{"version":"https://jsonfeed.org/version/1.1","title":"Steven Schaefer","home_page_url":"https://stevenschaefer.net/","feed_url":"https://stevenschaefer.net/feed.json","items":[{"id":"https://stevenschaefer.net/representable-grammar.html","url":"https://stevenschaefer.net/representable-grammar.html","title":"Representable Grammars","content_html":"

For any string 𝑀, we can define a representable grammar βŒˆπ‘€βŒ‰ which matches exactly the string 𝑀 and nothing else. The parse trees for a representable grammar are proofs that the string is exactly equal to 𝑀:

\n
βŒˆπ‘€βŒ‰(𝑒)=(𝑒≑𝑀)
","date_published":"2026-08-24T15:31:37Z","tags":["intrinsically-correct","lambekd","parsing"]},{"id":"https://stevenschaefer.net/predecessors-simplify-later.html","url":"https://stevenschaefer.net/predecessors-simplify-later.html","title":"Predecessors simplify later","content_html":"

A predecessor of π‘₯ is a top element of its strict downset: a strict morphism 𝜌:𝑝→π‘₯ through which every strict morphism into π‘₯ factors uniquely. Equivalently, π–²π—π—‹π—‚π–Όπ—π–£π—ˆπ—π—‡(π‘₯)≅𝗒𝑝 β€” the downset is representable.

\n

The Yoneda lemma then collapses later to evaluation:

\n
βŠ³π‘ƒ(π‘₯)=π–―π—Œπ—π’žοΈ€(π–²π—π—‹π—‚π–Όπ—π–£π—ˆπ—π—‡(π‘₯),𝑃)β‰…π–―π—Œπ—π’žοΈ€(𝗒𝑝,𝑃)≅𝑃(𝑝),
\n

with 𝗇𝖾𝗑𝗍 becoming restriction along 𝜌. The name is from the naturals: every strict map into 𝑛+1 factors through 𝑛→𝑛+1, so on πœ” β€” in the topos of trees β€” later is just the shift

\n
βŠ³π‘ƒ(0)β‰…βŠ€,βŠ³π‘ƒ(𝑛+1)≅𝑃(𝑛),
\n

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

\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.

","date_published":"2026-07-20T18:20:09Z","tags":["category-theory","guarded-recursion","presheaf","well-founded"]},{"id":"https://stevenschaefer.net/comparison-functor.html","url":"https://stevenschaefer.net/comparison-functor.html","title":"The comparison functor of an adjunction","content_html":"

An adjunction πΉβŠ£π‘ˆ with 𝐹:π’žοΈ€β†’π’ŸοΈ€ and π‘ˆ:π’ŸοΈ€β†’π’žοΈ€ induces a monad 𝑇=π‘ˆβˆ˜πΉ on π’žοΈ€. Write πœ€:πΉβˆ˜π‘ˆβ‡’π–¨π–½ for the counit of the adjunction. Every object 𝑑 of π’ŸοΈ€ then induces a 𝑇-algebra carried by the object π‘ˆπ‘‘, witnessed by the map

\n
π‘ˆπœ€π‘‘:π‘ˆπΉπ‘ˆπ‘‘β†’π‘ˆπ‘‘.
\n

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

\n
𝐾:π’ŸοΈ€β†’EM(𝑇),
\n

the comparison functor of the adjunction.

\n

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 πΉβŠ£π‘ˆ is (co)monadic.

","date_published":"2026-07-19T18:48:31Z","tags":["adjunction","category-theory","eilenberg-moore"]},{"id":"https://stevenschaefer.net/lob-family.html","url":"https://stevenschaefer.net/lob-family.html","title":"LΓΆb induction on families","content_html":"

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

\n
πœ‘π‘₯:⊳Fam𝐴(π‘₯)→𝐴(π‘₯)for each π‘₯,
\n

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

\n

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.

","date_published":"2026-07-19T06:01:19Z","tags":["category-theory","families","guarded-recursion","presheaf","well-founded"]},{"id":"https://stevenschaefer.net/later-family.html","url":"https://stevenschaefer.net/later-family.html","title":"Later on families","content_html":"

Conjugation with the adjunction between presheaves and families π‘ˆβŠ£Cofree lets us induce a later construction on families from the one on presheaves,

\n
⊳Fam=π‘ˆβˆ˜βŠ³βˆ˜Cofree:Fam(π’žοΈ€)β†’Fam(π’žοΈ€).
\n

Concretely, later on families evaluates to

\n
⊳Fam𝐴(π‘₯)β‰…βˆπ‘¦β‰Ίπ‘₯Β βˆπ‘“:𝑦→π‘₯𝐴(𝑦)
","date_published":"2026-07-19T06:01:18Z","tags":["category-theory","families","guarded-recursion","presheaf","right-adjoint","well-founded"]},{"id":"https://stevenschaefer.net/locally-contractive-functor.html","url":"https://stevenschaefer.net/locally-contractive-functor.html","title":"Locally contractive endofunctors","content_html":"

Write π‘‹β‡’π‘Œ for the presheaf of morphisms π‘‹β†’π‘Œ. An endofunctor 𝐹 on presheaves is locally contractive when its action on morphisms factors through later: there is a map

\n
𝐹𝛿:⊳(π‘‹β‡’π‘Œ)β†’(πΉπ‘‹β‡’πΉπ‘Œ)
","date_published":"2026-07-19T06:01:17Z","tags":["category-theory","guarded-recursion","presheaf","well-founded"]},{"id":"https://stevenschaefer.net/lob-presheaf.html","url":"https://stevenschaefer.net/lob-presheaf.html","title":"LΓΆb induction for presheaves on a direct category","content_html":"

Let π’žοΈ€ be a direct category and 𝑃 a presheaf on it. Every map

\n
πœ‘:βŠ³π‘ƒβ†’π‘ƒ
\n

has a fixed point: a global element π—…ΓΆπ–»πœ‘:βŠ€β†’π‘ƒ with

\n
π—…ΓΆπ–»πœ‘=π—…ΓΆπ–»πœ‘β‹†π—‡π–Ύπ—‘π—β‹†πœ‘,
\n

and this fixed point is unique.

\n

The hypothesis says: the value of 𝑃 at any object is determined by its values over the strict past β€” πœ‘ turns a coherent family over the strict downset of π‘₯ into a value at π‘₯. The proof is recursion along the well-founded β‰Ί: at each π‘₯, the section already constructed over the past assembles into an element of (βŠ³π‘ƒ)(π‘₯), and πœ‘ extends it to π‘₯.

","date_published":"2026-07-19T06:01:15Z","tags":["category-theory","guarded-recursion","presheaf","well-founded"]},{"id":"https://stevenschaefer.net/earlier-presheaf.html","url":"https://stevenschaefer.net/earlier-presheaf.html","title":"Earlier on presheaves","content_html":"

Later takes a limit over smaller indices. Dually, the earlier modality takes a colimit over larger indices: an element of βŠ²π‘ƒ at π‘₯ is a 𝑃-element sitting at some object 𝑦 strictly above π‘₯, carried down along a chosen morphism π‘₯→𝑦.

\n

Earlier is left adjoint to later:

\n
⊲⊣⊳
\n

Under this adjunction, 𝗇𝖾𝗑𝗍:π‘ƒβ†’βŠ³π‘ƒ corresponds to

\n
𝗉𝗋𝖾𝗏:βŠ²π‘ƒβ†’π‘ƒ,
\n

.

","date_published":"2026-07-19T06:01:14Z","tags":["category-theory","guarded-recursion","left-adjoint","presheaf","well-founded"]},{"id":"https://stevenschaefer.net/later-presheaf.html","url":"https://stevenschaefer.net/later-presheaf.html","title":"Later on presheaves","content_html":"

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

\n

out of the strict downset.

\n
(βŠ³π‘ƒ)(π‘₯)=π–―π—Œπ—π’žοΈ€(π–²π—π—‹π—‚π–Όπ—π–£π—ˆπ—π—‡(π‘₯),𝑃).
\n

An element of βŠ³π‘ƒ at π‘₯ is a coherent choice of 𝑃-elements at all objects strictly smaller than π‘₯.

\n

Restriction in βŠ³π‘ƒ along 𝑓:𝑦→π‘₯ precomposes with the induced map π–²π—π—‹π—‚π–Όπ—π–£π—ˆπ—π—‡(𝑦)β†’π–²π—π—‹π—‚π–Όπ—π–£π—ˆπ—π—‡(π‘₯). At an object of minimal degree the strict downset is empty, so βŠ³π‘ƒ is trivial there.

\n

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

\n
𝗇𝖾𝗑𝗍:π‘ƒβ†’βŠ³π‘ƒ
\n

that sends an element 𝑝 over π‘₯ to the family of all its restrictions along morphisms from strictly lower objects.

","date_published":"2026-07-19T05:58:58Z","tags":["category-theory","guarded-recursion","presheaf","right-adjoint","well-founded"]},{"id":"https://stevenschaefer.net/strict-downset-maximal.html","url":"https://stevenschaefer.net/strict-downset-maximal.html","title":"Maximality of the strict downset among proper sieves","content_html":"

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:

\n

If the direct structure is reflecting, then every proper sieve 𝑆 on π‘₯ refines into the strict downset:

\n
π‘†βŠ†π–²π—π—‹π—‚π–Όπ—π–£π—ˆπ—π—‡(π‘₯).
\n

Suppose π‘“βˆˆπ‘† with 𝑓:𝑦→π‘₯ of equal degree. By reflection 𝑓 has a section 𝑠, and closure under precomposition gives 𝑠⋆𝑓=𝗂𝖽π‘₯βˆˆπ‘†, contradicting properness.

\n

So every morphism in 𝑆 strictly raises degree. That is, every morphism in 𝑆 is also a member of π–²π—π—‹π—‚π–Όπ—π–£π—ˆπ—π—‡(π‘₯).

","date_published":"2026-07-19T05:58:57Z","tags":["category-theory","ideal","presheaf","subobject"]},{"id":"https://stevenschaefer.net/strict-downset-sieve.html","url":"https://stevenschaefer.net/strict-downset-sieve.html","title":"The strict downset sieve of a direct category","content_html":"

Let π’žοΈ€ carry a direct structure. The strict downset π–²π—π—‹π—‚π–Όπ—π–£π—ˆπ—π—‡(π‘₯) of an object π‘₯ is the presheaf of morphisms into π‘₯ from strictly lower objects:

\n
π–²π—π—‹π—‚π–Όπ—π–£π—ˆπ—π—‡(π‘₯)(𝑦)={𝑓:𝑦→π‘₯βˆ£π‘¦β‰Ίπ‘₯},
\n

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

\n
\n

The evident inclusion π–²π—π—‹π—‚π–Όπ—π–£π—ˆπ—π—‡(π‘₯)β†£γ‚ˆπ‘₯ makes π–²π—π—‹π—‚π–Όπ—π–£π—ˆπ—π—‡(π‘₯) a sieve on π‘₯. It is moreover a proper sieve, as it exlcudes the identity.

","date_published":"2026-07-19T05:58:56Z","tags":["category-theory","ideal","presheaf","subobject"]},{"id":"https://stevenschaefer.net/proper-maximal-sieve.html","url":"https://stevenschaefer.net/proper-maximal-sieve.html","title":"Proper and maximal sieves","content_html":"

The representable 𝗒π‘₯ is itself a sieve on π‘₯. A sieve on π‘₯ is proper when it is not equal to the representable.

\n

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

","date_published":"2026-07-19T05:58:54Z","tags":["category-theory","ideal","presheaf","subobject"]},{"id":"https://stevenschaefer.net/sieve.html","url":"https://stevenschaefer.net/sieve.html","title":"Sieves","content_html":"

A sieve on an object π‘₯ of π’žοΈ€ is a subobject of the representable presheaf 𝗒π‘₯: a presheaf 𝑆 with a monic morphism 𝑆↣𝗒π‘₯. Sieves are a generalization from the notion of ideal found in ring theory to category theory.

\n

A morphism 𝑓:𝑦→π‘₯ belongs to 𝑆, written π‘†βˆ‹π‘“, 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:

\n
π‘†βˆ‹π‘“β‡’π‘†βˆ‹π‘”β‹†π‘“for every 𝑔:𝑧→𝑦.
\n

A sieve is thus a β€œdownward closed” collection of morphisms into π‘₯.

\n

Sieves on π‘₯ are ordered by refinement: π‘†βŠ†π‘‡ when every morphism belonging to 𝑆 belongs to 𝑇.

","date_published":"2026-07-19T05:58:53Z","tags":["category-theory","ideal","presheaf","subobject"]},{"id":"https://stevenschaefer.net/total-category.html","url":"https://stevenschaefer.net/total-category.html","title":"Total category","content_html":"

The total category of a displayed category over π’žοΈ€ collects the displayed data into a single category. Its objects are pairs of an object π‘₯ of π’žοΈ€ with an object over it, and its morphisms are pairs of a morphism 𝑓:π‘₯→𝑦 with a displayed morphism over 𝑓.

\n

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

","date_published":"2026-07-19T05:58:51Z","tags":["displayed-category-theory"]},{"id":"https://stevenschaefer.net/well-founded-poset-as-thin.html","url":"https://stevenschaefer.net/well-founded-poset-as-thin.html","title":"Well-founded posets are thin direct categories","content_html":"

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

","date_published":"2026-07-19T05:51:54Z","tags":["category-theory","well-founded"]},{"id":"https://stevenschaefer.net/direct-category.html","url":"https://stevenschaefer.net/direct-category.html","title":"Direct categories","content_html":"

A well-founded order is a set 𝐷 with a proposition-valued transitive relation < admitting no infinite descent: every element is accessible. Write π‘Žβ‰€π‘ for (π‘Ž<𝑏)∨(π‘Ž=𝑏).

\n

A direct structure on a category π’žοΈ€ over (𝐷,<) is a functor

\n
deg:π’žοΈ€β†’(𝐷,≀)
\n

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 β€” 𝑓:π‘₯→𝑦 forces degπ‘₯≀deg𝑦.

\n

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

\n
π‘₯β‰Ίπ‘¦βŸΊdegπ‘₯<deg𝑦.
\n

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.

\n
\n

The degrees order the objects, while the morphisms of π’žοΈ€ say how an object sits over its predecessors.

","date_published":"2026-07-19T05:51:05Z","tags":["category-theory","well-founded"]},{"id":"https://stevenschaefer.net/monadicity-comonadicity.html","url":"https://stevenschaefer.net/monadicity-comonadicity.html","title":"Monadicity and comonadicity","content_html":"

An adjunction πΉβŠ£π‘ˆ with 𝐹:π’žοΈ€β†’π’ŸοΈ€ and π‘ˆ:π’ŸοΈ€β†’π’žοΈ€ induces a monad 𝑇=π‘ˆβˆ˜πΉ on π’žοΈ€, and a comparison functor

\n
𝐾:π’ŸοΈ€β†’EM(𝑇)
\n

sending each object of π’ŸοΈ€ to the 𝑇-algebra it carries.

\n

The functor π‘ˆ is monadic when 𝐾 is an equivalence: the adjunction exhibits π’ŸοΈ€ as objects of π’žοΈ€ equipped with algebraic structure for 𝑇, the Eilenberg–Moore category.

\n

Comonadicity is monadicity in the opposite category: a left adjoint 𝐿:π’ŸοΈ€β†’π’žοΈ€ with right adjoint 𝑅 induces a comonad π‘Š=πΏβˆ˜π‘… on π’žοΈ€, a comparison π’ŸοΈ€β†’coEM(π‘Š) into the co-Eilenberg–Moore category, and 𝐿 is comonadic when this comparison is an equivalence.

","date_published":"2026-07-19T05:48:18Z","tags":["adjunction","category-theory","eilenberg-moore"]},{"id":"https://stevenschaefer.net/presheaves-monadic-comonadic-over-families.html","url":"https://stevenschaefer.net/presheaves-monadic-comonadic-over-families.html","title":"Presheaves are monadic and comonadic over families","content_html":"

The adjoint triple FreeβŠ£π‘ˆβŠ£Cofree induces a monad 𝑇=π‘ˆβˆ˜Free and a comonad π‘Š=π‘ˆβˆ˜Cofree on Fam(π’žοΈ€).

\n

Both comparison functors are equivalences: presheaves are the Eilenberg–Moore algebras of 𝑇 and the co-Eilenberg–Moore coalgebras of π‘Š,

\n
π–―π—Œπ—π’žοΈ€β‰ƒEM(𝑇)π–―π—Œπ—π’žοΈ€β‰ƒcoEM(π‘Š).
\n

So presheaves are both monadic and comonadic over families.

\n

Reading the algebra structure concretely: a 𝑇-algebra on a family 𝐴 is a map βˆ‘π‘¦π’žοΈ€(π‘₯,𝑦)×𝐴(𝑦)→𝐴(π‘₯) for each π‘₯, subject to the monad algebra laws β€” that is, exactly a functorial action of restriction.

\n

The comonadic reading is the same structure seen from the element’s side: a π‘Š-coalgebra is a map 𝐴(π‘₯)β†’βˆπ‘¦π’žοΈ€(𝑦,π‘₯)→𝐴(𝑦), giving each value its restriction along every morphism into π‘₯. Where the monad says restriction acts on values, the comonad says a value already carries all of its restrictions β€” and the coalgebra laws say it does so coherently.

","date_published":"2026-07-19T05:45:10Z","tags":["adjunction","category-theory","families","monad","presheaf"]},{"id":"https://stevenschaefer.net/presheaf-family-adjoint-triple.html","url":"https://stevenschaefer.net/presheaf-family-adjoint-triple.html","title":"The adjoint triple between presheaves and families","content_html":"

A family over π’žοΈ€ is a set 𝐴(π‘₯) for each object π‘₯, with no action of morphisms. Families form a category Fam(π’žοΈ€): a morphism 𝐴→𝐡 is a function 𝐴(π‘₯)→𝐡(π‘₯) for each π‘₯.

\n

Forgetting the restriction maps of a presheaf gives a functor

\n
π‘ˆ:π–―π—Œπ—π’žοΈ€β†’Fam(π’žοΈ€).
\n

It has both a left and a right adjoint,

\n
FreeβŠ£π‘ˆβŠ£Cofree.
\n
\n

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

\n
Cofree(𝐴)(π‘₯)=βˆπ‘¦π’žοΈ€(𝑦,π‘₯)→𝐴(𝑦),
\n

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

\n
Free(𝐴)(π‘₯)=βˆ‘π‘¦π’žοΈ€(π‘₯,𝑦)×𝐴(𝑦),
\n

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

","date_published":"2026-07-19T05:33:22Z","tags":["adjunction","category-theory","families","presheaf"]},{"id":"https://stevenschaefer.net/co-eilenberg-moore-displayed.html","url":"https://stevenschaefer.net/co-eilenberg-moore-displayed.html","title":"The co-Eilenberg–Moore category as a displayed category","content_html":"

A comonad on π’žοΈ€ is a monad π‘Š on π’žοΈ€op. Everything about its coalgebras is then inherited from the Eilenberg–Moore construction, instantiated at the opposite category β€” nothing is defined twice.

\n

Algebras of π‘Š over π’žοΈ€op are coalgebras π›Ύβˆˆπ’žοΈ€(π‘₯,π‘Šπ‘₯) of the underlying endofunctor, and the monad algebra laws, read in π’žοΈ€op, are the comonad coalgebra laws β€” the unit and multiplication of π‘Š, viewed in π’žοΈ€, are the counit πœ€ and comultiplication 𝛿:

\n
π›Ύβ‹†πœ€π‘₯=𝗂𝖽π‘₯𝛾⋆𝛿π‘₯=π›Ύβ‹†π‘Šπ›Ύ.
\n

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

\n
coEM(π‘Š)=(EM(π‘Š))op.
","date_published":"2026-07-19T05:28:28Z","tags":["F-algebra","F-coalgebra","category-theory","displayed-category-theory","eilenberg-moore"]},{"id":"https://stevenschaefer.net/eilenberg-moore-displayed.html","url":"https://stevenschaefer.net/eilenberg-moore-displayed.html","title":"The Eilenberg–Moore category as a displayed category","content_html":"

Fix a monad (𝑇,πœ‚,πœ‡) on π’žοΈ€. Its Eilenberg–Moore category arises in two displayed layers. The first layer is the displayed category of algebras AlgStr(𝑇) of the underlying endofunctor.

\n

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

\n
πœ‚π‘₯⋆𝛼=𝗂𝖽π‘₯πœ‡π‘₯⋆𝛼=𝑇𝛼⋆𝛼.
\n

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

\n
EM(𝑇)=∫EMStr(𝑇).
","date_published":"2026-07-19T05:23:11Z","tags":["F-algebra","category-theory","displayed-category-theory","eilenberg-moore"]},{"id":"https://stevenschaefer.net/terminal-coalgebra.html","url":"https://stevenschaefer.net/terminal-coalgebra.html","title":"Terminal coalgebra","content_html":"

Fix an endofunctor 𝐹:π’žοΈ€β†’π’žοΈ€. A terminal 𝐹-coalgebra, written 𝜈𝐹, is a terminal object of the category of coalgebras Coalg(𝐹) β€” equivalently, an initial algebra for 𝐹op.

\n

Unfolding the universal property: a terminal coalgebra is a coalgebra (𝜈𝐹,out) such that every coalgebra (π‘₯,𝛾) admits a unique morphism unfold𝛾:π‘₯β†’πœˆπΉ satisfying

\n
(unfold𝛾)⋆out=𝛾⋆𝐹(unfold𝛾).
","date_published":"2026-07-19T05:17:44Z","tags":["F-algebra","F-coalgebra","category-theory"]},{"id":"https://stevenschaefer.net/coalgebras-displayed.html","url":"https://stevenschaefer.net/coalgebras-displayed.html","title":"Coalgebras as a displayed category","content_html":"

Fix an endofunctor 𝐹:π’žοΈ€β†’π’žοΈ€. Coalgebras require no new construction: a coalgebra is an algebra in the opposite category. Define

\n
CoalgStr(𝐹)=AlgStr(𝐹op),
\n

a displayed category over π’žοΈ€op, where 𝐹op:π’žοΈ€opβ†’π’žοΈ€op is 𝐹 acting on the opposite category.

\n

Concretely, over an object π‘₯ a displayed object is a structure map

\n
π›Ύβˆˆπ’žοΈ€(π‘₯,𝐹π‘₯).
\n

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

\n
Coalg(𝐹)=(∫CoalgStr(𝐹))op.
\n

The outer opposite returns morphisms to the direction of π’žοΈ€: a morphism (π‘₯,𝛾)β†’(𝑦,𝛿) is a map 𝑓:π‘₯→𝑦 with

\n
𝑓⋆𝛿=𝛾⋆𝐹𝑓.
","date_published":"2026-07-19T05:17:42Z","tags":["F-algebra","category-theory","displayed-category-theory"]},{"id":"https://stevenschaefer.net/initial-algebra.html","url":"https://stevenschaefer.net/initial-algebra.html","title":"Initial algebra","content_html":"

Fix an endofunctor 𝐹:π’žοΈ€β†’π’žοΈ€. An initial 𝐹-algebra, written πœ‡πΉ, is an initial object of the category of algebras Alg(𝐹).

\n

Unfolding the universal property: an initial algebra is an algebra (πœ‡πΉ,in) such that every algebra (π‘₯,𝛼) admits a unique morphism fold𝛼:πœ‡πΉβ†’π‘₯ satisfying

\n
in⋆(fold𝛼)=𝐹(fold𝛼)⋆𝛼.
","date_published":"2026-07-19T05:17:41Z","tags":["F-algebra","category-theory"]},{"id":"https://stevenschaefer.net/algebras-displayed.html","url":"https://stevenschaefer.net/algebras-displayed.html","title":"Algebras as a displayed category","content_html":"

Fix an endofunctor 𝐹:π’žοΈ€β†’π’žοΈ€. The 𝐹-algebras form a displayed category AlgStr(𝐹) over π’žοΈ€.

\n

Over an object π‘₯, a displayed object of AlgStr(𝐹) is a structure map

\n
π›Όβˆˆπ’žοΈ€(𝐹π‘₯,π‘₯).
\n

Over 𝑓:π‘₯→𝑦, a displayed morphism from 𝛼 to 𝛽 is the proposition that 𝑓 is an algebra homomorphism:

\n
𝛼⋆𝑓=𝐹𝑓⋆𝛽.
\n
\n

The total category Alg(𝐹)=∫AlgStr(𝐹) is the category of 𝐹-algebras.

","date_published":"2026-07-19T04:54:07Z","tags":["F-algebra","category-theory","displayed-category-theory"]},{"id":"https://stevenschaefer.net/category.html","url":"https://stevenschaefer.net/category.html","title":"Category","content_html":"

A category π’žοΈ€ consists of

\n
    \n
  1. A type of objects π’žοΈ€0
    2. \n
  2. For each pair of objects π‘₯,𝑦:π’žοΈ€0 a set of morphisms π’žοΈ€(π‘₯,𝑦). We may simply write a morphism with an arrow, denote 𝑓:π’žοΈ€(π‘₯,𝑦) as 𝑓:π‘₯→𝑦 or π‘₯→𝑓𝑦 or similar
    3. \n
  3. A composition operation on morphisms. For 𝑓:π‘₯→𝑦 and 𝑔:𝑦→𝑧, there is a morphism 𝑓⋆𝑔:π‘₯→𝑧
    4. \n
  4. For each π‘₯:π’žοΈ€0, an identity morphism 𝗂𝖽π‘₯:π‘₯β†’π‘₯
    5. \n

  5. Left-unitality of composition: for all 𝑓:π‘₯→𝑦, an equality

    \n
    𝗂𝖽𝖫𝑓:𝗂𝖽π‘₯⋆𝑓=𝑓
    \n
    6. \n

  6. Right-unitality of composition: for all 𝑓:π‘₯→𝑦, an equality

    \n
    𝗂𝖽𝖱𝑓:𝑓⋆𝗂𝖽𝑦=𝑓
    \n
    7. \n

  7. Associativity of composition: for all 𝑓:π‘₯→𝑦, 𝑔:𝑦→𝑧, β„Ž:𝑧→𝑀, an equality

    \n
    π–Ίπ—Œπ—Œπ—ˆπ–Όπ‘“,𝑔,β„Ž:(𝑓⋆𝑔)β‹†β„Ž=𝑓⋆(π‘”β‹†β„Ž)
    \n
    8. \n
\n

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

\n

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.

","date_published":"2026-06-13T17:16:17Z","tags":["category-theory"]},{"id":"https://stevenschaefer.net/displayed-category.html","url":"https://stevenschaefer.net/displayed-category.html","title":"Displayed Category","content_html":"

A displayed category over a base category π’žοΈ€ packages the data of a category that β€œlies over” π’žοΈ€: each object and each morphism of π’žοΈ€ is equipped with a fiber of objects and morphisms displayed atop it. It consists of

\n
    \n
  1. For each object π‘₯:π’žοΈ€0, a type of displayed objects lying over π‘₯. We write for a displayed object over π‘₯.
    2. \n
  2. For each morphism 𝑓:π’žοΈ€(π‘₯,𝑦) and displayed objects and , a set of displayed morphisms lying over 𝑓. We write such a displayed morphism as or , subscripting the arrow with the base morphism 𝑓 it lies over.
    3. \n
  3. A displayed composition operation. For 𝑓:π‘₯→𝑦 and 𝑔:𝑦→𝑧 with displayed morphisms and , there is a displayed morphism lying over the composite 𝑓⋆𝑔.
    4. \n
  4. For each , a displayed identity lying over 𝗂𝖽π‘₯.
    5. \n

  5. Left-unitality, displayed: for all , a heterogeneous equality

    \n
    \n
    6. \n

  6. Right-unitality, displayed: for all , a heterogeneous equality

    \n
    \n
    7. \n

  7. Associativity, displayed: for all , , over 𝑓:π‘₯→𝑦, 𝑔:𝑦→𝑧, β„Ž:𝑧→𝑀, a heterogeneous equality

    \n
    \n
    8. \n
\n

Because the type of displayed morphisms depends on the base morphism 𝑓, the two sides of each displayed law inhabit different displayed hom-sets β€” those indexed by the two sides of the corresponding base-category equation. Each displayed law is therefore a heterogeneous equality π‘Ž=𝑝𝑏: a path from π‘Ž to 𝑏 lying over the base path 𝑝, rather than an equation within a single fixed set.

\n

Concretely, this definition models the one used in the Cubical standard library [1].

\n

A displayed category is to a category as a dependent type is to a context. In this way, displayed categories are effectively dependent categories, as we present the structure of as a category parametrized by the structure of π’žοΈ€.

\n

Displayed categories were introduced by Ahrens and Lumsdaine [2]. The point is that a displayed category over π’žοΈ€ is equivalent to the data of a category π’ŸοΈ€ together with a functor 𝐹:π’ŸοΈ€β†’π’žοΈ€, but presented as families indexed by the objects and morphisms of π’žοΈ€ β€” so that constructions like Grothendieck fibrations can be defined without ever invoking equality of objects.

","date_published":"2026-06-13T17:14:29Z","tags":["displayed-category-theory"]},{"id":"https://stevenschaefer.net/monad-in-a-bicategory.html","url":"https://stevenschaefer.net/monad-in-a-bicategory.html","title":"Monad in a bicategory","content_html":"

Fix a bicategory 𝒦︀, with composition ⋆, identity 1-cells 1π‘₯, associator 𝛼, and unitors πœ†,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.

\n

A monad in 𝒦︀ consists of

\n
    \n
  1. a 0-cell π‘₯, the object the monad acts on;
    2. \n
  2. an endo-1-cell 𝑑:𝒦︀(π‘₯,π‘₯);
    3. \n
  3. a multiplication 2-cell πœ‡:𝑑⋆𝑑⇒𝑑;
    4. \n
  4. a unit 2-cell πœ‚:1π‘₯⇒𝑑;
    5. \n

  5. such that πœ‡ is associative: the following diagram of 2-cells commutes in 𝒦︀(π‘₯,π‘₯), where the top map is the associator that rebrackets the threefold composite:

    \n
    \n
    6. \n

  6. and such that πœ‡ and πœ‚ satisfy the unit laws: the following two diagrams commute in 𝒦︀(π‘₯,π‘₯), where the hypotenuses are the left and right unitors:

    \n
    \n
    7. \n
\n

Taking 𝒦︀ to be the bicategory of categories, functors, and natural transformations recovers an ordinary monad on a category: 𝑑 is the endofunctor, πœ‡ the multiplication, and πœ‚ the unit, with the coherence cells all identities.

","date_published":"2026-06-11T02:55:17Z","tags":["bicategory","category-theory","monad"]},{"id":"https://stevenschaefer.net/bicategory.html","url":"https://stevenschaefer.net/bicategory.html","title":"Bicategory","content_html":"

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.

\n

A bicategory 𝒦︀ consists of

\n
    \n
  1. A type of objects 𝒦︀0, or 0-cells
    2. \n
  2. For all π‘₯,𝑦:𝒦︀0, a category 𝒦︀1(π‘₯,𝑦). We may elide the subscript and simply write this as 𝒦︀(π‘₯,𝑦). Refer to the objects of 𝒦︀(π‘₯,𝑦) as 1-cells between π‘₯ and 𝑦, and we may write 𝑓:𝒦︀(π‘₯,𝑦) as 𝑓:π‘₯→𝑦 or π‘₯→𝑓𝑦. For 𝑓,𝑔:𝒦︀(π‘₯,𝑦), refer to the morphisms in 𝒦︀(π‘₯,𝑦) between 𝑓 and 𝑔 as 2-cells and write the morphism 𝛼:(𝒦︀(π‘₯,𝑦))(𝑓,𝑔) as 𝛼:𝑓⇒𝑔 or 𝑓⇒𝛼𝑔
    3. \n
  3. For each π‘₯:𝒦︀0, an identity 1-cell 1π‘₯:𝒦︀(π‘₯,π‘₯)
    4. \n
  4. For all π‘₯,𝑦,𝑧:𝒦︀0, a composition functor 𝒦︀⋆π‘₯,𝑦,𝑧:𝒦︀(π‘₯,𝑦)×𝒦︀(𝑦,𝑧)→𝒦︀(π‘₯,𝑧). For 1-cells 𝑓:π‘₯→𝑦 and 𝑔:𝑦→𝑧, write their composite as 𝑓⋆𝑔:π‘₯→𝑧
    5. \n

  5. For all 𝑀,π‘₯,𝑦,𝑧:𝒦︀0, a natural isomorphism, the associator 𝛼 between the two composite functors 𝒦︀(𝑀,π‘₯)×𝒦︀(π‘₯,𝑦)×𝒦︀(𝑦,𝑧)→𝒦︀(𝑀,𝑧) that compose the leftmost, respectively rightmost, pair first:

    \n
    𝒦︀⋆𝑀,𝑦,π‘§βˆ˜(𝒦︀⋆𝑀,π‘₯,𝑦×id)⇒𝒦︀⋆𝑀,π‘₯,π‘§βˆ˜(id×𝒦︀⋆π‘₯,𝑦,𝑧)
    \n

    Its component at 1-cells 𝑓,𝑔,β„Ž is the invertible 2-cell

    \n
    𝛼𝑓,𝑔,β„Ž:(𝑓⋆𝑔)β‹†β„Žβ‡’π‘“β‹†(π‘”β‹†β„Ž)
    \n
    6. \n

  6. For all π‘₯,𝑦:𝒦︀0, natural isomorphisms, the left unitor πœ† and right unitor 𝜌, each between an endofunctor of 𝒦︀(π‘₯,𝑦) and the identity functor:

    \n
    𝒦︀⋆π‘₯,π‘₯,π‘¦βˆ˜βŸ¨1π‘₯,idβŸ©β‡’id𝒦︀⋆π‘₯,𝑦,π‘¦βˆ˜βŸ¨id,1π‘¦βŸ©β‡’id
    \n

    where 1π‘₯ and 1𝑦 in the pairings βŸ¨βˆ’,βˆ’βŸ© denote the constant functors at the identity 1-cells. The components at a 1-cell 𝑓:𝒦︀(π‘₯,𝑦) are the invertible 2-cells

    \n
    πœ†π‘“:1π‘₯β‹†π‘“β‡’π‘“πœŒπ‘“:𝑓⋆1𝑦⇒𝑓
    \n
    7. \n

  7. such that for all 𝑓:𝒦︀(π‘₯,𝑦) and 𝑔:𝒦︀(𝑦,𝑧) the triangle below commutes in 𝒦︀(π‘₯,𝑧):

    \n
    \n
    8. \n

  8. and such that for all composable 1-cells 𝑓,𝑔,β„Ž,π‘˜ the pentagon below commutes:

    \n
    \n
    9. \n
","date_published":"2026-06-11T02:12:57Z","tags":["bicategory","category-theory"]},{"id":"https://stevenschaefer.net/freely-transported-terms.html","url":"https://stevenschaefer.net/freely-transported-terms.html","title":"Freely transported terms in dependent type theory","content_html":"

Given 𝐴 and 𝐡:π΄β†’π“π²π©πž we can make sense of transported terms along equalities between indices in 𝐴. Say, with

\n
π—Œπ—Žπ–»π—Œπ—:(𝑝:π‘Ž=π‘Žβ€²)β†’π΅π‘Žβ†’π΅π‘Žβ€²
\n

for π‘Ž,π‘Žβ€²:𝐴.

\n

For instance, if 𝑝:π‘Ž=π‘Žβ€² and 𝑏:π΅π‘Ž then π—Œπ—Žπ–»π—Œπ—π‘π‘:π΅π‘Žβ€²

\n

To avoid landing in transport hell, I suspect that it may be preferable to work inside of a description of freely transported terms instead of taking semantic transports. The hypothesis is that by using descriptions of formal transport rather than actually computing a transport, we may defer the computation of an actual transport until the end of a construction. So instead of working with π΅π‘Ž directly, perhaps we may work with

\n
π–₯π—‹π–Ύπ–Ύπ–²π—Žπ–»π—Œπ—π΅π‘Žβ‰”βˆ‘(π‘Žβ€²:𝐴)βˆ‘(𝑝:π‘Ž=π‘Žβ€²)π΅π‘Žβ€²
\n

I think that this is very closely related to the Fording trick, as a map out of π–₯π—‹π–Ύπ–Ύπ–²π—Žπ–»π—Œπ—π΅π‘Ž,

\n
𝑓:βˆ‘(π‘Žβ€²:𝐴)βˆ‘(𝑝:π‘Ž=π‘Žβ€²)π΅π‘Žβ€²β†’πΆ
\n

can instead be described as a map,

\n
𝑔:(π‘Žβ€²:𝐴)β†’(𝑝:π‘Ž=π‘Žβ€²)β†’π΅π‘Žβ€²β†’πΆ
\n

Both this and the fording trick use the Coyoneda lemma to represent an dependent type family.

","date_published":"2026-06-03T21:38:21Z","tags":["cubical","type-theory"]},{"id":"https://stevenschaefer.net/onpls-2026-talk.html","url":"https://stevenschaefer.net/onpls-2026-talk.html","title":"Lessons from Mechanizing Categorical Logic in Cubical Agda","content_html":"
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.
","date_published":"2026-06-01T00:00:00Z","tags":["agda","cubical","cubical-categorical-logic","logical-relations"]},{"id":"https://stevenschaefer.net/thin-category.html","url":"https://stevenschaefer.net/thin-category.html","title":"Thin Category","content_html":"

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

","date_published":"2026-02-26T06:18:45Z","tags":["category-theory"]},{"id":"https://stevenschaefer.net/weakening-category.html","url":"https://stevenschaefer.net/weakening-category.html","title":"Weakening a Category","content_html":"

For categories π’žοΈ€ and π’ŸοΈ€, we can define the weakening of π’ŸοΈ€ over π’žοΈ€ as a displayed category over π’žοΈ€ which trivially displays a copy of π’ŸοΈ€ over each object of π’žοΈ€.

\n
(𝗐𝖾𝖺𝗄𝖾𝗇(π’žοΈ€,π’ŸοΈ€))π‘β‰”π’ŸοΈ€
","date_published":"2026-02-26T05:50:24Z","tags":["displayed-category-theory"]},{"id":"https://stevenschaefer.net/product-as-total-category.html","url":"https://stevenschaefer.net/product-as-total-category.html","title":"Products of Categories as Total Categories","content_html":"

Given categories π’žοΈ€ and π’ŸοΈ€, π’žοΈ€Γ—π’ŸοΈ€ is equivalent to the total category of weakening.

","date_published":"2026-02-26T05:48:35Z","tags":["category-theory","displayed-category-theory"]},{"id":"https://stevenschaefer.net/displayed-total-category.html","url":"https://stevenschaefer.net/displayed-total-category.html","title":"Displayed Total Category","content_html":"

Given a displayed category over a category π’žοΈ€ and another displayed category over , the total category of , we can define , the displayed total category of , as a displayed category over π’žοΈ€.

\n
","date_published":"2026-02-26T05:13:07Z","tags":["displayed-category-theory"]},{"id":"https://stevenschaefer.net/reindexing.html","url":"https://stevenschaefer.net/reindexing.html","title":"Reindexing a Displayed Category","content_html":"

Given a displayed category over a category π’žοΈ€ and a functor 𝐹:π’ŸοΈ€β†’π’žοΈ€, the reindexing of along 𝐹 is a displayed category over π’ŸοΈ€. It is simply the portion of over the image of 𝐹.

\n
","date_published":"2026-02-26T04:52:32Z","tags":["displayed-category-theory"]},{"id":"https://stevenschaefer.net/quiver.html","url":"https://stevenschaefer.net/quiver.html","title":"Quiver","content_html":"

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.

","date_published":"2026-02-26T04:14:27Z","tags":["category-theory"]},{"id":"https://stevenschaefer.net/free-monoidal-category-elimination.html","url":"https://stevenschaefer.net/free-monoidal-category-elimination.html","title":"Global Elimination Principle for the Free Monoidal Category","content_html":"

Given any displayed monoidal category 𝑀𝙳 over π–₯π—‹π–Ύπ–Ύπ–¬π—ˆπ—‡(𝑋) with an interpretation πœ„:𝑋⇝𝑀𝙳, we may construct a global section π–₯π—‹π–Ύπ–Ύπ–¬π—ˆπ—‡(𝑋)→𝑀𝙳. We refer to this as the global elimination principle of π–₯π—‹π–Ύπ–Ύπ–¬π—ˆπ—‡(𝑋).

","date_published":"2026-02-26T04:09:08Z","tags":["category-theory","displayed-category-theory","metatheory"]},{"id":"https://stevenschaefer.net/free-monoidal-category.html","url":"https://stevenschaefer.net/free-monoidal-category.html","title":"Free Monoidal Category over a Set","content_html":"

Fix a set 𝑋. The objects of the free monoidal category over 𝑋, π–₯π—‹π–Ύπ–Ύπ–¬π—ˆπ—‡(𝑋), are generated inductively by the elements of 𝑋 and a unit element 𝐼 over a binary operation βŠ—. The morphisms are given by a quotient-inductive type. They are generated by associators, unitors, and identity over composition and parallel action over βŠ— 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.

\n
\n
\n

Definition 1. Global Elimination Principle for the Free Monoidal Category free-monoidal-category-elimination

\n
\n
\n

Given any displayed monoidal category 𝑀𝙳 over π–₯π—‹π–Ύπ–Ύπ–¬π—ˆπ—‡(𝑋) with an interpretation πœ„:𝑋⇝𝑀𝙳, we may construct a global section π–₯π—‹π–Ύπ–Ύπ–¬π—ˆπ—‡(𝑋)→𝑀𝙳. We refer to this as the global elimination principle of π–₯π—‹π–Ύπ–Ύπ–¬π—ˆπ—‡(𝑋).

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","date_published":"2026-02-26T03:39:50Z","tags":["category-theory","metatheory"]},{"id":"https://stevenschaefer.net/element-of-presheaf.html","url":"https://stevenschaefer.net/element-of-presheaf.html","title":"Element of a Presheaf","content_html":"

An element of a presheaf 𝑃 at an object 𝑐 is an element π‘₯ of the set 𝑃𝑐.

","date_published":"2025-08-25T15:50:15Z","tags":["category-theory","presheaf"]},{"id":"https://stevenschaefer.net/universal-element.html","url":"https://stevenschaefer.net/universal-element.html","title":"Universal Element of a Presheaf","content_html":"

A universal element of a presheaf 𝑃 on a category 𝐢 is an element π‘₯βˆˆπ‘ƒπ‘, where 𝑐 is some object of 𝐢, demonstrating that 𝑃 is representable by 𝑐.

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π‘ƒβ‰…γ‚ˆπ‘
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(Slightly) more concretely, a universal element is captured by the following three pieces of data

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

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

","date_published":"2025-08-25T15:42:48Z","tags":["category-theory","presheaf","universal-property"]},{"id":"https://stevenschaefer.net/universal-property.html","url":"https://stevenschaefer.net/universal-property.html","title":"What is a universal property, really?","content_html":"

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.

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

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

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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?

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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.

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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×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→𝑐×𝑑, we must simultaneously provide elements of γ‚ˆπ‘ and γ‚ˆπ‘‘ at 𝑏.

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Using the product of presheaves, this means we are providing a single element of the presheaf (γ‚ˆπ‘)Γ—(γ‚ˆπ‘‘). Quite nicely, the universal element of this presheaf provides the object of 𝐢 that is the product of 𝑐 and 𝑑. The universal element, provided that it exists, contains the following data:

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Recall that the product of presheaves is computed pointwise in the category of sets, so if we expand the type of the element π‘₯ above we find that π‘₯ is a pair of maps 𝑝→𝑐 and 𝑝→𝑑.

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(γ‚ˆπ‘Γ—γ‚ˆπ‘‘)𝑝≅(γ‚ˆπ‘π‘)Γ—(γ‚ˆπ‘‘π‘)
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The first part of this pair is precisely πœ‹1. Correspondingly, the second part of this pair is πœ‹2. Finally, the universality of the element π‘₯ (i.e. the proof that any other element factors through π‘₯) captures our commutative diagram from above.

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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.

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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.

","date_published":"2025-08-25T15:17:14Z","tags":["category-theory","presheaf","universal-property"]},{"id":"https://stevenschaefer.net/pldi-2025-talk.html","url":"https://stevenschaefer.net/pldi-2025-talk.html","title":"Intrinsic Verification of Parsers and Formal Grammar Theory in Dependent Lambek Calculus","content_html":"
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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.

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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.

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","date_published":"2025-06-19T00:00:00Z","tags":["intrinsically-correct","lambekd","parsing"]},{"id":"https://stevenschaefer.net/cloudflare-parsing-error.html","url":"https://stevenschaefer.net/cloudflare-parsing-error.html","title":"Cloudflare Parsing Error","content_html":"

An error in the Cloudflare HTML parser would allow uninitialized memory to be dumped when there were imbalanced HTML tags.

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This is indeed a case where a verified parser would have alleviated the issue.

","date_published":"2025-04-29T02:14:12Z","tags":["bug","cve","parsing","security"]},{"id":"https://stevenschaefer.net/groebner-basis.html","url":"https://stevenschaefer.net/groebner-basis.html","title":"GrΓΆbner Basis","content_html":"

Let 𝐾 be a ring and 𝐾[π‘₯0,…,π‘₯𝑛] a polynomial ring over it. Suppose πΌβŠ‚πΎ[π‘₯0,…,π‘₯𝑛] is an ideal. A GrΓΆbner basis for 𝐼 is a generating set of polynomials for the ideal that is minimal with respect to a given ordering on the monomials π‘₯0,…,π‘₯𝑛.

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Given an ideal 𝐼, a GrΓΆbner basis for 𝐼 may be found via Buchberger’s algorithm. Intuitively, Buchberger’s algorithm attempts to solve a system of polynomial equations by iterated polynomial division to eliminate variables. At any given point, there is a degree of freedom in which what variable will be eliminated by the next division. The algorithm attempts to eliminate variables with respect to the monomial ordering.

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Buchberger’s algorithm may be viewed simultaneously as a generalization of the Quine-McCluskey Boolean minimization algorithm and as a special case of the Knuth-Bendix algorithm.

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The complexity of Buchberger’s algorithm is a little unwieldy to estimate in general. However, just like SAT solvers there are enough optimizations to execute Buchberger reasonably fast in practice. For instance, it is fast enough to handle several hundreds of polynomials, each having hundreds of terms with very large coefficients.

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There are some very fun applications of this approach, such as Solving Sudoku with Algebra. This idea has also been applied to inferring polynomial loop invariants.

","date_published":"2025-02-19T17:10:30Z","tags":["grobner-bases"]},{"id":"https://stevenschaefer.net/groebner-loop-invariants.html","url":"https://stevenschaefer.net/groebner-loop-invariants.html","title":"GrΓΆbner Bases for Inferring Polynomial Loop Invariants","content_html":"

When the tools in I4: Incremental inference of inductive invariants for verification of distributed protocols and On Symmetry and Quantification: A New Approach to Verify Distributed Protocols search for an inductive invariant of a distributed system, the search procedure instantiates a series of small finite models and tries to infer from their truth tables a series of logical formulae that hold over those finite models. These formulae are found by running the Quine-McCluskey algorithm for minimization of Boolean functions. The prime implicants found by Quine-McCluskey have a latent symmetry that can be abstracted into quantified formulae. There are only so many small numbers, and so small finite models may propose formulae that do not hold at larger sizes. However, if you find a formula that holds at size 𝑛 as well as size 𝑛+1, then it is likely a good candidate to hold at all sizes. You need to be a little careful if your protocol is indexed by several variables, but mostly this general idea holds when abstracting to a protocol of unbounded size. Further discussion of this idea can be found in SAT-based quantified symmetric minimization of the reachable states of distributed protocols: An update.

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My observation was that Quine-McCluskey is just a special instance of Buchberger’s algorithm for computing GrΓΆbner Bases. That is, you can describe Boolean formulae as polynomials over the field with two elements, and in this translation Quine-McCluskey and Buchberger each compute the same data. This observation isn’t new in and of itself, but it does open up an opportunity to generalize the invariant search procedure that is used above.

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The place I went looking to apply this idea was in the search of polynomial loop invariants. If a loop had an invariant that is expressible as a polynomial relation between the program variables, then you could apply the same idea as above to infer the loop invariant.

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Suppose π‘₯1,…,π‘₯𝑛 are the variables in scope of program 𝑆 and 𝑆 contains a loop that we want to infer an invariant for. Denote the value of π‘₯𝑖 at the 𝑗-th loop iteration by π‘₯𝑖,𝑗. The invariant search procedure proceeds intuitively as the following: we will keep track of the minimal set of polynomials that could interpolate between all of the variable assignments that we have witnessed thus far. Formally this is kept track of by the ideal of polynomials. At the 𝑗-th loop iteration we add a new generator to the ideal which corresponds to the assignments π‘₯1,𝑗,…,π‘₯𝑛,𝑗. The GrΓΆbner basis for this ideal provides the minimal data needed to generate all the assignments witnessed thus far, so if this process saturates then the GrΓΆbner basis encodes a polynomial loop invariant. The nice thing about polynomials is that they have finite degree which guarantees that this process does indeed saturate (provided that the degree of the invariant is smaller than the number of loop iterations).

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I was so excited to find this idea. I’d felt like it was my first good idea in grad school. Then I read Automatic Generation of Polynomial Loop Invariants: Algebraic Foundations and found out someone had done this 20 years ago. I still wonder from time to time if there is room to further refine this idea or perhaps further generalize it. For instance, there is further generalization beyond Quine-McCluskey or Buchberger to the Knuth-Bendix algorithm, which seems to be a more general instance of both of these algorithms. So perhaps this search procedure can be weakened to an even more general class? Although, I’m not yet familiar much with the Knuth-Bendix algorithm.

","date_published":"2025-02-19T15:41:11Z","tags":["grobner-bases","halfbaked","inductive-invariants","loop-invariants"]},{"id":"https://stevenschaefer.net/translation-invariance-verification.html","url":"https://stevenschaefer.net/translation-invariance-verification.html","title":"A Method for Verifying Translational Invariance of Image Processing Neural Networks","content_html":"

My understanding for how one may prove a safety property 𝑃 for a neural network 𝑁 is as follows. First, express 𝑃 as an input-output property. That is, choose some region 𝐷 in the domain of 𝑁 to represent the inputs of interest. Further choose some region 𝐴 in the codomain of 𝑁 to describe safe outputs. Then describe 𝑃 as the property:

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βˆ€π‘₯∈𝐷.𝑁(π‘₯)∈𝐴
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For instance, this is more or less how Reluplex, Marabou, and 𝛼𝛽-CROWN each work. Squinting my eyes, this is the only such method for verifying a safety property for 𝑁. That is, this is the only way to get a 100% guarantee that 𝑃 holds rather than some high measure of confidence.

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I believe in this formalism I have an idea for how to express the property β€œπ‘ is translationally invariant” where 𝑁 is an image classifying net.

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To this end, we need a continuous artifact that captures what it means to translate an image. We can express an image as a matrix of pixels 𝐼 (or perhaps several parallel matrices if we care about color channels, but stick to a single grayscale matrix for now). To shift 𝐼 over by a single pixel, we may left-multiply 𝐼 by the shift matrix 𝑆, where 𝑆 is the matrix filled with zeros and has 1β€²s on the subdiagonal. Note that all the directions of shifting 𝐼 are captured by the combinations of left/right multiplication by 𝑆, 𝑆𝑇.

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Shifting by multiple pixels is now expressed by the matrix 𝑆𝑛𝐼, however this is still a discrete dynamical system. We don’t have a continuous object by which we can test our safety property. My initial thought to continuousify this system was to express some sort of exponential flow by 𝑆. That is, consider the matrix

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π‘†π‘‘πΌβ‰”π‘’π‘‘π—…π—ˆπ—€(𝑆)
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𝑆𝑑𝐼 then captures what it means to translate the image 𝐼 over by 𝑑, a real-valued β€œamount of shifting”. This choice of continuous artifact could then be used for a verification effort, however there are some issues related to numerical stability because 𝑆 isn’t invertible. Because it is not invertible, π—…π—ˆπ—€(𝑆) doesn’t actually exist. So to make the above construction work, we need to mildly perturb 𝑆 and take a pseudoinverse. This sort of works and makes it so 𝑆𝑑𝐼 does capture some real valued shift, but it is only accurate when 𝑑 is small. So this is maybe problematic for our verification effort. This may be resolvable by chopping up the problem into subproblems, each of which is thin enough for the current iteration is accurate enough on that subproblem. However, there are two big things to consider.

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  1. The exponential approach above is probably too complicated. It seems
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likely that you may be able to take some (sequence of) linear interpolation(s) between the 𝑆𝑛𝐼’s. This will still be some continuous object that captures a real-valued shift and it will be much more stable than the approach above.

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  1. Even if we sort out which continuous object represents translation
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of an image, I cannot for the life of me train any neural network that is translationally invariant. So the proof method is useless if there is nothing that it would ever apply to.

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Precisely in this last point, I have mostly focused on trying to train a small CNN for MNIST handwritten digit classification that preserves the output class for small, reasonable translations of the digit. I’ve used data augmentation to predispose the network to being translationally invariant, and even though I can get a high degree of invariance, I cannot get a network that is invariant for all of the examples even in the training set or a reserved testing set.

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It may be the case that the method I propose for measuring this invariance could be used to adversarially train a network to have better invariance. It may also be the case that all CNNs are bound to suffer from small degrees of translational sensitivity. I cannot find the citation at the moment, but there was a paper that suggested that CNNs suffer from weird issues of translational sensitivity that relate to the size of the convolutional window. So maybe this approach is doomed to fail anyway.

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On the whole, I will say that machine learning verification almost sounds like an oxymoron. That is, if you have the expressivity to properly state a sophisticated safety property, then you likely understand the problem enough to not need to resort to machine learning in the first place. So almost tautologically, it seems that there cannot be satisfying verification of neural nets, as the tasks of machine learning and verification live on very different epistemic foundations.

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The related works I could liberate from Zotero may be found below.

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1 Adequate Losses via Quantitative Linear Logic capucci-2026-adequatearXiv Β· 2026

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As neural components are increasingly embedded in existing symbolic software – including safety-critical systems – the question arises of how to specify and enforce the safety of the newly introduced neural parts. Unlike traditional logical specifications, these must be amenable not only to the standard Boolean interpretation, but also to training and optimisation. The latter calls for a quantitative interpretation of the logical syntax, subject to further requirements such as smoothness and differentiability. Moreover, the qualitative and quantitative sides of the logic must share a unifying proof-theoretic and categorical semantics. Finally, the new logic should link cleanly to the substructural and program logics that underpin the verification of existing symbolic programs. In this paper, we present a logic that ticks all of these boxes. We introduce a family of calculi, pQLL, indexed by a hardness degree 𝑝, prove a cut-elimination theorem for them, and establish completeness with respect to enriched residuated β€˜soft’ lattices. At 𝑝=∞, pQLL reduces to multiplicative additive linear logic (MALL), and provability in pQLL converges to provability in MALL as π‘β†’βˆž. We express optimisation objectives in the syntax of this logic and prove the quantitative adequacy of neuro-symbolic loss functions – a result that has eluded the neuro-symbolic machine learning community for nearly a decade.
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DOI Β· arXiv
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2 Quantitative Linear Logic for Neuro-Symbolic Learning and Verification flinkow-2026-quantitativearXiv Β· 2026

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Differentiable Logics are deployed in neuro-symbolic learning tasks as a way of embedding logical constraints in the training objective of neural networks. A differentiable logic consists of a syntax to write logical properties and a semantics to interpret them as real-valued functions to be folded in the loss function. A defining trade-off of the field is that between logical properties of the connectives, and analytic concerns for the semantics, with both aspects being relevant in applications. At one extreme we find fuzzy logics, that have well-established algebraic and proof-theoretic foundations, and at the other ad-hoc differentiable logics like Fischer’s DL2, conceived for deep learning applications. However, no satisfactory foundation has emerged yet. We propose a resolution to this long-standing tension via a novel logic, Quantitative Linear Logic (QLL), with foundational ambitions. Our design is driven by naturality – the idea that, since logical constraints are translated to losses, the semantics of the connectives should be pertinent operations used in ML practice (that is, sum and log-sum-exp) on additive quantities (like logits). We then judge the result on two aspects: logical adequacy – that they satisfy most of the standard logical laws of Linear Logic; and empirical effectiveness – test-time performance (as measured by adversarial attacks) is well-correlated to the actual verification of the logical constraints (as measured by off-the-shelf neural network verifiers), which makes QLL stand out among SoTA techniques.
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DOI Β· arXiv
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3 Quantifiers for Differentiable Logics in Rocq (Extended Abstract) marulandagiraldo-2025-quantifiersSAIV Β· 2025

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4 Fundamental Components of Deep Learning: A category-theoretic approach gavranovicFundamentalComponentsDeep2024

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Deep learning, despite its remarkable achievements, is still a young field. Like the early stages of many scientific disciplines, it is marked by the discovery of new phenomena, ad-hoc design decisions, and the lack of a uniform and compositional mathematical foundation. From the intricacies of the implementation of backpropagation, through a growing zoo of neural network architectures, to the new and poorly understood phenomena such as double descent, scaling laws or in-context learning, there are few unifying principles in deep learning. This thesis develops a novel mathematical foundation for deep learning based on the language of category theory. We develop a new framework that is a) end-to-end, b) unform, and c) not merely descriptive, but prescriptive, meaning it is amenable to direct implementation in programming languages with sufficient features. We also systematise many existing approaches, placing many existing constructions and concepts from the literature under the same umbrella. In Part I we identify and model two main properties of deep learning systems parametricity and bidirectionality by we expand on the previously defined construction of actegories and Para to study the former, and define weighted optics to study the latter. Combining them yields parametric weighted optics, a categorical model of artificial neural networks, and more. Part II justifies the abstractions from Part I, applying them to model backpropagation, architectures, and supervised learning. We provide a lens-theoretic axiomatisation of differentiation, covering not just smooth spaces, but discrete settings of boolean circuits as well. We survey existing, and develop new categorical models of neural network architectures. We formalise the notion of optimisers and lastly, combine all the existing concepts together, providing a uniform and compositional framework for supervised learning.
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5 Architecture-Preserving Provable Repair of Deep Neural Networks taoArchitecturePreservingProvableRepair2023PLDI Β· 2023

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Deep neural networks (DNNs) are becoming increasingly important components of software, and are considered the state-of-the-art solution for a number of problems, such as image recognition. However, DNNs are far from infallible, and incorrect behavior of DNNs can have disastrous real-world consequences. This paper addresses the problem of architecture-preserving V-polytope provable repair of DNNs. A V-polytope defines a convex bounded polytope using its vertex representation. V-polytope provable repair guarantees that the repaired DNN satisfies the given specification on the infinite set of points in the given V-polytope. An architecture-preserving repair only modifies the parameters of the DNN, without modifying its architecture. The repair has the flexibility to modify multiple layers of the DNN, and runs in polynomial time. It supports DNNs with activation functions that have some linear pieces, as well as fully-connected, convolutional, pooling and residual layers. To the best our knowledge, this is the first provable repair approach that has all of these features. We implement our approach in a tool called APRNN. Using MNIST, ImageNet, and ACAS Xu DNNs, we show that it has better efficiency, scalability, and generalization compared to PRDNN and REASSURE, prior provable repair methods that are not architecture preserving. CCS Concepts: β€’ Computing methodologies β†’ Neural networks; β€’ Theory of computation β†’ Linear programming; β€’ Software and its engineering β†’ Software post-development issues.
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PDF Β· DOI Β· pldb
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6 Compiling Higher-Order Specifications to SMT Solvers: How to Deal with Rejection Constructively daggitt-2023-compilingCPP Β· 2023

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PDF Β· DOI Β· pldb
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7 Vehicle: Interfacing Neural Network Verifiers with Interactive Theorem Provers daggitt-2022-vehiclearXiv Β· 2022

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Verification of neural networks is currently a hot topic in automated theorem proving. Progress has been rapid and there are now a wide range of tools available that can verify properties of networks with hundreds of thousands of nodes. In theory this opens the door to the verification of larger control systems that make use of neural network components. However, although work has managed to incorporate the results of these verifiers to prove larger properties of individual systems, there is currently no general methodology for bridging the gap between verifiers and interactive theorem provers (ITPs). In this paper we present Vehicle, our solution to this problem. Vehicle is equipped with an expressive domain specific language for stating neural network specifications which can be compiled to both verifiers and ITPs. It overcomes previous issues with maintainability and scalability in similar ITP formalisations by using a standard ONNX file as the single canonical representation of the network. We demonstrate its utility by using it to connect the neural network verifier Marabou to Agda and then formally verifying that a car steered by a neural network never leaves the road, even in the face of an unpredictable cross wind and imperfect sensors. The network has over 20,000 nodes, and therefore this proof represents an improvement of 3 orders of magnitude over prior proofs about neural network enhanced systems in ITPs.
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8 PRIMA: General and Precise Neural Network Certification via Scalable Convex Hull Approximations mullerPRIMAGeneralPrecise2022POPL Β· 2022

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Formal verification of neural networks is critical for their safe adoption in real-world applications. However, designing a precise and scalable verifier which can handle different activation functions, realistic network architectures and relevant specifications remains an open and difficult challenge. In this paper, we take a major step forward in addressing this challenge and present a new verification framework, called PRIMA. PRIMA is both (i) general: it handles any non-linear activation function, and (ii) precise: it computes precise convex abstractions involving multiple neurons via novel convex hull approximation algorithms that leverage concepts from computational geometry. The algorithms have polynomial complexity, yield fewer constraints, and minimize precision loss. We evaluate the effectiveness of PRIMA on a variety of challenging tasks from prior work. Our results show that PRIMA is significantly more precise than the state-of-the-art, verifying robustness to input perturbations for up to 20%, 30%, and 34% more images than existing work on ReLU-, Sigmoid-, and Tanh-based networks, respectively. Further, PRIMA enables, for the first time, the precise verification of a realistic neural network for autonomous driving within a few minutes.
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PDF Β· DOI Β· arXiv Β· pldb
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9 Provable repair of deep neural networks sotoudehProvableRepairDeep2021PLDI Β· 2021

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Deep Neural Networks (DNNs) have grown in popularity over the past decade and are now being used in safety-critical domains such as aircraft collision avoidance. This has motivated a large number of techniques for finding unsafe behavior in DNNs. In contrast, this paper tackles the problem of correcting a DNN once unsafe behavior is found. We introduce the provable repair problem, which is the problem of repairing a network N to construct a new network Nβ€² that satisfies a given specification. If the safety specification is over a finite set of points, our Provable Point Repair algorithm can find a provably minimal repair satisfying the specification, regardless of the activation functions used. For safety specifications addressing convex polytopes containing infinitely many points, our Provable Polytope Repair algorithm can find a provably minimal repair satisfying the specification for DNNs using piecewise-linear activation functions. The key insight behind both of these algorithms is the introduction of a Decoupled DNN architecture, which allows us to reduce provable repair to a linear programming problem. Our experimental results demonstrate the efficiency and effectiveness of our Provable Repair algorithms on a variety of challenging tasks.
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PDF Β· DOI Β· pldb
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10 Tracking translation invariance in CNNs myburghTrackingTranslationInvariance2021arXiv Β· 2021

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Although Convolutional Neural Networks (CNNs) are widely used, their translation invariance (ability to deal with translated inputs) is still subject to some controversy. We explore this question using translation-sensitivity maps to quantify how sensitive a standard CNN is to a translated input. We propose the use of Cosine Similarity as sensitivity metric over Euclidean Distance, and discuss the importance of restricting the dimensionality of either of these metrics when comparing architectures. Our main focus is to investigate the effect of different architectural components of a standard CNN on that network’s sensitivity to translation. By varying convolutional kernel sizes and amounts of zero padding, we control the size of the feature maps produced, allowing us to quantify the extent to which these elements influence translation invariance. We also measure translation invariance at different locations within the CNN to determine the extent to which convolutional and fully connected layers, respectively, contribute to the translation invariance of a CNN as a whole. Our analysis indicates that both convolutional kernel size and feature map size have a systematic influence on translation invariance. We also see that convolutional layers contribute less than expected to translation invariance, when not specifically forced to do so.
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Web Β· arXiv
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11 Towards Verified Artificial Intelligence seshiaVerifiedArtificialIntelligence2020arXiv Β· 2020

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Verified artificial intelligence (AI) is the goal of designing AI-based systems that have strong, ideally provable, assurances of correctness with respect to mathematically-specified requirements. This paper considers Verified AI from a formal methods perspective. We describe five challenges for achieving Verified AI, and five corresponding principles for addressing these challenges.
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DOI Β· arXiv
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12 Verification of Deep Convolutional Neural Networks Using ImageStars tranVerificationDeepConvolutional2020CAV Β· 2020

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Convolutional Neural Networks (CNN) have redefined stateof-the-art in many real-world applications, such as facial recognition, image classification, human pose estimation, and semantic segmentation. Despite their success, CNNs are vulnerable to adversarial attacks, where slight changes to their inputs may lead to sharp changes in their output in even well-trained networks. Set-based analysis methods can detect or prove the absence of bounded adversarial attacks, which can then be used to evaluate the effectiveness of neural network training methodology. Unfortunately, existing verification approaches have limited scalability in terms of the size of networks that can be analyzed.
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PDF Β· DOI Β· arXiv Β· pldb
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13 Stride and Translation Invariance in CNNs moutonStrideTranslationInvariance2020Artificial Intelligence Research Β· 2020

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Convolutional Neural Networks have become the standard for image classification tasks, however, these architectures are not invariant to translations of the input image. This lack of invariance is attributed to the use of stride which ignores the sampling theorem, and fully connected layers which lack spatial reasoning. We show that stride can greatly benefit translation invariance given that it is combined with sufficient similarity between neighbouring pixels, a characteristic which we refer to as local homogeneity. We also observe that this characteristic is dataset-specific and dictates the relationship between pooling kernel size and stride required for translation invariance. Furthermore we find that a trade-off exists between generalization and translation invariance in the case of pooling kernel size, as larger kernel sizes lead to better invariance but poorer generalization. Finally we explore the efficacy of other solutions proposed, namely global average pooling, anti-aliasing, and data augmentation, both empirically and through the lens of local homogeneity.
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DOI Β· arXiv
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14 Why do deep convolutional networks generalize so poorly to small image transformations? azulayWhyDeepConvolutional2019

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Convolutional Neural Networks (CNNs) are commonly assumed to be invariant to small image transformations: either because of the convolutional architecture or because they were trained using data augmentation. Recently, several authors have shown that this is not the case: small translations or rescalings of the input image can drastically change the network’s prediction. In this paper, we quantify this phenomena and ask why neither the convolutional architecture nor data augmentation are sufficient to achieve the desired invariance. Specifically, we show that the convolutional architecture does not give invariance since architectures ignore the classical sampling theorem, and data augmentation does not give invariance because the CNNs learn to be invariant to transformations only for images that are very similar to typical images from the training set. We discuss two possible solutions to this problem: (1) antialiasing the intermediate representations and (2) increasing data augmentation and show that they provide only a partial solution at best. Taken together, our results indicate that the problem of insuring invariance to small image transformations in neural networks while preserving high accuracy remains unsolved.
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DOI
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15 Formal Verification of CNN-based Perception Systems kouvarosFormalVerificationCNNbased2018arXiv Β· 2018

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We address the problem of verifying neural-based perception systems implemented by convolutional neural networks. We define a notion of local robustness based on affine and photometric transformations. We show the notion cannot be captured by previously employed notions of robustness. The method proposed is based on reachability analysis for feed-forward neural networks and relies on MILP encodings of both the CNNs and transformations under question. We present an implementation and discuss the experimental results obtained for a CNN trained from the MNIST data set.
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DOI Β· arXiv
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","date_published":"2025-02-19T14:40:25Z","tags":["halfbaked","machine-learning-verification"]},{"id":"https://stevenschaefer.net/german-potato-salad.html","url":"https://stevenschaefer.net/german-potato-salad.html","title":"German Potato Salad","content_html":"
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Ingredients

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  • 5 pounds potatoes
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  • 1 pound bacon
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  • 1 cup sour cream
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  • 2 cups mayo
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  • 1 cup cold water
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  • 1 tbsp sugar
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  • 1 onion
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Method

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  1. Peel and cube potatoes.
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  2. Boil potatoes until fork tender, but still slightly firm.
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  3. Cook bacon, then crumble. Reserve bacon grease.
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  4. Dice onion and cook in bacon grease until brown.
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  5. Mix bacon, onion, potatoes, mayo, sour cream, cold water, 1 tbsp salt, 1 tbsp pepper, and sugar in an oven-safe bowl.
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  6. Bake for 60 minutes at 350 Fahrenheit. Broil for 5-7 minutes, or until golden brown.
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Notes

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This is another cherished recipe from my mother.

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As a child, I was extraordinarily averse to mayo, sour cream, cottage cheese, etc. I didn’t understand how delicious this recipe was until I was about 20 or so. I can’t believe I had missed out for so long.

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","date_published":"2025-02-19T04:45:53Z","tags":["recipe"]},{"id":"https://stevenschaefer.net/golabki.html","url":"https://stevenschaefer.net/golabki.html","title":"GoΕ‚Δ…bki (Stuffed Cabbage)","content_html":"
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Ingredients

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  • 5 pounds cheap beef
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  • 1 pound bacon
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  • 4 large onions
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  • 3 cans tomato bisque
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  • 3 large cans stewed tomatoes
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  • 4 large cabbages
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  • 3 large eggs
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Method

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  1. Cook bacon and reserve grease. Crumble bacon, then add bacon and grease to the 5 pounds of beef.
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  2. Squeeze liquid out of stewed tomatoes and reserve the liquid. Add the tomato solids into the beef.
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  3. Add egg and rice to beef, mix until homogenous.
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  4. Bring a pot of water to boil. Core the cabbage then add to the pot.
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  5. Wiggle the leaves off the heads of cabbage. Don’t pull the leaves off forcefully, they should release easily. Put the removed leaves aside to cool. Keep any leaves that are too small or have torn.
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  6. Preheat the oven to 350 Fahrenheit.
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  7. Begin to assemble rolls of cabbage. Put a serving of meat in the middle of a leaf of cabbage, then roll tightly like a burrito. Begin layering the completed rolls in a large roasting pan.
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  8. After one layer of rolls is filled, add salt, pepper, tomato bisque, the reserve tomato liquid, and the torn scraps of cabbage leaves on top.
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  9. Repeat until the pan is full or you have run out of ingredients.
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  10. Roast for 4 hours. Time permitting, reduce the temperature to 275 halfway through roasting. Ideally the lower and slower, the better.
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The Annoying Bit That I Wish Recipe Writers Put on the Bottom

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This is a cherished recipe passed from my grandfather, to my mother, and then to me. I have many fond memories of eating this as a child, which then evolved to fond memories making it with my mom, and now live on as fond memories making it with friends and family.

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To whoever finds themselves reading this, I hope this recipe brings you as much as it has brought me.

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","date_published":"2025-02-19T04:02:25Z","tags":["recipe"]},{"id":"https://stevenschaefer.net/constant-presheaf.html","url":"https://stevenschaefer.net/constant-presheaf.html","title":"Constant Presheaf","content_html":"

The constant presheaf of a set 𝑋 on a category 𝐢 is a functor Ξ”(𝑋):πΆπ‘œπ‘β†’π’πžπ­ such that

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Ξ”(𝑋)𝑐≔𝑋
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I often call this the discrete presheaf for 𝑋, but I don’t know if that’s standard.

","date_published":"2025-02-12T20:21:29Z","tags":["category-theory","presheaf"]}]}