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Passing from a category $C$ to its presheaf category $PSh(C) := [C^{op},Set]$ may be regarded as the operation of “freely adjoining colimits to $C$”.
A slightly more precise version of this statement is that the Yoneda embedding
is the free cocompletion of $C$.
The universal property of the Yoneda embedding is expressed in terms of the Yoneda extension of any functor $F : C \to D$ to a category $D$ with colimits.
The rough statement is that the Yoneda embedding
of a small category $S$ into the category $Set^{S^{op}}$ of presheaves on $S$ is universal among functors from $S$ into cocomplete categories. Technically, this should be understood in an appropriate 2-categorical sense: given a functor $F: S \to D$ where $D$ is (small-)cocomplete, there exists a unique (up to isomorphism) cocontinuous extension
called the Yoneda extension, meaning that $\hat{F} y_S \cong F$ and $\hat{F}$ preserves small colimits.
Put slightly differently: let $Cocomp$ denote the 2-category of cocomplete categories, cocontinuous functors, and natural transformations between them. Then for cocomplete $D$, the Yoneda embedding $y S: S \to Set^{S^{op}}$ induces by restriction a functor
that is an equivalence for each $D$, one that is 2-natural in $D$. In fact we show that the Yoneda extension $F \mapsto \hat{F}$ gives a functor $ext_{y S}D: Cat(S, D) \to Cocomp(Set^{S^{op}}, D)$ that is a left adjoint to $res_{y S}D$, giving a 2-natural adjoint equivalence.
As a first step, we construct the desired extension $\hat{F}$ as a left adjoint to the functor
Being a left adjoint, $\hat{F}$ is cocontinuous.
An explicit formula for the left adjoint is given by the weighted colimit, coend, or tensor product of functors formula
where $S \cdot d$ is notation for copowering (or tensoring) an object $d$ of $D$ by a set $S$ (in this case, a coproduct of an $S$-indexed set of copies of $D$). This formula recurs frequently throughout this wiki; see also nerve, Day convolution.
This “free cocompletion” property generalizes to enriched category theory. If $V$ is complete, cocomplete, symmetric monoidal closed, and $S$ is a small $V$-enriched category, then the enriched presheaf category $V^{S^{op}}$ is a free $V$-cocompletion of $S$. The explicit meaning is analogous to the case where $V = Set$, where all ordinary category concepts are replaced by their $V$-enriched analogues; in particular, the notion of “$V$-cocontinuous functor” referes to preservation of enriched weighted colimits (not just ordinary conical colimits).
If $C$ is not small, then its free cocompletion still exists, but it is not the category of all presheaves on $C$. Rather, it is the category of small presheaves on $C$, i.e. presheaves that are small colimits of representables.
For $\mathcal{C}$ a small category, its Yoneda embedding $\mathcal{C} \overset{y}{\hookrightarrow} [\mathcal{C}^{op}, Set]$ exhibits the category of presheaves $[\mathcal{C}^{op}, Set]$ as the free co-completion of $\mathcal{C}$, in that it is a universal morphism (as in this Def. but “up to natural isomorphism”) into a cocomplete category, in that:
for $\mathcal{D}$ any cocomplete category;
for $F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ any functor;
there is a functor $\widetilde F \;\colon\; [\mathcal{C}^{op}, Set] \longrightarrow \mathcal{D}$, unique up to natural isomorphism, such that
$\widetilde F$ extends $F$ through the Yoneda embedding, in that the following diagram commutes, up to natural isomorphism:
The last condition says that $\widetilde F$ is fixed on representable presheaves, up to isomorphism, by
and in fact naturally so:
But the co-Yoneda lemma expresses every presheaf $\mathbf{X} \in [\mathcal{C}^{op}, Set]$ as a colimit of representable presheaves
Since $\tilde F$ is required to preserve any colimit and hence these particular colimits, (1) implies that $\widetilde F$ is fixed to act, up to isomorphism, as
(where the colimit (a coend) on the right is computed in $\mathcal{D}$!).
In general, given a locally small (but possibly large) category $\mathcal{A}$, we can define a locally small category of small presheaves on $\mathcal{A}$ as the full subcategory of the functor category $Fun(\mathcal{A}^{\mathrm{op}},\operatorname{Set})$ spanned by those functors that are colimits of small diagrams of representables. Equivalently, a small presheaf is a functor $F:\mathcal{A}^{\mathrm{op}} \to \operatorname{Set}$ such that there exists a functor $\gamma:\mathcal{C} \to \mathcal{A}$ with $\mathcal{C}$ small and a presheaf $F': \mathcal{C}^{\mathrm{op}}\to \operatorname{Set}$ such that
This process of free cocompletion defines a functor $L$ from the 2-category
to the 2-category
where, just to be clear, ‘cocomplete’ means having all small colimits, as usual. Moreover, this functor $L$ is part of a pseudoadjunction where the right adjoint $R: Cocomp \to CAT$ assigns each locally small coccomplete category its underlying category, and so on.
In the case where $\mathcal{A}^{\mathrm{op}}$ is locally presentable, the category of small presheaves on $\mathcal{A}$ is equivalent to the category of accessible functors $\mathcal{A}^{\mathrm{op}}\to \operatorname{Set}$. In this situation, the category of small functors has many nice properties and is often said to be almost a topos. In particular, under this assumption, the category of small presheaves is complete, cocomplete, and Cartesian-closed.
See Day-Lack for more details on all these matters. They handle the more general case of enriched categories.
David Corfield: So is this ‘free cocompletion’ part of an adjunction between the category of categories and the category of cocomplete categories (modulo size worries?). Or should we think of it as part of a pseudoadjunction between 2-categories? (I would start a page on that, but how are naming conventions going in this area?)
John Baez: Equations between functors tends to hold only up to natural isomorphism. So, your first guess should not be that there’s an adjunction between the categories $CAT$ and $Cocomp$, but rather, a pseudoadjunction between the 2-categories $CAT$ and $Cocomp$.
This means there’s a forgetful 2-functor:
together with a ‘free cocompletion’ 2-functor:
And, it mean there’s an equivalence of categories
for every $C \in CAT$, $D \in Cocomp$. And finally, it means that this equivalence is pseudonatural as a function of $C$ and $D$.
If we have a pseudonatural equivalence of categories
instead of a natural isomorphism of sets, then we say we have a ‘pseudoadjunction’ instead of an adjunction. A pseudoadjunction is the right generalization of adjunction when we go to 2-categories; if we were feeling in a modern mood we might just say ‘adjunction’ and expect people to know we meant ‘pseudo’.
The details have been worked out by Day-Lack in even more generality: they consider enriched categories. Note that the free cocompletion of a small category $C$ is the category of all presheaves on $C$, but for a more general locally small category it’s just the category of small presheaves.
In their work on species, Fiore, Gambino, Hyland and Winskel confronted the size issues in another way. In one draft of this paper they had a very artful and sophisticated device for dealing with this size problem. In the latest draft they seem to have sidestepped it entirely: you’ll see they discuss the ‘free symmetric monoidal category on a category’ pseudomonad, but never the ‘free cocomplete category on a category’ pseudomonad, even though they do use the $\widehat{C}$ construction all over the place. Somehow they’ve managed to avoid the need to consider this construction as a pseudomonad!
Fiore, Gambino, Hyland and Winskel later exhibited the free cocompletion of a small category as a relative pseudomonad from the 2-category Cat of small categories into the 2-category CAT of locally-small categories (via the 2-category COC of locally-small cocomplete categories). See FGHW for more details.
One can ask for the notion of free cocompletion in the wider context of higher category theory.
free cocompletions
Brian Day, Steve Lack, Limits of small functors (web)
Fiore, Gambino, Hyland and Winskel, Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures, (web)
This reference might also give helpful clues:
A pedagogical explanation of the universal property of the Yoneda embedding is given starting on page 7. On page 8 there’s an explanation with lots of pictures how a presheaf is an “instruction for how to build a colimit”. Then on p. 9 the universal morphism that we are looking for here is identified as the one that “takes the instructions for building a colimit and actually builds it”.
(This text, by the way, contains various other gems. A pity that it is left unfinished.)
Last revised on October 15, 2021 at 10:24:00. See the history of this page for a list of all contributions to it.