⋆ 𝑓 \Rightarrow 𝑓 𝜌_{𝑓} : 𝑓 ⋆ 1_{𝑦} \Rightarrow 𝑓$
such that for all $𝑓 : 𝒦︀ (𝑥, 𝑦)$ and $𝑔 : 𝒦︀ (𝑦, 𝑧)$ the triangle below commutes in $𝒦︀ (𝑥, 𝑧)$ :
and such that for all composable 1-cells $𝑓, 𝑔, ℎ, 𝑘$ the pentagon below commutes:

Definition. Monad in a bicategory monad-in-a-bicategory

Fix a bicategory $𝒦︀$ , with composition $⋆$ , identity 1-cells $1_{𝑥}$ , associator $𝛼$ , and unitors $𝜆, 𝜌$ . A monad in $𝒦︀$ internalises the usual notion of monad: it is an endo-1-cell carrying a multiplication and a unit that satisfy the monoid laws up to the coherence cells of the bicategory.

A monad in $𝒦︀$ consists of

a 0-cell $𝑥$ , the object the monad acts on;
an endo-1-cell $𝑡 : 𝒦︀ (𝑥, 𝑥)$ ;
a multiplication 2-cell $𝜇 : 𝑡 ⋆ 𝑡 \Rightarrow 𝑡$ ;
a unit 2-cell $𝜂 : 1_{𝑥} \Rightarrow 𝑡$ ;
such that $𝜇$ is associative: the following diagram of 2-cells commutes in $𝒦︀ (𝑥, 𝑥)$ , where the top map is the associator that rebrackets the threefold composite:
and such that $𝜇$ and $𝜂$ satisfy the unit laws: the following two diagrams commute in $𝒦︀ (𝑥, 𝑥)$ , where the hypotenuses are the left and right unitors:

Taking $𝒦︀$ to be the bicategory of categories, functors, and natural transformations recovers an ordinary monad on a category: $𝑡$ is the endofunctor, $𝜇$ the multiplication, and $𝜂$ the unit, with the coherence cells all identities.

Definition. Free Monoidal Category over a Set free-monoidal-category

Fix a set $𝑋$ . The objects of the free monoidal category over $𝑋$ , $𝖥𝗋𝖾𝖾𝖬𝗈𝗇 (𝑋)$ , are generated inductively by the elements of $𝑋$ and a unit element $𝐼$ over a binary operation $\otimes$ . The morphisms are given by a quotient-inductive type. They are generated by associators, unitors, and identity over composition and parallel action over $\otimes$ then quotiented by associativity and composition equation to satisfy the category laws, equations constraining the associators/unitors to be natural isomorphisms, and pentagon/triangle equations to satiate the axioms of a monoidal category.

Definition 0.1. Global Elimination Principle for the Free Monoidal Category free-monoidal-category-elimination

Given any displayed monoidal category $𝑀^{𝙳}$ over $𝖥𝗋𝖾𝖾𝖬𝗈𝗇 (𝑋)$ with an interpretation $𝜄 : 𝑋 ⇝ 𝑀^{𝙳}$ , we may construct a global section $𝖥𝗋𝖾𝖾𝖬𝗈𝗇 (𝑋) \to 𝑀^{𝙳}$ . We refer to this as the global elimination principle of $𝖥𝗋𝖾𝖾𝖬𝗈𝗇 (𝑋)$ .

Definition. Global Elimination Principle for the Free Monoidal Category free-monoidal-category-elimination

Products of Categories as Total Categories product-as-total-category

Given categories $𝒞︀$ and $𝒟︀$ , $𝒞︀ \times 𝒟︀$ is equivalent to the total category of weakening.

Definition. Quiver quiver

A quiver is just a directed graph presented via a type of objects, a type of edges, and two projection functions that pick out source and target of an edge.

Definition. Thin Category thin-category

A category $𝐶$ is thin if there is at most one morphism between any two objects.

Definition. Element of a Presheaf element-of-presheaf

An element of a presheaf $𝑃$ at an object $𝑐$ is an element $𝑥$ of the set $𝑃 𝑐$ .

Definition. Universal Element of a Presheaf universal-element

A universal element of a presheaf $𝑃$ on a category $𝐶$ is an element $𝑥 \in 𝑃 𝑐$ , where $𝑐$ is some object of $𝐶$ , demonstrating that $𝑃$ is representable by $𝑐$ .

𝑃 ≅ よ 𝑐

(Slightly) more concretely, a universal element is captured by the following three pieces of data

An object $𝑐$ of $𝐶$
An element $𝑥 \in 𝑃 𝑐$
A proof that the map sending a morphism $𝑓 : 𝑏 \to 𝑐$ to $(𝑃 𝑓) (𝑥)$ is an equivalence

This third point states that morphisms from $𝑏$ into $𝑐$ are uniquely determined by an element of $𝑃$ at the domain $𝑏$ .

Or equivalently, universal elements are terminal in the category of elements.

What is a universal property, really? universal-property

Universal properties are a convenient method for defining an object in a category up to isomorphism. Rather than giving a concrete, bottom-up construction of an object, we can instead uniquely specify its behavior.

Consider the example of products in a category $𝐶$ . We say that the product of $𝑐$ and $𝑑$ is any object $𝑝$ of $𝐶$ such that the following diagram commutes.

We say that $𝑝$ satisfies the universal property of the product of $𝑐$ and $𝑑$ .

Surely this matches our set-based intuition of what a product should behave like. Similarly, we can sketch out constructions of other universal properties like initial objects, terminal objects, exponentials, etc. However, what is precisely meant by the term universal property?

The notion of a universal property is made precise by the notion of a universal element of a presheaf. That is, an object satisfies a universal property if we can build a universal element of the appropriate presheaf at that object.

Let’s look at the universal element characterization of the products example. Note that a map into a product is determined by a map into each component. To map into $𝑐 \times 𝑑$ , we need both a map into $𝑐$ and a map into $𝑑$ , as in the above diagram. That is, to build a map $𝑏 \to 𝑐 \times 𝑑$ , we must simultaneously provide elements of $よ 𝑐$ and $よ 𝑑$ at $𝑏$ .

Using the product of presheaves, this means we are providing a single element of the presheaf $(よ 𝑐) \times (よ 𝑑)$ . Quite nicely, the universal element of this presheaf provides the object of $𝐶$ that is the product of $𝑐$ and $𝑑$ . The universal element, provided that it exists, contains the following data:

An object $𝑝$
An element $𝑥 \in (よ 𝑐 \times よ 𝑑) 𝑝$
A proof that the map sending $𝑓 : 𝑏 \to 𝑝$ to a pair of maps $𝑓_{1} : 𝑏 \to 𝑐$ , $𝑓_{2} : 𝑏 \to 𝑑$ is an equivalence. Therefore, any element of $よ 𝑐 \times よ 𝑑$ factors through $𝑥$

Recall that the product of presheaves is computed pointwise in the category of sets, so if we expand the type of the element $𝑥$ above we find that $𝑥$ is a pair of maps $𝑝 \to 𝑐$ and $𝑝 \to 𝑑$ .

(よ 𝑐 \times よ 𝑑) 𝑝 ≅ (よ 𝑐 𝑝) \times (よ 𝑑 𝑝)

The first part of this pair is precisely $𝜋_{1}$ . Correspondingly, the second part of this pair is $𝜋_{2}$ . Finally, the universality of the element $𝑥$ (i.e. the proof that any other element factors through $𝑥$ ) captures our commutative diagram from above.

This is a very rough sketch of what a universal property is, and has elided for now an important application of the Yoneda lemma. In any case, all a universal property is really saying is that a particular presheaf is representable; and, rather elegantly, a universal element of a presheaf is convenient packaging of that representability proof.

In summary, universal properties are not as ad-hoc as they may initially seem, and the language of presheaves provides a reusable and precise definition that can be instantiated to describe a very large class of properties.

Definition. Constant Presheaf constant-presheaf

The constant presheaf of a set $𝑋$ on a category $𝐶$ is a functor $Δ (𝑋) : 𝐶^{𝑜 𝑝} \to 𝐒𝐞𝐭$ such that

Δ (𝑋) 𝑐 ≔ 𝑋

I often call this the discrete presheaf for $𝑋$ , but I don’t know if that’s standard.

Finite Cardinal Arithmetic in a Topos finite-cardinal-arithmetic-topos

In a topos with a natural numbers object $(𝑁, 𝑧, 𝑠)$ , you can define finite cardinals as objects that arise as the pullback along a morphism $𝑝 : ⊤ \to 𝑁$ of the generic finite cardinal. Write $[𝑝]$ for the cardinal corresponding to $𝑝$ .

The above characterization is a little obtuse and does warrant some more explanation. One way to make it more concrete is that $[𝑧] = ⊥$ and $[𝑠 \circ 𝑛] = ⊤ \oplus [𝑛]$ , and that finite cardinals warrant a nice induction principle. If $𝑃$ is a property expressible in the internal language such that $⊥$ satisfies $𝑃$ , and that whenever $𝐴$ satisfies $𝑃$ then $⊤ \oplus 𝐴$ satisfies $𝑃$ ; then every finite cardinal satisfies $𝑃$ . That is, $𝑃$ forms a $(𝑧, 𝑠)$ -closed subobject of $𝑁$ and thus $𝑃$ is all of $𝑁$ .

My working mental model is in a presheaf topos, where the natural numbers object can be defined explicitly as $Δ (ℕ)$ .

Definition 0.2. Constant Presheaf constant-presheaf

The constant presheaf of a set $𝑋$ on a category $𝐶$ is a functor $Δ (𝑋) : 𝐶^{𝑜 𝑝} \to 𝐒𝐞𝐭$ such that

Δ (𝑋) 𝑐 ≔ 𝑋

I often call this the discrete presheaf for $𝑋$ , but I don’t know if that’s standard.

The finite cardinals with respect to $Δ (ℕ)$ can be characterized then as,

[0] ≔ ⊥

[𝗌𝗎𝖼 (𝑛)] ≔ ⊤ \oplus [𝑛]

These obey nice algebraic properties.

[𝑛 𝑚] ≅ [𝑛] \times [𝑚]

[𝑛 + 𝑚] ≅ [𝑛] \oplus [𝑚]

[𝑛^{𝑚}] ≅ [𝑚] \to [𝑛]

Definition. Subobject subobject

In a category $𝒞︀$ , a subobject of $𝑐$ is an isomorphism class of monomorphisms into $𝑐$ .

Definition. Terminal Object terminal-object

An object $𝑐$ in a category $𝒞︀$ is terminal if there is a unique morphism from any other object into it.

Because terminal objects are unique up to unique isomorphism (as are all universal objects) we often just write $⊤$ to refer to the terminal object. Likewise, $! : 𝑑 \to ⊤$ refers to the unique morphism into $⊤$ .

Definition. Equalizer equalizer

Let $𝑐$ and $𝑑$ be objects in a category $𝒞︀$ with two parallel morphisms $𝑓, 𝑔 : 𝑐 \to 𝑑$ . The equalizer of $𝑓$ and $𝑔$ , if it exists, is the universal object $𝖾𝗊$ with the following property:

There is a morphism $𝜋 : 𝖾𝗊 \to 𝑐$
$𝑓 \circ 𝜋 = 𝑔 \circ 𝜋$

Constructing Equalizers in Type Theory equalizers-in-type-theory

In the presence of $Σ$ -types, one may construct all equalizers. Given types $𝐴$ and $𝐵$ with functions $𝑓, 𝑔 : 𝐴 \to 𝐵$ , the equalizer may be constructed as

{𝖾𝗊}_{𝑓, 𝑔} ≔ \sum_{𝑎 : 𝐴} (𝑓 (𝑎) = 𝑔 (𝑎))

Definition. Subobject Classifier subobject-classifier

Subsets $𝐴$ of a set $𝑋$ may classically be identified with a characteristic map $𝜒_{𝐴} : 𝑋 \to 𝖻𝗈𝗈𝗅$ . Intuitively, for every $𝑥 : 𝑋$ , $𝜒_{𝐴}$ gives a truth value to the statement “ $𝑥$ is in the subset $𝐴$ ”. In this manner, the domain of the characteristic map, $𝖻𝗈𝗈𝗅$ , classifies the subsets of $𝑋$ .

Generalizing over this principle, in a category $𝒞︀$ an object $Ω$ is a subobject classifier if maps into it from some object $𝑐$ likewise uniquely identify a subobject of $𝑐$ .

We can understand $Ω$ to behave like an object of truth values that are not necessarily boolean valued. A morphism $𝑝 : 𝑐 \to Ω$ can be thought of like a predicate on $𝑐$ . If $𝑐$ were a set, this would precisely be the characteristic function on it. However, this idea can generalize beyond sets. For instance in the category of graphs, $Ω$ is a cleverly constructed graph such that any graph homomorphism $𝑔 \to Ω$ into it picks out a unique subgraph of $𝑔$ .

There are always two (suggestively named) disjoint “points” of $Ω$ , thought of as morphisms out of the terminal object $𝗍𝗋𝗎𝖾, 𝖿𝖺𝗅𝗌𝖾 : ⊤ \to Ω$ . I think if we’re being careful, $𝗍𝗋𝗎𝖾$ may properly be the “subobject classifier” but I always use the term to refer to $Ω$ itself.

A subobject $𝜙 : 𝑑 ↪ 𝑐$ induces a unique characteristic morphism $𝑝_{𝜙} : 𝑐 \to Ω$ such that

𝑝_{𝜙} \circ 𝜙 = 𝗍𝗋𝗎𝖾 \circ!_{𝑑}

Moreover, the appropriate square must be a pullback.

Definition. Closed Monoidal Structure closed-monoidal-category

A monoidal category is left closed if for each $𝑌 \in 𝐶$ , the functor $- \otimes 𝑌 : 𝐶 \to 𝐶$ has a right adjoint $𝑌 ⊸ - : 𝐶 \to 𝐶$ forming the left internal-hom out of $𝑌$ .

That is, for all $𝑋, 𝑌, 𝑍$ , there is a natural isomorphism $𝐶 [𝑋 \otimes 𝑌, 𝑍] ≅ 𝐶 [𝑋, [𝑌, 𝑍]]$

There is an obvious right-handed variant $𝑌 ⟜ -$ that is right adjoint to $𝑌 \otimes -$ .

If $𝐶$ is both right and left closed, the monoidal category is simply called closed (or perhaps biclosed).

Definition. Monoidal Category monoidal-category

A monoidal category is a category $𝒞︀$ together with

a functor $\otimes : 𝒞︀ \times 𝒞︀ \to 𝒞︀$ , the tensor product;
an object $𝐼$ of $𝒞︀$ , the unit;
natural isomorphisms
$𝛼_{𝑥, 𝑦, 𝑧} : (𝑥 \otimes 𝑦) \otimes 𝑧 \to 𝑥 \otimes (𝑦 \otimes 𝑧) 𝜆_{𝑥} : 𝐼 \otimes 𝑥 \to 𝑥 𝜌_{𝑥} : 𝑥 \otimes 𝐼 \to 𝑥$
the associator, left unitor and right unitor;

such that the triangle and the pentagon below commute for all objects $𝑤, 𝑥, 𝑦, 𝑧$ .

Definition. Category of Elements category-of-elements

Let $𝑃$ be a presheaf on a category $𝒞︀$ . The category of elements of $𝑃$ is the displayed category over $𝒞︀$ whose displayed objects over $𝑐$ are the elements $𝑝 \in 𝑃 𝑐$ , and whose displayed morphisms over $𝑓 : 𝑐 \to 𝑑$ from $𝑝$ to $𝑞$ are proofs that $(𝑃 𝑓) (𝑞) = 𝑝$ .

Since $𝑃 𝑐$ is a set, there is at most one displayed morphism over each $𝑓$ between given elements: a morphism of elements is a morphism of $𝒞︀$ that happens to carry $𝑞$ back to $𝑝$ . Its total category is the classical category of elements $\int 𝑃$ , and a universal element of $𝑃$ is exactly a terminal object of $\int 𝑃$ .

Helianthus

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Helianthus is a Zettelkasten notetaking tool written in Rust. Helianthus is the binomial name for the sunflower, after the Greek helios (sun) + anthos (flower).

It owes a lot to Forester and Kodama.

It is admittedly a vibecoded tool for personal use, and I have no intention of maintaining it for others to use. This is simply meant to be a tool for my personal organization. I’m usually not well organized, and a big contributor to this is a lack of organizational systems for myself. When met with the obstacle of cognitive burden twoard planning, note taking, and the like, I am inclined to shy away. I don’t love that this is the case, but I am trying to accept that it is. To the end of being a bit more put together than the day before, I will take any help in that direction (even AI-slopped help). Use at your own risk.

Tensor of Grammars

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For formal grammars $𝐴$ and $𝐵$ , their tensor $𝐴 \otimes 𝐵$ represents the concatenation of the languages they describe. A parse for $𝐴 \otimes 𝐵$ at a string $𝑤$ consists of a splitting of $𝑤$ into a prefix $𝑢$ and suffix $𝑣$ , along with a parse of $𝑢$ in $𝐴$ and a parse of $𝑣$ in $𝐵$ .

Formally, it is defined as:

(𝐴 \otimes 𝐵) (𝑤) = \sum_{𝑤 = 𝑢 𝑣} 𝐴 (𝑢) \times 𝐵 (𝑣)

This operation is exactly the Day convolution of $𝐴$ and $𝐵$ , where we view formal grammars as presheaves over the monoid of strings.

Grammar Residuals

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For formal grammars $𝐴$ and $𝐵$ , the residual $𝐴 ⊸ 𝐵$ (sometimes called the lollipop or linear implication) is the right adjoint to the grammar tensor $𝐴 \otimes 𝐵$ . It describes strings that, when prefixed by a string matching $𝐴$ , will match $𝐵$ .

Formally, it is defined as:

(𝐴 ⊸ 𝐵) (𝑣) = Π_{𝑢} 𝐴 (𝑢) \to 𝐵 (𝑢 \cdot 𝑣)

Intuitively, a parse of $𝐴 ⊸ 𝐵$ at a string $𝑣$ is a function that takes any prefix string $𝑢$ and a parse of $𝑢$ in $𝐴$ , and produces a parse of the full concatenated string $𝑢 \cdot 𝑣$ in $𝐵$ .

Grammar Quotients

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For formal grammars $𝐴$ and $𝐵$ , the quotient of $𝐵$ by $𝐴$ is the grammar of what is left of $𝐵$ once an $𝐴$ has been read off the front:

(𝐷_{𝐴} 𝐵) (𝑣) = Σ_{𝑢} 𝐴 (𝑢) \times 𝐵 (𝑢 \cdot 𝑣)

A parse of the quotient consists of an $𝐴$ -parses at prefix $𝑢$ and a $𝐵$ -parse of the full string $𝑢 \cdot 𝑣$ .

This operation has a right adjoint:

(\sqrt[𝐴]{𝐶}) (𝑤) = Π_{𝑢, 𝑣} (𝑢 \cdot 𝑣 = 𝑤) \to 𝐴 (𝑢) \to 𝐶 (𝑣)

Intuitively, $\sqrt[𝐴]{𝐶}$ parses $𝑤$ if: for all splittings of $𝑤$ into a prefix and suffix, if the prefix matches $𝐴$ then the suffix must match $𝐶$ .

Together, these form the adjunction:

𝐷_{𝐴} 𝐵 ⊢ 𝐶 ⟺ 𝐵 ⊢ \sqrt[𝐴]{𝐶}

This is an instance of the Day quotient over the discrete monoidal category of strings.

Formal Grammar

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For an alphabet $Σ$ , let $Σ^{*}$ be the free monoid of strings over $Σ$ . A formal grammar $𝐴$ is a family of types indexed by strings:

𝐴 : Σ^{*} \to 𝖳𝗒𝗉𝖾

This definition views formal grammars directly as indexed families of types over strings (which equivalently form presheaves on the discrete category of strings). This view forms the central notion of the Dependent Lambek Calculus. For a given string $𝑤$ , the type $𝐴 (𝑤)$ represents the type of all valid parse trees for $𝑤$ according to the grammar $𝐴$ . If the grammar cannot parse $𝑤$ , then $𝐴 (𝑤)$ is the empty type.

Dependent Tensor of Grammars

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The tensor of grammars,

(𝐴 \otimes 𝐵) (𝑤) = \sum_{𝑤 = 𝑢 𝑣} 𝐴 (𝑢) \times 𝐵 (𝑣)

is simply typed, so $𝐵$ cannot see what $𝐴$ matched.

Just as we have the generalization from the pair type $𝐴 \times 𝐵$ to the dependent pair $Σ_{𝑎 : 𝐴} 𝐵 𝑎$ , we can define a dependent tensor operation. It is a $Σ$ -like generalization over the ordinary tensor, in which the second grammar depends on a parse tree of the first.

Write $𝖳𝗋𝖾𝖾 (𝐴) = \sum_{𝑤} 𝐴 (𝑤)$ for the type of parse trees of $𝐴$ , each paired with the string it parses.

For a grammar $𝐴$ and a family of grammars $𝐵 : 𝖳𝗋𝖾𝖾 (𝐴) \to 𝐆𝐫$ , the dependent tensor is:

(𝐴 \otimes^{𝖣} 𝐵) (𝑤) = \sum_{𝑤 = 𝑢 𝑣} \sum_{𝑎 : 𝐴 (𝑢)} 𝐵 (𝑢, 𝑎) (𝑣)

We may also write it as $(𝑎 \in 𝐴) \otimes 𝐵 (𝑎)$ .

When $𝐵$ is a constant family, the dependent tensor reduces to the ordinary tensor $𝐴 \otimes 𝐵$ .

Dependence runs left to right here, which fits left-to-right parsing. The other handedness, in which the first grammar depends on a parse of the second, is also definable.

Relation to Day Convolution

Just as the ordinary tensor is given by Day convolution, I think there is a similar dependent Day convolution for which this operation is an instance. My best guess is for any $𝑋$ presheaf on a monoidal category $𝒞︀$ and $𝑌$ a functor from the category of elements of $𝑋$ into presheaves on $𝒞︀$ ,

$(𝑋 \otimes^{𝖣} 𝑌) 𝑐 = \int^{𝑢, 𝑣} 𝒞︀ [𝑐, 𝑢 \otimes 𝑣] \times \int^{𝑥 : 𝑋 𝑢} 𝑌 (𝑢, 𝑥) 𝑣$

I’m not positive on this though, and I would hope to also extend this to enrichments that aren’t $𝐒𝐞𝐭$ but I don’t see how that could make sense given that I have quantified over the element $𝑥 : 𝑋 𝑢$ in the coend rather than using the tensor of the enriching category.

The Quotient and Residual Coincidence in Day Convolution

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The Day quotient, $𝐷_{𝐴} 𝐵$ , takes $𝐴$ to be a covariant functor and $𝐵$ to be a contravariant one. While the residual $𝐴 ⊸ 𝐵$ takes both to be contravariant.

This apparent variance mismatch disappears when we restrict to the groupoid core $𝐺 = 𝖼𝗈𝗋𝖾 (𝐶)$ , where variance is trivialized since $𝐺 ≅ 𝐺^{op}$ .

𝗆𝗄𝖯𝗌𝗁 (𝐹) (𝑐) = \int^{𝑠 \in 𝐺} 𝐶 [𝑐, 𝑠] \times 𝐹 (𝑠)

𝗆𝗄𝖢𝗈𝗉𝗌𝗁 (𝐹) (𝑐) = \int^{𝑠 \in 𝐺} 𝐶 [𝑠, 𝑐] \times 𝐹 (𝑠)

By substituting these extensions into the definitions of the residual and the quotient, we obtain a general coincidence for any $𝐹 : 𝐺 \to 𝐒𝐞𝐭$ and $𝐵 : 𝐶^{op} \to 𝐒𝐞𝐭$ :

𝗆𝗄𝖯𝗌𝗁 (𝐹) ⊸ 𝐵 ≅ 𝐷_{𝗆𝗄𝖢𝗈𝗉𝗌𝗁 (𝐹)} 𝐵

This equivalence states that computing the residual against the presheaf extension of $𝐹$ is perfectly isomorphic to taking the quotient by its covariant extension.

The Representable Case

Instantiating the above theorem at a representable functor on the core, $𝐹 = 𝐺 [𝑥, -]$ : the co-Yoneda lemma says that the left Kan extensions compute to the representables on $𝐶$ :

𝗆𝗄𝖯𝗌𝗁 (𝐺 [𝑥, -]) ≅ 𝐶 [-, 𝑥]

𝗆𝗄𝖢𝗈𝗉𝗌𝗁 (𝐺 [𝑥, -]) ≅ 𝐶 [𝑥, -]

Applying the general coincidence theorem, the left and right adjoints coincide precisely on (opposite-variance) representables:

(𝐶 [-, 𝑥] ⊸ 𝐵) (𝑐) ≅ 𝐵 (𝑥 \otimes 𝑐) ≅ (𝐷_{𝐶 [𝑥, -]} 𝐵) (𝑐)

When $𝐶$ is the discrete monoidal category of strings, this recovers the derivative of formal grammars.

In nominal sets, I suspect that this construction also describes name abstraction and the freshness quantifier, although I have not check all of the details of the proof.

The Quotient and its Right Adjoint in Day Convolution

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Let $𝐶$ be a small monoidal category. Recall that for presheaves $𝐴, 𝐵 : 𝐶^{op} \to 𝐒𝐞𝐭$ , the Day convolution provides a closed monoidal structure:

(𝐴 \otimes 𝐵) (𝑐) = \int^{𝑢, 𝑣} 𝐶 [𝑐, 𝑢 \otimes 𝑣] \times 𝐴 (𝑢) \times 𝐵 (𝑣)

(𝐴 ⊸ 𝐵) (𝑐) = \int_{𝑢} 𝐴 (𝑢) \to 𝐵 (𝑢 \otimes 𝑐)

which forms an adjunction $𝐴 \otimes - ⊣ 𝐴 ⊸ -$ .

For covariant functors $𝐴 : 𝐶 \to 𝐒𝐞𝐭$ and presheaves $𝐵 : 𝐶^{op} \to 𝐒𝐞𝐭$ , we can define the quotient and its right adjoint, each of which is a presheaf on $𝐶$ :

(𝐷_{𝐴} 𝐵) (𝑐) = \int^{𝑢} 𝐴 (𝑢) \times 𝐵 (𝑢 \otimes 𝑐)

(\sqrt[𝐴]{𝐵}) (𝑐) = \int_{𝑢, 𝑣} 𝐶 [𝑢 \otimes 𝑣, 𝑐] \to (𝐴 (𝑢) \to 𝐵 (𝑣))

These form the adjunction $𝐷_{𝐴} ⊣ \sqrt[𝐴]{-}$ .

Day Convolution

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(𝐴 \otimes_{Day} 𝐵) 𝑐 = \int^{𝑢, 𝑣} 𝒞︀ (𝑐, 𝑢 \otimes_{𝒞︀} 𝑣) \otimes_{𝒱︀} 𝐴 𝑢 \otimes_{𝒱︀} 𝐵 𝑣,

with unit the representable $𝒞︀ (-, 𝐼)$ .

Day convolution makes the enriched presheaf category $[{𝒞︀}^{op}, 𝒱︀]$ a monoidal category, symmetric when $𝒞︀$ is [1]. It is moreover closed.

As a Kan Extension

In $𝐒𝐞𝐭$

When $𝒱︀ = 𝐒𝐞𝐭$ , we recover the ordinary Day convolution of presheaves $𝐴, 𝐵 : {𝒞︀}^{op} \to 𝐒𝐞𝐭$ , where the formula simplifies to:

(𝐴 \otimes_{Day} 𝐵) 𝑐 = \int^{𝑢, 𝑣} 𝒞︀ [𝑐, 𝑢 \otimes_{𝒞︀} 𝑣] \times 𝐴 𝑢 \times 𝐵 𝑣 .

Proof that Day Convolution is Closed

2026-10-01T00:00:00Z

For $𝑋, 𝐵 : {𝒞︀}^{op} \to 𝒱︀$ , the enriched hom in $[{𝒞︀}^{op}, 𝒱︀]$ is given by the end:

\begin{aligned} \int_{𝑐} ((𝐴 \otimes_{Day} 𝑋) 𝑐 ⊸_{𝒱︀} 𝐵 𝑐) & ≅ \int_{𝑐} ((\int^{𝑢, 𝑣} 𝒞︀ (𝑐, 𝑢 \otimes_{𝒞︀} 𝑣) \otimes_{𝒱︀} 𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} 𝐵 𝑐) \\ ≅ \int_{𝑐} \int_{𝑢, 𝑣} ((𝒞︀ (𝑐, 𝑢 \otimes_{𝒞︀} 𝑣) \otimes_{𝒱︀} 𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} 𝐵 𝑐) \\ ≅ \int_{𝑢, 𝑣} ((𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} \int_{𝑐} (𝒞︀ (𝑐, 𝑢 \otimes_{𝒞︀} 𝑣) ⊸_{𝒱︀} 𝐵 𝑐)) \\ ≅ \int_{𝑢, 𝑣} ((𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} 𝐵 (𝑢 \otimes_{𝒞︀} 𝑣)) \\ ≅ \int_{𝑣} (𝑋 𝑣 ⊸_{𝒱︀} \int_{𝑢} (𝐴 𝑢 ⊸_{𝒱︀} 𝐵 (𝑢 \otimes_{𝒞︀} 𝑣))) \\ ≅ \int_{𝑣} (𝑋 𝑣 ⊸_{𝒱︀} (𝐴 ⊸ 𝐵) 𝑣) . \end{aligned}

∎

Correct-by-Construction Parsing

2026-10-01T00:00:00Z

Parsers are some of the most important pieces of software that we have. You have likely relied on dozens of parsers to even read this message. Anytime someone writes a piece of high-level code, it must be parsed into a representation that is workable for the machine.

Because parsers are so ubiquituous, the correctness of so many other programs is contingent on a bug-free parser. Despite this fundamental reliance, parsers are often unverified. Buggy parsers have led to many critical security vulnerabilities. Even in otherwise verified software development, parsing is usually unverified (or the final component to become verified).

To address this gap, we develop Lambek ${\begin{matrix} \end{matrix}}^{𝙳}$ , a domain-specific programming language for writing intrinsically correct parsers. That is, if a parser written in Lambek ${\begin{matrix} \end{matrix}}^{𝙳}$ compiles then we guarantee the resulting program is sound with respect to its parsing semantics.

I first explored this idea in Intrinsic Verification of Parsers and Formal Grammar Theory in Dependent Lambek Calculus, and it will be the focus of my dissertation.

Notes and sketches of ongoing work in Lambek ${\begin{matrix} \end{matrix}}^{𝙳}$ may be found below and throughout this site.

30 entries

Definition 1. The Quotient and its Right Adjoint in Day Convolution day-quotient

Let $𝐶$ be a small monoidal category. Recall that for presheaves $𝐴, 𝐵 : 𝐶^{op} \to 𝐒𝐞𝐭$ , the Day convolution provides a closed monoidal structure:

(𝐴 \otimes 𝐵) (𝑐) = \int^{𝑢, 𝑣} 𝐶 [𝑐, 𝑢 \otimes 𝑣] \times 𝐴 (𝑢) \times 𝐵 (𝑣)

(𝐴 ⊸ 𝐵) (𝑐) = \int_{𝑢} 𝐴 (𝑢) \to 𝐵 (𝑢 \otimes 𝑐)

which forms an adjunction $𝐴 \otimes - ⊣ 𝐴 ⊸ -$ .

For covariant functors $𝐴 : 𝐶 \to 𝐒𝐞𝐭$ and presheaves $𝐵 : 𝐶^{op} \to 𝐒𝐞𝐭$ , we can define the quotient and its right adjoint, each of which is a presheaf on $𝐶$ :

(𝐷_{𝐴} 𝐵) (𝑐) = \int^{𝑢} 𝐴 (𝑢) \times 𝐵 (𝑢 \otimes 𝑐)

(\sqrt[𝐴]{𝐵}) (𝑐) = \int_{𝑢, 𝑣} 𝐶 [𝑢 \otimes 𝑣, 𝑐] \to (𝐴 (𝑢) \to 𝐵 (𝑣))

These form the adjunction $𝐷_{𝐴} ⊣ \sqrt[𝐴]{-}$ .

Definition 2. Dependent Tensor of Grammars dependent-tensor

The tensor of grammars,

(𝐴 \otimes 𝐵) (𝑤) = \sum_{𝑤 = 𝑢 𝑣} 𝐴 (𝑢) \times 𝐵 (𝑣)

is simply typed, so $𝐵$ cannot see what $𝐴$ matched.

Write $𝖳𝗋𝖾𝖾 (𝐴) = \sum_{𝑤} 𝐴 (𝑤)$ for the type of parse trees of $𝐴$ , each paired with the string it parses.

For a grammar $𝐴$ and a family of grammars $𝐵 : 𝖳𝗋𝖾𝖾 (𝐴) \to 𝐆𝐫$ , the dependent tensor is:

(𝐴 \otimes^{𝖣} 𝐵) (𝑤) = \sum_{𝑤 = 𝑢 𝑣} \sum_{𝑎 : 𝐴 (𝑢)} 𝐵 (𝑢, 𝑎) (𝑣)

We may also write it as $(𝑎 \in 𝐴) \otimes 𝐵 (𝑎)$ .

When $𝐵$ is a constant family, the dependent tensor reduces to the ordinary tensor $𝐴 \otimes 𝐵$ .

Dependence runs left to right here, which fits left-to-right parsing. The other handedness, in which the first grammar depends on a parse of the second, is also definable.

Relation to Day Convolution

$(𝑋 \otimes^{𝖣} 𝑌) 𝑐 = \int^{𝑢, 𝑣} 𝒞︀ [𝑐, 𝑢 \otimes 𝑣] \times \int^{𝑥 : 𝑋 𝑢} 𝑌 (𝑢, 𝑥) 𝑣$

Definition 3. Formal Grammar formal-grammar

For an alphabet $Σ$ , let $Σ^{*}$ be the free monoid of strings over $Σ$ . A formal grammar $𝐴$ is a family of types indexed by strings:

𝐴 : Σ^{*} \to 𝖳𝗒𝗉𝖾

Definition 4. Grammar Quotients grammar-quotients

For formal grammars $𝐴$ and $𝐵$ , the quotient of $𝐵$ by $𝐴$ is the grammar of what is left of $𝐵$ once an $𝐴$ has been read off the front:

(𝐷_{𝐴} 𝐵) (𝑣) = Σ_{𝑢} 𝐴 (𝑢) \times 𝐵 (𝑢 \cdot 𝑣)

A parse of the quotient consists of an $𝐴$ -parses at prefix $𝑢$ and a $𝐵$ -parse of the full string $𝑢 \cdot 𝑣$ .

This operation has a right adjoint:

(\sqrt[𝐴]{𝐶}) (𝑤) = Π_{𝑢, 𝑣} (𝑢 \cdot 𝑣 = 𝑤) \to 𝐴 (𝑢) \to 𝐶 (𝑣)

Intuitively, $\sqrt[𝐴]{𝐶}$ parses $𝑤$ if: for all splittings of $𝑤$ into a prefix and suffix, if the prefix matches $𝐴$ then the suffix must match $𝐶$ .

Together, these form the adjunction:

𝐷_{𝐴} 𝐵 ⊢ 𝐶 ⟺ 𝐵 ⊢ \sqrt[𝐴]{𝐶}

This is an instance of the Day quotient over the discrete monoidal category of strings.

Definition 5. Grammar Residuals grammar-residual

Formally, it is defined as:

(𝐴 ⊸ 𝐵) (𝑣) = Π_{𝑢} 𝐴 (𝑢) \to 𝐵 (𝑢 \cdot 𝑣)

Definition 6. Tensor of Grammars grammar-tensor

Formally, it is defined as:

(𝐴 \otimes 𝐵) (𝑤) = \sum_{𝑤 = 𝑢 𝑣} 𝐴 (𝑢) \times 𝐵 (𝑣)

This operation is exactly the Day convolution of $𝐴$ and $𝐵$ , where we view formal grammars as presheaves over the monoid of strings.

Definition 7. Types over an Algebraic Theory theory-type

Fix a finitary algebraic theory $𝒯︀$ : sorts $𝑆$ , a signature $𝜎$ , equations, and a set $𝑉$ of generators. Write $𝑴$ for the free $𝒯︀$ -model on $𝑉$ and $| 𝑴 |_{𝑠}$ for its carrier at sort $𝑠$ .

A $𝒯︀$ -type of sort $𝑠$ is a family $𝐴 : | 𝑴 |_{𝑠} \to 𝖳𝗒𝗉𝖾$ .

For $𝒯︀$ -types $𝐴$ and $𝐵$ of sort $𝑠$ we have the additives, defined pointwise,

(𝐴 & 𝐵) (𝑚) = 𝐴 (𝑚) \times 𝐵 (𝑚), (𝐴 \oplus 𝐵) (𝑚) = 𝐴 (𝑚) + 𝐵 (𝑚), (𝐴 \Rightarrow 𝐵) (𝑚) = 𝐴 (𝑚) \to 𝐵 (𝑚),

along with $⊤$ , $⊥$ , and their indexed versions.

Each operation $𝑜 : 𝑠_{0}, \dots, 𝑠_{𝑛 - 1} \to 𝑠$ of $𝒯︀$ gives a multiplicative, defined by Day convolution,

\otimes [𝑜] (𝐴_{0}, \dots, 𝐴_{𝑛 - 1}) (𝑚) = Σ_{𝑜 (𝑚_{0}, \dots, 𝑚_{𝑛 - 1}) = 𝑚} \times Π_{𝑖 < 𝑛} 𝐴_{𝑖} (𝑚_{𝑖}) .

Each lifted operation also has closed structure: a residual in each of its arguments. Day algebras [1] give a more general convolutional definition.

When $𝒯︀$ is the theory of monoids and $𝑉$ is an alphabet, $𝒯︀$ -types are formal grammars. Multiplication gives $\otimes$ , the unit gives $𝜀$ , and we recover Lambek ${\begin{matrix} \end{matrix}}^{𝙳}$ .

Definition 8. Representable Grammars representable-grammar

For any string $𝑤$ , we can define a representable grammar $⌈ 𝑤 ⌉$ which matches exactly the string $𝑤$ and nothing else. The parse trees for a representable grammar are proofs that the string is exactly equal to $𝑤$ :

⌈ 𝑤 ⌉ (𝑢) = (𝑢 \equiv 𝑤)

9 Intrinsic Verification of Parsers and Formal Grammar Theory in Dependent Lambek Calculus pldi-2025-talkSteven Schaefer, Nathan Varner, Pedro H. Azevedo de Amorim, Max S. New · PLDI · June 19, 2025

We present Dependent Lambek Calculus, a domain-specific dependent type theory for verified parsing and formal grammar theory. In Dependent Lambek Calculus, linear types are used as a syntax for formal grammars, and parsers can be written as linear terms. The linear typing restriction provides a form of intrinsic verification that a parser yields only valid parse trees for the input string. We demonstrate the expressivity of this system by showing that the combination of inductive linear types and dependency on non-linear data can be used to encode commonly used grammar formalisms such as regular and context-free grammars as well as traces of various types of automata. Using these encodings, we define parsers for regular expressions using deterministic automata, as well as examples of verified parsers of context-free grammars.

We present a denotational semantics of our type theory that interprets the types as a mathematical notion of formal grammars. Based on this denotational semantics, we have made a prototype implementation of Dependent Lambek Calculus using a shallow embedding in the Agda proof assistant. All of our examples parsers have been implemented in this prototype implementation.

Event · Slides · Video

10 Intrinsic Verification of Parsers and Formal Grammar Theory in Dependent Lambek Calculus intrinsic-verification-of-parsersPLDI · 2025

We present Dependent Lambek Calculus (Lambek ${\begin{matrix} \end{matrix}}^{𝙳}$ ), a domain-specific dependent type theory for verified parsing and formal grammar theory. In Lambek ${\begin{matrix} \end{matrix}}^{𝙳}$ , linear types are used as a syntax for formal grammars, and parsers can be written as linear terms. The linear typing restriction provides a form of intrinsic verification that a parser yields only valid parse trees for the input string. We demonstrate the expressivity of this system by showing that the combination of inductive linear types and dependency on non-linear data can be used to encode commonly used grammar formalisms such as regular and context-free grammars as well as traces of various types of automata. Using these encodings, we define parsers for regular expressions using deterministic automata, as well as examples of verified parsers of context-free grammars.

We present a denotational semantics of our type theory that interprets the linear types as functions from strings to sets of abstract parse trees and terms as parse transformers. Based on this denotational semantics, we have made a prototype implementation of Lambek ${\begin{matrix} \end{matrix}}^{𝙳}$ using a shallow embedding in the Agda proof assistant. All of our examples parsers have been implemented in this prototype implementation.

PDF · DOI · arXiv (extended version) · Source code · pldb

Definition 11. A Grammar of Finite Cardinals finite-cardinal-grammar

Define a grammar for each finite cardinal $𝑛 : ℕ$ .

[0] ≔ ⊥

[𝗌𝗎𝖼 (𝑛)] ≔ ⊤ \oplus [𝑛]

Then define the grammar of all finite cardinals as

𝖥𝗂𝗇 ≔ ⨁_{𝑛 : ℕ} [𝑛]

Definition 12. Finite Unambiguity finite-unambiguity

Define a grammar $𝐴$ to be finitely unambiguous if $𝐴 & 𝐴 ≅ 𝐴$ .

There is likely a better name for this.

13 Finite Unambiguity is Not Equivalent to Unambiguity finite-unambiguity-not-unambiguity

For a while I believed finite unambiguity to be equivalent to the other definitions of unambiguity.

Definition 13.1. Finite Unambiguity finite-unambiguity

Define a grammar $𝐴$ to be finitely unambiguous if $𝐴 & 𝐴 ≅ 𝐴$ .

There is likely a better name for this.

Definition 13.2. Unambiguity as Subterminality unambiguity-as-subterminality

A grammar $𝐴$ is unambiguous if the unique map into the terminal object is a monomorphism. That is, $𝐴$ is a subobject of $⊤$ .

Definition 13.3. Unambiguity as Unique Map into Codomain unambiguity-as-unique-map

A grammar $𝐴$ is unambiguous if for all grammars $𝐵$ and maps $𝑒, 𝑒' : 𝐴 ⊢ 𝐵$ we have $𝑒 = 𝑒'$ .

In a category with terminal objects, this is equivalent to unambiguity defined via subterminality.

Definition 13.3.1. Unambiguity as Subterminality unambiguity-as-subterminality

A grammar $𝐴$ is unambiguous if the unique map into the terminal object is a monomorphism. That is, $𝐴$ is a subobject of $⊤$ .

We may use an analogy from the category of sets, however I had missed the infinite case when translating this idea to grammars.

It is true that if a grammar is unambiguous, then it is finitely unambiguous. However, the converse does not hold unless the grammar has finitely many parse trees for each string. This finiteness condition is a semantic one. If this proof of unambiguity were to be internalized, then it could maybe be captured through the lens of some grammar of finite cardinals.

Let $𝐴 ≔ ⨁_{𝑛 : ℕ} ⊤$ . $𝐴$ is finitely unambiguous but not unambiguous. The isomorphism between $𝐴$ and $𝐴 & 𝐴$ amounts to building a bijection between $ℕ$ and $ℕ \times ℕ$ , which is straightforward.

14 What Connectives Preserve Finiteness? finiteness-preserving-connectives

In order to bridge the gap between finite unambiguity and unambiguity, we can try to restrict to grammars that have finitely many parse trees. We’d expect this to address the concern semantically, as that fixes the problem when the parses are interpreted in $𝐒𝐞𝐭$ .

We can define when a grammar is finite, and then try to prove that finiteness is preserved on some sane operations on grammars. For instance, $&$ , $\oplus$ , and $\to$ each preserve finiteness. I haven’t proven this myself (which would be a useful exercise), but the following is substantiated in any topos with a natural numbers object [johnstone-2002].

14.1 Finite Cardinal Arithmetic in a Topos finite-cardinal-arithmetic-topos

My working mental model is in a presheaf topos, where the natural numbers object can be defined explicitly as $Δ (ℕ)$ .

Definition 14.1.1. Constant Presheaf constant-presheaf

The constant presheaf of a set $𝑋$ on a category $𝐶$ is a functor $Δ (𝑋) : 𝐶^{𝑜 𝑝} \to 𝐒𝐞𝐭$ such that

Δ (𝑋) 𝑐 ≔ 𝑋

I often call this the discrete presheaf for $𝑋$ , but I don’t know if that’s standard.

The finite cardinals with respect to $Δ (ℕ)$ can be characterized then as,

[0] ≔ ⊥

[𝗌𝗎𝖼 (𝑛)] ≔ ⊤ \oplus [𝑛]

These obey nice algebraic properties.

[𝑛 𝑚] ≅ [𝑛] \times [𝑚]

[𝑛 + 𝑚] ≅ [𝑛] \oplus [𝑚]

[𝑛^{𝑚}] ≅ [𝑚] \to [𝑛]

The case of $\otimes$ is more problematic. Semantically, for all $𝑤 : String$ we may bound the size of the set of parse trees

| (𝐴 \otimes 𝐵) 𝑤 | \leq Σ_{𝑤_{1} ++ 𝑤_{2} = 𝑤} | 𝐴 𝑤_{1} | | 𝐵 𝑤_{2} |

Precisely knowing this bound isn’t too important, but certainly it exists. We could capture this behavior by just adding an axiom that if $𝐴$ and $𝐵$ are each finite, then $𝐴 \otimes 𝐵$ is finite. Although, I’d rather not add an axiom.

We could directly try to prove that $\otimes$ preserves finiteness. The statement would follow from showing that $\otimes$ preserves monomorphisms. That is, if $𝑓 : 𝐴 ↪ 𝐶$ and $𝑔 : 𝐵 ↪ 𝐷$ , it suffices to show that $𝑓 \otimes 𝑔 : 𝐴 \otimes 𝐵 \to 𝐶 \otimes 𝐷$ is a monomorphism. This would imply finiteness, because you may then apply this for $𝐶$ and $𝐷$ as $𝖥𝗂𝗇$ . However, you would also need to show that $𝖥𝗂𝗇 \otimes 𝖥𝗂𝗇$ is finite, which isn’t immediately clear.

15 Internal Finiteness internally-finite-grammar

Define a grammar $𝐴$ to be finite (or perhaps subfinite) if it is a subobject of grammar of finite cardinals.

𝐴 ↪ 𝖥𝗂𝗇

Where $𝖥𝗂𝗇$ is defined as follows.

Definition 15.1. A Grammar of Finite Cardinals finite-cardinal-grammar

Define a grammar for each finite cardinal $𝑛 : ℕ$ .

[0] ≔ ⊥

[𝗌𝗎𝖼 (𝑛)] ≔ ⊤ \oplus [𝑛]

Then define the grammar of all finite cardinals as

𝖥𝗂𝗇 ≔ ⨁_{𝑛 : ℕ} [𝑛]

16 Star Continuity is a Semantic Property star-continuity-semantic-property

Star continuity in Dependent Lambek Calculus can often be a convenient proof technique, but it’s important to remember that this shouldn’t be the first line of defense.

Star continuity holds in the Agda model, but does it hold in the syntactic model? I believe that it does because of the presence of the indexed coproducts. So perhaps it isn’t so sinister after all. It is worth noting that much of the reasoning performed by inducting on the length of a Kleene star isn’t very elegant. If a proof necessitates star continuity, then it doesn’t seem to be aided greatly by the type system.

Definition 17. Unambiguity as Subterminality unambiguity-as-subterminality

A grammar $𝐴$ is unambiguous if the unique map into the terminal object is a monomorphism. That is, $𝐴$ is a subobject of $⊤$ .

Definition 18. Unambiguity as Unique Map into Codomain unambiguity-as-unique-map

A grammar $𝐴$ is unambiguous if for all grammars $𝐵$ and maps $𝑒, 𝑒' : 𝐴 ⊢ 𝐵$ we have $𝑒 = 𝑒'$ .

In a category with terminal objects, this is equivalent to unambiguity defined via subterminality.

Definition 18.1. Unambiguity as Subterminality unambiguity-as-subterminality

A grammar $𝐴$ is unambiguous if the unique map into the terminal object is a monomorphism. That is, $𝐴$ is a subobject of $⊤$ .

Definition 19. Unambiguity via the Diagonal Being an Isomorphism unambiguity-via-diagonal

Finite unambiguity does not serve as an adequate definition of unambiguity that is equivalent to unambiguity as subterminality and unambiguity as a unique map. However, the definition attempted via finite unambiguity can be refined to something that is equivalent to these.

A grammar $𝐴$ is unambiguous if $Δ : 𝐴 ⊢ 𝐴 & 𝐴$ is an isomorphism. Equivalently, $𝐴$ is unambiguous if $𝜋_{1} : 𝐴 & 𝐴 ⊢ 𝐴$ and $𝜋_{2} : 𝐴 & 𝐴 ⊢ 𝐴$ are equal.

20 Subobject Classifier in Dependent Lambek Calculus subobject-classifier-for-grammars

Previously in Agda we had constructed equalizers in Lambek ${\begin{matrix} \end{matrix}}^{𝙳}$ using sigma types. Further, equalizers form subobjects, which may be comprehended as maps from a type into a subobject classifier.

I believe in our formalization we want to generalize this idea to a broader class of subobjects, maybe even all of them. That is, we could define a subobject classifying grammar $Ω$ . Then we can internally define any predicate on a grammar $𝑔$ via a term $𝑝 : 𝑔 ⊢ Ω$ .

That is, forall $𝑤 : String$ we have $⟦ Ω ⟧ (𝑤) ≔ 𝗁𝖯𝗋𝗈𝗉$ , and

⟦ 𝗌𝗎𝖻𝗀𝗋𝖺𝗆𝗆𝖺𝗋 (𝑔) (𝑝) ⟧ (𝑤) ≔ \sum_{𝑥 : ⟦ 𝑔 ⟧ (𝑤)} ⟦ 𝑝 ⟧ (𝑥)

The code for this is nearly identical to the definition of equalizers as sigma types, and I have even built a translation of the equalizers code that is defined with this as its foundation. I have not yet tested if either implementation is preferable.

It isn’t yet clear how to best expose this sort of construct syntactically. I suppose you could assume some subobject classifying grammar, but then I don’t know if you can then reap the benefits of the propositions internally. That is, how do you reflect the proposition that two terms are equal in the internal language of propositions rather than the external one?

If nothing else, this code gives me a reusable interface to axiomatize smaller, sandboxed ways in which I’d like to internalize certain types of propositions (such as “two terms are equal” or “does not begin with the character $𝑐$ ”).

The hope for this code is that it lets me inductively prove the follow last soundness of Kleene star.

Definition 21. Kleene Star in Dependent Lambek Calculus kleene-star

For a grammar $𝐴$ , the Kleene star $𝐴^{*}$ is defined as a least-fixed point,

𝜇 𝑥 . 𝜖 \oplus (𝐴 \otimes 𝑥)

Definition 22. Star Continuity in Dependent Lambek Calculus star-continuity-in-dependent-lambek

For a grammar $𝐴$ , the Kleene star $𝐴^{*}$ is isomorphic to an indexed coproduct.

𝐴^{*} ≅ ⨁_{𝑛 : ℕ} 𝐴^{\otimes 𝑛}

That is, we may view the parses of $𝐴^{*}$ like a linear list comprising parses of $𝐴$ concatenated together. Further, for each of these lists we may know the precise length.

When viewing Dependent Lambek Calculus as a model of Kleene algebra, this is precisely the statement that star continuity holds.

Definition 23. First Sets in Dependent Lambek Calculus first-set-in-dependent-lambek

The first set of a grammar $𝐴$ may be captured in Lambek ${\begin{matrix} \end{matrix}}^{𝙳}$ via the following proposition:

𝑐 \notin 𝖥𝗂𝗋𝗌𝗍 (𝐴) ≔ 𝐴 & (' 𝑐' \otimes ⊤) ⊢ ⊥

Or perhaps with ones of the grammars

𝐴 \Rightarrow \neg (' 𝑐' \otimes ⊤),' 𝑐' \otimes ⊤ \Rightarrow \neg 𝐴

Definition 24. FollowLast Sets in Dependent Lambek Calculus followlast-set-in-dependent-lambek

The followlast set of a grammar $𝐴$ may be captured in Lambek ${\begin{matrix} \end{matrix}}^{𝙳}$ via the following proposition:

𝑐 \notin 𝖥𝖫𝖺𝗌𝗍 (𝐴) ≔ 𝐴 & (𝐴 \otimes' 𝑐' \otimes ⊤) ⊢ ⊥

Or perhaps with ones of the grammars

𝐴 \Rightarrow \neg (𝐴 \otimes' 𝑐' \otimes ⊤), 𝐴 \otimes' 𝑐' \otimes ⊤ \Rightarrow \neg 𝐴

Definition 25. Nullability in Dependent Lambek Calculus nullability-in-dependent-lambek

The nullability ( $𝖭𝗎𝗅𝗅 (\cdot)$ ) of a grammar $𝐴$ may be captured in Lambek ${\begin{matrix} \end{matrix}}^{𝙳}$ via the following proposition:

\neg 𝖭𝗎𝗅𝗅 (𝐴) ≔ 𝐴 & 𝜖 ⊢ ⊥

Or perhaps with one of the grammars

𝐴 \Rightarrow \neg 𝜀, 𝜀 \Rightarrow \neg 𝐴

26 Intrinsic Verification of Parsers and Formal Grammar Theory in Dependent Lambek Calculus mwpls-2024-talkSteven Schaefer, Nathan Varner, Pedro H. Azevedo de Amorim, Max S. New · MWPLS · November 22, 2024

Event · Slides

Definition 27. The Derivative of a Grammar grammar-derivative

When a grammar quotient $𝐷_{𝐴} 𝐵$ is taken with respect to a representable grammar $𝐴 = ⌈ 𝑤 ⌉$ (matching exactly the string $𝑤$ ), it coincides with the residual $𝐴 ⊸ 𝐵$ , as established by the Quotient and Residual Coincidence.

This special case is known as the Brzozowski derivative of $𝐵$ by the string $𝑤$ , often written as $𝜕_{𝑤} 𝐵$ . We have the following coincidence:

(⌈ 𝑤 ⌉ ⊸ 𝐵) (𝑣) ≅ 𝐵 (𝑤 \cdot 𝑣) ≅ (𝐷_{⌈ 𝑤 ⌉} 𝐵) (𝑣)

Definition 28. Later on Grammars grammar-later

Write $𝑢 ⊏ 𝑣$ when $𝑢$ is a proper suffix of $𝑣$ , that is, $𝑣 = 𝑥 ++ 𝑢$ with $𝑥 \neq 𝜀$ . The later of a grammar $𝐴$ is

(▷ 𝐴) 𝑣 = \prod_{𝑢 ⊏ 𝑣} 𝐴 𝑢 .

A parse of $▷ 𝐴$ over $𝑣$ is a parse of $𝐴$ over every proper suffix of $𝑣$ . In particular $(▷ 𝐴) 𝜀$ is a singleton.

This is later on families for strings under the proper-suffix order. That order is well-founded because it strictly decreases length, so it is a thin direct category. There is at most one map $𝑢 \to 𝑣$ , so the product over maps from the strict past has one factor per proper suffix. Guarded recursion is modelled by presheaves on $𝜔$ , the topos of trees, and more generally by sheaves over a well-founded base [1]. Here the later acts on families, which is what grammars are, and there is no clock.

Later is the right adjoint of the proper-suffix derivative. Let $𝖭𝖤𝖲 = ⨁_{𝑥 \neq 𝜀} ⌈ 𝑥 ⌉$ be the grammar of non-empty strings. Then the derivative $(𝜕_{𝖭𝖤𝖲} 𝐵) 𝑢 ≅ \sum_{𝑥 \neq 𝜀} 𝐵 (𝑥 ++ 𝑢)$ has the right adjoint

\sqrt[𝖭𝖤𝖲]{𝐶} 𝑣 ≅ \prod_{𝑥 ++ 𝑢 = 𝑣} 𝖭𝖤𝖲 𝑥 \to 𝐶 𝑢 ≅ (▷ 𝐶) 𝑣,

so $𝜕_{𝖭𝖤𝖲} - ⊣ ▷$ . Splitting $𝖭𝖤𝖲$ into its summands gives the form of $▷$ that the calculus can define:

▷ 𝐴 ≅ \underset{𝑤 \neq 𝜀}{&} \sqrt[⌈ 𝑤 ⌉]{𝐴}, \sqrt[⌈ 𝑤 ⌉]{𝐴} ≅ (⌈ 𝑤 ⌉ \otimes ⊤) \Rightarrow (⌈ 𝑤 ⌉ \otimes 𝐴) .

The component at $𝑤$ says: if the string begins with $𝑤$ , then the rest parses as $𝐴$ .

The restriction to non-empty $𝑤$ is what makes this a later. At $𝑤 = 𝜀$ the component is $\sqrt[⌈ 𝜀 ⌉]{𝐴} ≅ 𝐴$ . Including it would give a projection $▷ 𝐴 ⊢ 𝐴$ , and Löb would then prove every grammar. For the same reason the later is a product over suffixes. A sum such as $⨁_{𝑐} ⌈ 𝑐 ⌉ ⊸ 𝐴$ is empty at $𝜀$ , and at $𝐴 = ⊥$ it is $⊥$ . The identity step $⊥ ⊢ ⊥$ would then give Löb a proof of $⊤ ⊢ ⊥$ .

In the Agda this is ▷ in Grammar/Later/Base.agda, which is defined as the indexed conjunction of √l-string. The mirror image ▷r, over proper prefixes, is defined in the same way. See also the bilateral later and the later along an arbitrary well-founded order.

Theorem 29. Löb Induction for Grammars grammar-lob

For every grammar $𝐴$ and every term $𝜑 : ▷ 𝐴 ⊢ 𝐴$ , where $▷$ is the later on grammars, there is a unique global parse $𝗅ö𝖻 𝜑 : ⊤ ⊢ 𝐴$ with

𝗅ö𝖻 𝜑 = 𝜑 \circ 𝗇𝖾𝗑𝗍 (𝗅ö𝖻 𝜑) .

Here $𝗇𝖾𝗑𝗍 𝑡 : ⊤ ⊢ ▷ 𝐴$ restricts a global parse $𝑡$ to every proper suffix.

The proof is recursion on the length of the string. At $𝑣$ , the parses already built at the proper suffixes of $𝑣$ form an element of $(▷ 𝐴) 𝑣$ , and $𝜑$ turns it into a parse at $𝑣$ . Uniqueness is Löb for families over the proper-suffix order. In the Agda, lob in Grammar/Later/Base.agda is this recursion, done by well-founded induction on length.

To prove an entailment $𝐵 ⊢ 𝐶$ this way, apply Löb to $𝐴 = 𝐵 \Rightarrow 𝐶$ . The hypothesis $▷ (𝐵 \Rightarrow 𝐶)$ is the induction hypothesis at every proper suffix. It becomes usable once a non-nullable grammar has been consumed. If $𝜀 & 𝐵 ⊢ ⊥$ , then

(𝐵 \otimes ⊤) & ▷ 𝐴 ⊢ 𝐵 \otimes 𝐴

because a parse of $𝐵 \otimes ⊤$ over $𝑣$ splits $𝑣 = 𝑥 ++ 𝑢$ with $𝑥$ non-empty, so $𝑢 ⊏ 𝑣$ . This is ▷-app-NE in Grammar/Later/Properties.agda. Induction on a Kleene star is the standard use.

Theorem 30. Next on Grammars Is Presheaf Structure grammar-next-presheaf

For presheaves, $𝗇𝖾𝗑𝗍 : 𝑃 \to ▷ 𝑃$ restricts along the strict past, as in later on presheaves. A grammar is only a family over strings, so a map $𝗇𝖾𝗑𝗍 : 𝐴 ⊢ ▷ 𝐴$ into the later on grammars is extra data. It sends a parse over $𝑣$ to parses over every proper suffix of $𝑣$ .

Let $□ 𝐴 = 𝐴 & ▷ 𝐴$ , so that $(□ 𝐴) 𝑣 ≅ \prod_{𝑢 ⊑ 𝑣} 𝐴 𝑢$ . This is the comonad $𝑈 \circ Cofree$ for the suffix order, and presheaves are comonadic over families. The counit law forces the first component of a coalgebra $𝐴 ⊢ □ 𝐴$ to be the identity. Hence:

a presheaf on strings under the suffix order is the same as a grammar $𝐴$ with a map $𝗇𝖾𝗑𝗍 : 𝐴 ⊢ ▷ 𝐴$ that satisfies coassociativity. Restricting to $𝑢'$ and then to $𝑢 ⊏ 𝑢'$ must agree with restricting to $𝑢$ directly;
for a proposition-valued grammar (a language), coassociativity is automatic, and $𝗇𝖾𝗑𝗍$ exists exactly when the language is closed under suffixes. For example, $⊤$ has one, but $⌈ 𝑎 ⌉$ does not, since $(▷ ⌈ 𝑎 ⌉) 𝑎 ≅ ⌈ 𝑎 ⌉ 𝜀$ is empty.

Löb does not need $𝗇𝖾𝗑𝗍$ on $𝐴$ . The fixed-point equation only restricts a global parse $⊤ ⊢ 𝐴$ , and a global parse can always be restricted. In the Agda, the grammar-level IsCoalgebra record, with only the field next, is in Grammar/Later/Coalgebra.agda on the guarded branch.

On presheaves the strict downset of $𝑐 ++ 𝑡$ is represented by $𝑡$ , because every proper suffix of $𝑐 ++ 𝑡$ is a suffix of $𝑡$ . So predecessors simplify later to $(▷ 𝑃) (𝑐 ++ 𝑡) ≅ 𝑃 𝑡$ . On grammars there is no such simplification, since the factors at different suffixes are unrelated.

Day Convolution is Closed

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(𝐴 ⊸ 𝐵) 𝑐 = \int_{𝑢} 𝐴 𝑢 ⊸_{𝒱︀} 𝐵 (𝑢 \otimes_{𝒞︀} 𝑐) .

Then, for Day convolution $\otimes_{Day}$ ,

𝐴 \otimes_{Day} - ⊣ 𝐴 ⊸_{Day} - .

Proof. Proof that Day Convolution is Closed day-closed-structure-proof

For $𝑋, 𝐵 : {𝒞︀}^{op} \to 𝒱︀$ , the enriched hom in $[{𝒞︀}^{op}, 𝒱︀]$ is given by the end:

\begin{aligned} \int_{𝑐} ((𝐴 \otimes_{Day} 𝑋) 𝑐 ⊸_{𝒱︀} 𝐵 𝑐) & ≅ \int_{𝑐} ((\int^{𝑢, 𝑣} 𝒞︀ (𝑐, 𝑢 \otimes_{𝒞︀} 𝑣) \otimes_{𝒱︀} 𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} 𝐵 𝑐) \\ ≅ \int_{𝑐} \int_{𝑢, 𝑣} ((𝒞︀ (𝑐, 𝑢 \otimes_{𝒞︀} 𝑣) \otimes_{𝒱︀} 𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} 𝐵 𝑐) \\ ≅ \int_{𝑢, 𝑣} ((𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} \int_{𝑐} (𝒞︀ (𝑐, 𝑢 \otimes_{𝒞︀} 𝑣) ⊸_{𝒱︀} 𝐵 𝑐)) \\ ≅ \int_{𝑢, 𝑣} ((𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} 𝐵 (𝑢 \otimes_{𝒞︀} 𝑣)) \\ ≅ \int_{𝑣} (𝑋 𝑣 ⊸_{𝒱︀} \int_{𝑢} (𝐴 𝑢 ⊸_{𝒱︀} 𝐵 (𝑢 \otimes_{𝒞︀} 𝑣))) \\ ≅ \int_{𝑣} (𝑋 𝑣 ⊸_{𝒱︀} (𝐴 ⊸ 𝐵) 𝑣) . \end{aligned}

∎

Total Bicategory

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The total bicategory of a displayed bicategory over $𝒦︀$ packages the base and the displayed data together, one dimension up from the total category of a displayed category.

Its 0-cells are pairs of a 0-cell of $𝒦︀$ and a displayed 0-cell over it.
Its hom-category from to is the total category of the displayed hom-category . So a 1-cell is a pair and a 2-cell is a pair .
Identities, composition, unitors and associator are pairs of the base structure and the displayed structure over it, and the triangle and pentagon hold because they hold in the base and, over that, in the displayed bicategory.

Projecting to first components is a pseudofunctor whose unit and composition comparisons are identity 2-cells.

Types over an Algebraic Theory

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A $𝒯︀$ -type of sort $𝑠$ is a family $𝐴 : | 𝑴 |_{𝑠} \to 𝖳𝗒𝗉𝖾$ .

For $𝒯︀$ -types $𝐴$ and $𝐵$ of sort $𝑠$ we have the additives, defined pointwise,

(𝐴 & 𝐵) (𝑚) = 𝐴 (𝑚) \times 𝐵 (𝑚), (𝐴 \oplus 𝐵) (𝑚) = 𝐴 (𝑚) + 𝐵 (𝑚), (𝐴 \Rightarrow 𝐵) (𝑚) = 𝐴 (𝑚) \to 𝐵 (𝑚),

along with $⊤$ , $⊥$ , and their indexed versions.

Each operation $𝑜 : 𝑠_{0}, \dots, 𝑠_{𝑛 - 1} \to 𝑠$ of $𝒯︀$ gives a multiplicative, defined by Day convolution,

\otimes [𝑜] (𝐴_{0}, \dots, 𝐴_{𝑛 - 1}) (𝑚) = Σ_{𝑜 (𝑚_{0}, \dots, 𝑚_{𝑛 - 1}) = 𝑚} \times Π_{𝑖 < 𝑛} 𝐴_{𝑖} (𝑚_{𝑖}) .

Each lifted operation also has closed structure: a residual in each of its arguments. Day algebras [1] give a more general convolutional definition.

Functorial egg?

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For the MPLSE Reading Group, this week we read egg: Fast and Extensible Equality Saturation. Pranav Srinivasan and I seemed to think that the $e$ -class analysis into any join semilattice looked a lot like a functor. However, I’m not quite sure what the domain of that functor ought to be.

Recursive coalgebra

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ℎ = 𝛾 ⋆ 𝐹 ℎ ⋆ 𝛼 .

Equivalently, the functor $𝖧𝗒𝗅𝗈 (𝛾, -)$ of the hylomorphism profunctor is constantly a singleton.

The dual notion is a corecursive algebra.

Pseudofunctor

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A pseudofunctor $𝐹 : ℬ︀ \to 𝒞︀$ is a lax functor whose unit and composition comparisons

𝐹_{𝑥}^{0} : 1_{𝐹 𝑥} \Rightarrow 𝐹 (1_{𝑥}) 𝐹_{𝑓, 𝑔}^{2} : 𝐹 𝑓 ⋆ 𝐹 𝑔 \Rightarrow 𝐹 (𝑓 ⋆ 𝑔)

are invertible 2-cells. So $𝐹$ preserves identities and composition up to coherent isomorphism.

Opposite Bicategory

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The opposite ${ℬ︀}^{op}$ of a bicategory $ℬ︀$ has the same 0-cells and reverses the 1-cells but not the 2-cells:

{ℬ︀}^{op} (𝑥, 𝑦) = ℬ︀ (𝑦, 𝑥) .

Duality saves work: a coherence lemma about $ℬ︀$ can often be obtained by instantiating a companion lemma at ${ℬ︀}^{op}$ , which swaps left and right.

Locally Discrete Bicategory

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A functor $𝐹 : 𝒞︀ \to 𝒟︀$ gives a pseudofunctor $𝖫𝖣 (𝐹) : 𝖫𝖣 (𝒞︀) \to 𝖫𝖣 (𝒟︀)$ , whose comparison 2-cells are the functor laws of $𝐹$ .

Lax and Pseudonatural Transformations

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Let $𝐹, 𝐺 : ℬ︀ \to 𝒞︀$ be lax functors. A lax natural transformation $𝜎 : 𝐹 \Rightarrow 𝐺$ consists of

for each 0-cell $𝑥$ , a 1-cell $𝜎_{𝑥} : 𝐹 𝑥 \to 𝐺 𝑥$ ;
for each 1-cell $𝑓 : 𝑥 \to 𝑦$ , a 2-cell filling the naturality square,
$𝜎_{𝑓} : 𝐹 𝑓 ⋆ 𝜎_{𝑦} \Rightarrow 𝜎_{𝑥} ⋆ 𝐺 𝑓;$
such that $𝜎_{𝑓}$ is natural in $𝑓$ : for a 2-cell $𝜃 : 𝑓 \Rightarrow 𝑔$ , $(𝐹 𝜃 ▷ 𝜎_{𝑦}) ⋆ 𝜎_{𝑔} = 𝜎_{𝑓} ⋆ (𝜎_{𝑥} ◁ 𝐺 𝜃)$ ;
and such that $𝜎$ respects the comparison cells of $𝐹$ and $𝐺$ : one law relating $𝜎_{1_{𝑥}}$ to $𝐹^{0}$ , $𝐺^{0}$ and the unitors, and one relating $𝜎_{𝑓 ⋆ 𝑔}$ to $𝜎_{𝑓}$ , $𝜎_{𝑔}$ , $𝐹^{2}$ , $𝐺^{2}$ and the associators.

Lax Functor

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A lax functor $𝐹 : ℬ︀ \to 𝒞︀$ between bicategories consists of

a map on 0-cells, $𝑥 \mapsto 𝐹 𝑥$ ;
for all $𝑥, 𝑦$ , a functor $𝐹_{𝑥, 𝑦} : ℬ︀ (𝑥, 𝑦) \to 𝒞︀ (𝐹 𝑥, 𝐹 𝑦)$ , acting on 1-cells and 2-cells;
a unit comparison, natural 2-cells $𝐹_{𝑥}^{0} : 1_{𝐹 𝑥} \Rightarrow 𝐹 (1_{𝑥})$ ;
a composition comparison, 2-cells $𝐹_{𝑓, 𝑔}^{2} : 𝐹 𝑓 ⋆ 𝐹 𝑔 \Rightarrow 𝐹 (𝑓 ⋆ 𝑔)$ natural in $𝑓$ and $𝑔$ ;
such that three coherence laws hold, one for each structure cell of $ℬ︀$ :
- left unit: $(𝐹^{0} ▷ 𝐹 𝑓) ⋆ 𝐹_{1, 𝑓}^{2} ⋆ 𝐹 (𝜆_{𝑓}) = 𝜆_{𝐹 𝑓}$ ;
- right unit: $(𝐹 𝑓 ◁ 𝐹^{0}) ⋆ 𝐹_{𝑓, 1}^{2} ⋆ 𝐹 (𝜌_{𝑓}) = 𝜌_{𝐹 𝑓}$ ;
- associativity: $(𝐹_{𝑓, 𝑔}^{2} ▷ 𝐹 ℎ) ⋆ 𝐹_{𝑓 ⋆ 𝑔, ℎ}^{2} ⋆ 𝐹 (𝛼_{𝑓, 𝑔, ℎ}) = 𝛼_{𝐹 𝑓, 𝐹 𝑔, 𝐹 ℎ} ⋆ (𝐹 𝑓 ◁ 𝐹_{𝑔, ℎ}^{2}) ⋆ 𝐹_{𝑓, 𝑔 ⋆ ℎ}^{2}$ .

Here $⋆$ between 2-cells is vertical composition, and $𝜃 ▷ ℎ$ and $ℎ ◁ 𝜃$ are whiskerings.

A lax functor whose comparison cells are invertible is a pseudofunctor.

The hylomorphism profunctor

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Fix an endofunctor $𝐹 : 𝒞︀ \to 𝒞︀$ . Hylomorphisms form a profunctor from coalgebras to algebras,

𝖧𝗒𝗅𝗈 : 𝖢𝗈𝖺𝗅𝗀 {(𝐹)}^{op} \times 𝖠𝗅𝗀 (𝐹) \to 𝐒𝐞𝐭,

where $𝖧𝗒𝗅𝗈 ((𝑋, 𝛾), (𝐵, 𝛼))$ is the set of $ℎ : 𝑋 \to 𝐵$ with $ℎ = 𝛾 ⋆ 𝐹 ℎ ⋆ 𝛼$ .

𝛿 ⋆ 𝐹 (𝑔 ⋆ ℎ ⋆ 𝑘) ⋆ 𝛼' = 𝛿 ⋆ 𝐹 𝑔 ⋆ 𝐹 ℎ ⋆ 𝛼 ⋆ 𝑘 = 𝑔 ⋆ 𝛾 ⋆ 𝐹 ℎ ⋆ 𝛼 ⋆ 𝑘 = 𝑔 ⋆ ℎ ⋆ 𝑘 .

When every value of $𝖧𝗒𝗅𝗈$ is a singleton, every divide-and-conquer specification over $𝐹$ has exactly one solution. This is what local contractivity guarantees.

Hylomorphism

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ℎ = 𝛾 ⋆ 𝐹 ℎ ⋆ 𝛼 .

Corecursive algebra

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The Bicategory of Categories

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The bicategory of categories $𝖢𝖠𝖳$ has

as 0-cells, categories (at a fixed pair of universe levels, for objects and for morphisms);
as hom-category $𝖢𝖠𝖳 (𝒞︀, 𝒟︀)$ , the functor category $[𝒞︀, 𝒟︀]$ , so 1-cells are functors and 2-cells are natural transformations;
as identity 1-cell, the identity functor;
as composition, $𝐹 ⋆ 𝐺 = 𝐺 \circ 𝐹$ on functors. On natural transformations $𝛽 : 𝐹 \Rightarrow 𝐹'$ and $𝛾 : 𝐺 \Rightarrow 𝐺'$ the horizontal composite is given directly by its components
${(𝛽 ⋆ 𝛾)}_{𝑐} = 𝐺 (𝛽_{𝑐}) ⋆ 𝛾_{𝐹' 𝑐} .$

A monad in $𝖢𝖠𝖳$ is an ordinary monad on a category, and a prestack is a pseudofunctor into $𝖢𝖠𝖳$ .

Representable Grammars

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⌈ 𝑤 ⌉ (𝑢) = (𝑢 \equiv 𝑤)

Predecessors simplify later

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The Yoneda lemma then collapses later to evaluation:

⊳ 𝑃 (𝑥) = 𝖯𝗌𝗁 𝒞︀ (𝖲𝗍𝗋𝗂𝖼𝗍𝖣𝗈𝗐𝗇 (𝑥), 𝑃) ≅ 𝖯𝗌𝗁 𝒞︀ (𝗒 𝑝, 𝑃) ≅ 𝑃 (𝑝),

⊳ 𝑃 (0) ≅ ⊤, ⊳ 𝑃 (𝑛 + 1) ≅ 𝑃 (𝑛),

and a Löb step is a base value together with a rule producing the value at $𝑛 + 1$ from the value at $𝑛$ .

When this predecessor exists, we can give a simpler description of later, as in the topos of trees, but this may not be possible in all direct categories.

The comparison functor of an adjunction

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𝑈 𝜀_{𝑑} : 𝑈 𝐹 𝑈 𝑑 \to 𝑈 𝑑 .

This assignment extends to a functor into the Eilenberg–Moore category,

𝐾 : 𝒟︀ \to EM (𝑇),

the comparison functor of the adjunction.

Löb induction on families

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Like later on families, the recursion principle for families is inherited from that on presheaves. Given a family $𝐴$ and a step

𝜑_{𝑥} : ⊳_{Fam} 𝐴 (𝑥) \to 𝐴 (𝑥) for each 𝑥,

the construction is a chain of transpositions with Löb for presheaves used in the middle:

Just as for presheaves, the fixed point constructed above is unique: the two transpositions are bijections, and the presheaf-level fixed point is already unique.

Later on families

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Conjugation with the adjunction between presheaves and families $𝑈 ⊣ Cofree$ lets us induce a later construction on families from the one on presheaves,

⊳_{Fam} = 𝑈 \circ ⊳ \circ Cofree : Fam (𝒞︀) \to Fam (𝒞︀) .

Concretely, later on families evaluates to

⊳_{Fam} 𝐴 (𝑥) ≅ \prod_{𝑦 ≺ 𝑥} \prod_{𝑓 : 𝑦 \to 𝑥} 𝐴 (𝑦)

Locally contractive endofunctors

2026-07-19T06:01:17Z

𝐹_{𝛿} : ⊳ (𝑋 \Rightarrow 𝑌) \to (𝐹 𝑋 \Rightarrow 𝐹 𝑌)

Löb induction for presheaves on a direct category

2026-07-19T06:01:15Z

Let $𝒞︀$ be a direct category and $𝑃$ a presheaf on it. Every map

𝜑 : ⊳ 𝑃 \to 𝑃

has a fixed point: a global element $𝗅ö𝖻 𝜑 : ⊤ \to 𝑃$ with

𝗅ö𝖻 𝜑 = 𝗅ö𝖻 𝜑 ⋆ 𝗇𝖾𝗑𝗍 ⋆ 𝜑,

and this fixed point is unique.

Earlier on presheaves

2026-07-19T06:01:14Z

Earlier is left adjoint to later:

⊲ ⊣ ⊳

Under this adjunction, $𝗇𝖾𝗑𝗍 : 𝑃 \to ⊳ 𝑃$ corresponds to

𝗉𝗋𝖾𝗏 : ⊲ 𝑃 \to 𝑃,

Later on presheaves

2026-07-19T05:58:58Z

The later modality for presheaves on a direct category is given by the presheaf of natural transformations

out of the strict downset.

(⊳ 𝑃) (𝑥) = 𝖯𝗌𝗁 𝒞︀ (𝖲𝗍𝗋𝗂𝖼𝗍𝖣𝗈𝗐𝗇 (𝑥), 𝑃) .

An element of $⊳ 𝑃$ at $𝑥$ is a coherent choice of $𝑃$ -elements at all objects strictly smaller than $𝑥$ .

Via functoriality, every presheaf restricts to smaller indices. Thus we may define the map

𝗇𝖾𝗑𝗍 : 𝑃 \to ⊳ 𝑃

that sends an element $𝑝$ over $𝑥$ to the family of all its restrictions along morphisms from strictly lower objects.

Maximality of the strict downset among proper sieves

2026-07-19T05:58:57Z

If the direct structure is reflecting, then every proper sieve $𝑆$ on $𝑥$ refines into the strict downset:

𝑆 \subseteq 𝖲𝗍𝗋𝗂𝖼𝗍𝖣𝗈𝗐𝗇 (𝑥) .

So every morphism in $𝑆$ strictly raises degree. That is, every morphism in $𝑆$ is also a member of $𝖲𝗍𝗋𝗂𝖼𝗍𝖣𝗈𝗐𝗇 (𝑥)$ .

The strict downset sieve of a direct category

2026-07-19T05:58:56Z

𝖲𝗍𝗋𝗂𝖼𝗍𝖣𝗈𝗐𝗇 (𝑥) (𝑦) = {𝑓 : 𝑦 \to 𝑥 ∣ 𝑦 ≺ 𝑥},

with restriction by precomposition — well defined since degrees are non-decreasing, so precomposing can only stay strictly below.

Proper and maximal sieves

2026-07-19T05:58:54Z

The representable $𝗒 𝑥$ is itself a sieve on $𝑥$ . A sieve on $𝑥$ is proper when it is not equal to the representable.

Say that a proper sieve is maximal when it contains all other proper sieves as a sub-sieve.

Sieves

2026-07-19T05:58:53Z

𝑆 ∋ 𝑓 \Rightarrow 𝑆 ∋ 𝑔 ⋆ 𝑓 for every 𝑔 : 𝑧 \to 𝑦 .

A sieve is thus a “downward closed” collection of morphisms into $𝑥$ .

Sieves on $𝑥$ are ordered by refinement: $𝑆 \subseteq 𝑇$ when every morphism belonging to $𝑆$ belongs to $𝑇$ .

Total category

2026-07-19T05:58:51Z

The total category of a displayed category over $𝒞︀$ collects the displayed data into a single category. Its objects are pairs of an object $𝑥$ of $𝒞︀$ with an object over it, and its morphisms are pairs of a morphism $𝑓 : 𝑥 \to 𝑦$ with a displayed morphism over $𝑓$ .

Projecting out the first components is a functor . Constructions presented displayed — algebras, Eilenberg–Moore categories — get their forgetful functor for free as this projection.

Well-founded posets are thin direct categories

2026-07-19T05:51:54Z

Every poset forms a thin category. Similarly, if the poset is well-founded then it induces a thin direct category.

Direct categories

2026-07-19T05:51:05Z

A direct structure on a category $𝒞︀$ over $(𝐷, <)$ is a functor

deg : 𝒞︀ \to (𝐷, \leq)

A direct structure equips the objects with a well-founded strict relation

𝑥 ≺ 𝑦 ⟺ deg 𝑥 < deg 𝑦 .

The degrees order the objects, while the morphisms of $𝒞︀$ say how an object sits over its predecessors.

Monadicity and comonadicity

2026-07-19T05:48:18Z

An adjunction $𝐹 ⊣ 𝑈$ with $𝐹 : 𝒞︀ \to 𝒟︀$ and $𝑈 : 𝒟︀ \to 𝒞︀$ induces a monad $𝑇 = 𝑈 \circ 𝐹$ on $𝒞︀$ , and a comparison functor

𝐾 : 𝒟︀ \to EM (𝑇)

sending each object of $𝒟︀$ to the $𝑇$ -algebra it carries.

The functor $𝑈$ is monadic when $𝐾$ is an equivalence: the adjunction exhibits $𝒟︀$ as objects of $𝒞︀$ equipped with algebraic structure for $𝑇$ , the Eilenberg–Moore category.

Presheaves are monadic and comonadic over families

2026-07-19T05:45:10Z

The adjoint triple $Free ⊣ 𝑈 ⊣ Cofree$ induces a monad $𝑇 = 𝑈 \circ Free$ and a comonad $𝑊 = 𝑈 \circ Cofree$ on $Fam (𝒞︀)$ .

Both comparison functors are equivalences: presheaves are the Eilenberg–Moore algebras of $𝑇$ and the co-Eilenberg–Moore coalgebras of $𝑊$ ,

𝖯𝗌𝗁 𝒞︀ ≃ EM (𝑇) 𝖯𝗌𝗁 𝒞︀ ≃ coEM (𝑊) .

So presheaves are both monadic and comonadic over families.

Steven Schaefer

Monoids for Day Convolution

Getting Organized Again: My First Weeknote

Notetaking in Helianthus

Sharing Notes

Learning about monoids and modules

The Schanuel Topos

Nominal Sets as Day Quotients

Nominal Sets

Navigating This Site

Mechanized Category Theory

Lessons from Mechanizing Categorical Logic in Cubical Agda onpls-2026-talkSteven Schaefer, Johnson He, Max S. New · ONPLS · June 1, 2026

Definition. Day Convolution day-convolution

As a Kan Extension

In 𝐒𝐞𝐭

Definition. The Quotient and its Right Adjoint in Day Convolution day-quotient

Definition. The Quotient and Residual Coincidence in Day Convolution day-quotient-coincidence

The Representable Case

Definition. Nominal Sets nominal-set

Definition. Nominal Sets as Day Quotients nominal-sets-quotient

Definition. The Schanuel Topos schanuel-topos

Theorem. Day Convolution is Closed day-closed-structure

Proof. Proof that Day Convolution is Closed day-closed-structure-proof

Definition. The Bicategory of Categories bicategory-of-categories

Definition. Corecursive algebra corecursive-algebra

Definition. Hylomorphism hylomorphism

Definition. The hylomorphism profunctor hylomorphism-profunctor

Definition. Lax Functor lax-functor

Definition. Lax and Pseudonatural Transformations lax-natural-transformation

Definition. Locally Discrete Bicategory locally-discrete-bicategory

Definition. Opposite Bicategory opposite-bicategory

Definition. Pseudofunctor pseudofunctor

Definition. Recursive coalgebra recursive-coalgebra

Definition. Total Bicategory total-bicategory

Predecessors simplify later predecessors-simplify-later

Algebras as a displayed category algebras-displayed

The co-Eilenberg–Moore category as a displayed category co-eilenberg-moore-displayed

Coalgebras as a displayed category coalgebras-displayed

Definition. The comparison functor of an adjunction comparison-functor

Definition. Direct categories direct-category

Definition. Earlier on presheaves earlier-presheaf

The Eilenberg–Moore category as a displayed category eilenberg-moore-displayed

Definition. Initial algebra initial-algebra

Definition. Later on families later-family

Definition. Later on presheaves later-presheaf

Theorem. Löb induction on families lob-family

Theorem. Löb induction for presheaves on a direct category lob-presheaf

Definition. Locally contractive endofunctors locally-contractive-functor

Definition. Monadicity and comonadicity monadicity-comonadicity

The adjoint triple between presheaves and families presheaf-family-adjoint-triple

Theorem. Presheaves are monadic and comonadic over families presheaves-monadic-comonadic-over-families

Definition. Proper and maximal sieves proper-maximal-sieve

Definition. Sieves sieve

Theorem. Maximality of the strict downset among proper sieves strict-downset-maximal

Definition. The strict downset sieve of a direct category strict-downset-sieve

Definition. Terminal coalgebra terminal-coalgebra

Well-founded posets are thin direct categories well-founded-poset-as-thin

Definition. Category category

Definition. Bicategory bicategory

Definition. Monad in a bicategory monad-in-a-bicategory

Definition. Free Monoidal Category over a Set free-monoidal-category

Definition 0.1. Global Elimination Principle for the Free Monoidal Category free-monoidal-category-elimination

Definition. Global Elimination Principle for the Free Monoidal Category free-monoidal-category-elimination

Products of Categories as Total Categories product-as-total-category

Definition. Quiver quiver

Definition. Thin Category thin-category

Definition. Element of a Presheaf element-of-presheaf

Definition. Universal Element of a Presheaf universal-element

What is a universal property, really? universal-property

Definition. Constant Presheaf constant-presheaf

Finite Cardinal Arithmetic in a Topos finite-cardinal-arithmetic-topos

Definition 0.2. Constant Presheaf constant-presheaf

Definition. Subobject subobject

Definition. Terminal Object terminal-object

Definition. Equalizer equalizer

Constructing Equalizers in Type Theory equalizers-in-type-theory

Definition. Subobject Classifier subobject-classifier

Definition. Closed Monoidal Structure closed-monoidal-category

Definition. Monoidal Category monoidal-category

Definition. Category of Elements category-of-elements

In $𝐒𝐞𝐭$

In $𝐒𝐞𝐭$