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

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

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.

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

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

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 $\text{deg}𝑥\le \text{deg}𝑦$.

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

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 $\mathrm{𝒞︀}$ say how an object sits over its predecessors.

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, $\text{𝗇𝖾𝗑𝗍}:𝑃\to ⊳𝑃$ corresponds to

.

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

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

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

Concretely, later on families evaluates to

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

out of the strict downset.

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 $\text{𝖲𝗍𝗋𝗂𝖼𝗍𝖣𝗈𝗐𝗇}(𝑦)\to \text{𝖲𝗍𝗋𝗂𝖼𝗍𝖣𝗈𝗐𝗇}(𝑥)$. 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

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

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

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.

Let $\mathrm{𝒞︀}$ be a direct category and $𝑃$ a presheaf on it. Every map

has a fixed point: a global element $\text{𝗅ö𝖻}𝜑:\top \to 𝑃$ with

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 $\prec$: at each $𝑥$, the section already constructed over the past assembles into an element of $(⊳𝑃)(𝑥)$, and $𝜑$ extends it to $𝑥$.

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

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

sending each object of $\mathrm{𝒟︀}$ to the $𝑇$-algebra it carries.

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

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

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

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

So presheaves are both monadic and comonadic over families.

Reading the algebra structure concretely: a $𝑇$-algebra on a family $𝐴$ is a map ${\sum }_{𝑦}\mathrm{𝒞︀}(𝑥,𝑦)\times 𝐴(𝑦)\to 𝐴(𝑥)$ 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 $𝐴(𝑥)\to {\prod }_{𝑦}\mathrm{𝒞︀}(𝑦,𝑥)\to 𝐴(𝑦)$, 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.

The representable $\text{𝗒}𝑥$ 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.

A sieve on an object $𝑥$ of $\mathrm{𝒞︀}$ is a subobject of the representable presheaf $\text{𝗒}𝑥$: a presheaf $𝑆$ with a monic morphism $𝑆\rightarrowtail \text{𝗒}𝑥$. 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 $𝑆\ni 𝑓$, 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:

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 $𝑇$.

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:

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

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