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I've read about free cocompletion of categories discussing on the adjunction between Cat and cocompleteCat (Cat: category of small categories, cocompleteCat: category of small cocomplete categories and cocontinuous functors) where the adjunction is about a free cocompletion functor F which sends each category C to PSh(C), the category of its presheaves. (http://ncatlab.org/nlab/show/free+cocompletion)

I wonder if the functor F have a left adjoint (even with some restrictions on its domain) and how it could be described.

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  • $\begingroup$ Your description of the target is incorrect. Categories of presheaves are almost never small, and in fact Freyd showed that a small cocomplete category is a preorder. There are genuine set-theoretic difficulties here; e.g. because of the point I just mentioned it's not at all obvious what the forgetful functor is supposed to be. $\endgroup$ – Qiaochu Yuan May 2 '15 at 20:34
  • $\begingroup$ This does not answer your question in any way. Just want to direct interested readers to the note Sheaves and Homotopy Theory by Daniel Dugger, where the cocompletion business is explained in very accessible terms with lots of intuition. Specifically, $F$ and its right adjoint, the forgetful functor from cocomplete categories to arbitrary categories, are discussed. The size issues which Qiaochu refers to are ignored though. $\endgroup$ – Ingo Blechschmidt May 2 '15 at 21:33
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    $\begingroup$ @Ingo: the size issues are genuine here and ignoring them makes it easy to say things that are false. If $C$ is not (essentially) small, it is not true, as Dugger appears to be claiming, that presheaves on $C$ is the free cocompletion of $C$ (provided we agree that colimits, by default, should be over small diagrams; if we don't, then it is not true that presheaves form a cocomplete category. Freyd's argument above also shows that a category which admits all large colimits is a preorder). $\endgroup$ – Qiaochu Yuan May 2 '15 at 21:44
  • $\begingroup$ Dugger's proof is incorrect because he never specifies whether colimits are supposed to be over small diagrams or over all diagrams, so he gets to say both "presheaves form a cocomplete category" (true over small diagrams) and "every presheaf is a colimit of representables" (true over large diagrams). The proof is correct if $C$ is (essentially) small; it establishes a universal property but does not describe an adjunction. $\endgroup$ – Qiaochu Yuan May 2 '15 at 21:51
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    $\begingroup$ The forgetful 2-functor from large categories with small colimits to large categories has a genuine left adjoint which sends a small category C to PSh(C). But this left adjoint is not a right adjoint: PSh(*) is not final, since it has many colimit-preserving endofunctors. $\endgroup$ – Marc Hoyois May 2 '15 at 22:02
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The free cocompletion functor $C \mapsto \widehat{C}$ (which, as Zhen Lin says, does not agree with the presheaf functor when $C$ is not essentially small) should in no reasonable sense have a left adjoint, since it is very far from preserving limits.

It already fails to preserve products: if $C, D$ are two small categories, then $\widehat{C \times D}$ is the "cocomplete tensor product" of $\widehat{C}$ and $\widehat{D}$, which is neither the product nor the coproduct in cocomplete categories. (This is closely analogous to the way in which the free abelian group functor sends a product to a tensor product, which is neither the product nor the coproduct in abelian groups.) In fact the product of $\widehat{C}$ and $\widehat{D}$ is $\widehat{C \sqcup D}$. So we already get a counterexample by taking $C$ and $D$ to be discrete categories with $3$ objects, say.

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  • $\begingroup$ the counterexample is good. thanks. $\endgroup$ – Sn K May 3 '15 at 4:15
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Here is the correct statement:

Let $\mathfrak{Cat}$ be the 2-category of locally small categories (and all functors) and let $\mathfrak{Cocomp}$ be the 2-category of locally small cocomplete categories (and all cocontinuous functors). Then the forgetful functor $\mathfrak{Cocomp} \to \mathfrak{Cat}$ has a left biadjoint. More precisely, for every locally small category $\mathcal{C}$, there is a locally small cocomplete category $\hat{\mathcal{C}}$ and a functor $\mathcal{C} \to \hat{\mathcal{C}}$ such that the induced functor $$\mathfrak{Cocomp} (\hat{\mathcal{C}}, \mathcal{D}) \to \mathfrak{Cat} (\mathcal{C}, \mathcal{D})$$ is an equivalence of categories for all locally small cocomplete $\mathcal{D}$.

In the case where $\mathcal{C}$ is essentially small, $\hat{\mathcal{C}}$ is equivalent to $[\mathcal{C}^\mathrm{op}, \mathbf{Set}]$; in general, it is a certain full subcategory.

(Note that we may take "locally small" in the weak sense of "hom-sets are small sets", without any condition on the set of objects.)

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In the case where $C$ is essentially small, $\hat{C}$ is equivalent to $[C^\text{op}, \mathbf{Set}]$; in general, it is a certain full subcategory.

Let me make this statement precise, as it is actually the tricky part.

A presheaf $H\colon C^\text{op} \to Set$ is called small if $H = \mathrm{Lan}_JF$ for some full inclusion $J\colon D \hookrightarrow C^\text{op}$. Equivalently, a presheaf is a small colimit of representables (in a bigger universe). Roughly speaking, $H$ is determined by a small full subcategory of $C$.

Then, the category $\mathcal{P}C$ of small presheaves with natural transformations forms the free cocompletion of $C$ via the usual Yoneda embedding. This gives you the left biadjoint to the forgetful functor $U\colon \mathbf{ConCAT} \to \mathbf{CAT}$ where $\mathbf{ConCAT}$ is the 2-category of large categories with colimits as objects, cocontinuous functors as 1-cells, and natural transformations as 2-cells; $\mathbf{CAT}$ the 2-category of large categories. In addition, the forgetful functor is also (pseudo-)monadic. The pseudomonad can be a strict 2-monad if a choice of colimits is given.

Small functors were introduced as accessible functors in (Kelly, 1982), and the monadicity of cocompletion can be found in (Kelly & Lack, 2000).

  • Kelly, Basic Concepts of Enriched Category Theory, 1982
  • Kelly and Lack, On the monadicity of categories with chosen colimits, 2000
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