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Presheaves on a category $C$ form the free co-completion, in the sense that every functor from $C$ to a cocomplete category extends in a unique way to the presheaf category.

If $C$ is equipped with a Grothendieck topology, then a similar statement holds if we ask that our cocontinuous extension also send some diagrams formed out of coverings to colimits.

A rough but pleasant analogy is that the presheaf category is like the free abelian monoid on a set with colimits replacing sums, and the sheaf category is what you get by quotienting by some linear relations.

From this point of view it is obvious that there must be a cocontinuous functor from presheaves to sheaves, which of course is sheafification. However from this perspective it seems somewhat surprising to me that the category of sheaves embeds as a full subcategory of presheaves.

Is there some reason to expect this to be the case from thinking only about cocompletions and related abstract nonsense?

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    $\begingroup$ Is it easy to see from this point of view that PSh ---> Sh is a localization? In the sense that there is an adjoint which is fully faithful (there might even be a slicker way of phrasing this). $\endgroup$ – Yosemite Sam Dec 1 '17 at 1:42
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    $\begingroup$ The adjoint functor theorem... $\endgroup$ – David Roberts Dec 1 '17 at 6:30
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    $\begingroup$ I've always found it similarly striking that the free completion of $C$ under $J$-colimits for some class $J$ of small diagrams is always computed as a full subcategory of the category of presheaves on $C$, namely the closure of the representables under $J$-colimits. $\endgroup$ – Tim Campion Dec 1 '17 at 19:07
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The universal properties you describe (by the way, you need to either restrict to the case that $C$ is essentially small or restrict to what are called small presheaves) tell you what cocontinuous functors out of sheaves or presheaves look like. But the embedding $\text{Sh}(C) \to \text{Psh}(C)$ of sheaves into presheaves isn't cocontinuous so the universal property isn't applicable here.

To my mind, the general point here is that if $F : C \to D$ is a functor and $D$ is a cocomplete category then the induced functor $\text{Psh}(C) \to D$ is always the left adjoint of a functor $D \to \text{Psh}(C)$, namely the restricted Yoneda embedding

$$D \ni d \mapsto \text{Hom}(F(-), d) \in \text{Psh}(C).$$

Taking $D = \text{Sh}(C)$ and $F : C \to \text{Sh}(C)$ the Yoneda embedding we get the right adjoint of sheafification, which is the inclusion of sheaves into presheaves. I suppose there's an additional question of why this adjoint is fully faithful.

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  • $\begingroup$ This is exactly what I was looking for. Thanks. $\endgroup$ – eon Dec 1 '17 at 6:44
  • $\begingroup$ I guess in light of Simon's answer I should restrict myself to the case that $C$ is essentially small to be safe. $\endgroup$ – Qiaochu Yuan Dec 1 '17 at 17:39
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One small additional remark to Qiaochu Yuan response and David Roberts comment to show that it is really the existence of an adjoint that is the important point here. (and that was really too long for a comment)

We can look at a more general situation, where the existence of an adjoint might fail: If $C$ is a non-small category, the "free co-completion of $C$" still exists: it is the category of presheaf over $C$ that are small colimits of representable (the presheaf of small co-finality). I will call them the "small presheaves" they always form a locally small category.

If I have a topology on $C$ then I can also try to construct a category of "small sheaves" that has the universal property you mentioned: I take the category of sheaves over $C$ which are the sheafification of small presheaves.

Depending on the topologies, several things might go wrong: the sheafification can be undefined, or one can get a non locally small category.

But for a well chosen topology it can happen that sheaficiation of small presheaves exists and that you get a locally small category this way. In this case this would be the solution to your universal problem.

In this situation the category of "small sheaves" that you constructed in still included in the category of all presheaves but is not necessarily included in the free co-completion. So you do have a sheafication functor from from the "free co-completion" to the "free co-completion with relation" as expected, but you no longer get this surprising functor in the other direction.

Let me give an explicit example where all of this happens:

Take $C$ to be the category of all ordinals (with morphisms given by the order relation) with the atomic topology (every non-empty sieve is a covering, in this case this is the same as saying that every map is a covering).

Then one can check the following:

1) A presheaf is an ordinal indexed collection of set with transition map $F(x) \leftarrow F(y)$ for $x<y$.

2) A presheaf on this category is small (a small colimit of representable) if and only it is empty after a certain rank.

3) A sheaf for this topology is a constant pre-sheaf. So no sheaf is a small presheaf.

4) The sheafification functor takes a presheaf $F$ to the constant presheaf whose value is the union of all the $F(a)$ quotiented by the action of the transition maps. It is well defined at least on the category of "small presheaf". and all sheaves are sheafification of a small presheaves.

5) Moreover, as expected, the category of small presheaves is universal among co-complete categories for functors $C \rightarrow E$ while the category of "small sheaves" is universal for functor $C \rightarrow E$ which send all maps to isos.

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  • $\begingroup$ In fact, my opinion is that in general, "the category of small sheaves" ought to be defined by this universal property, whether or not it is locally small or can be embedded into any category of presheaves; see tac.mta.ca/tac/volumes/27/7/27-07abs.html . $\endgroup$ – Mike Shulman Dec 3 '17 at 3:40
  • $\begingroup$ @Mike : Is that different from what I said ? I had the impression that the solution to this problem would always be the image of the category of small presheaves under the "sheafification functor" (which always exists if we look at presheaves with value in "large sets"), with the only problem that this might not be a locally small (in which case I would be hesitant to talk about small sheaves as i would prefer avoid having a "non locally small category of small widget", but that is only a matter of taste) $\endgroup$ – Simon Henry Dec 3 '17 at 13:08
  • $\begingroup$ Probably that's right; your phrasing in the answer suggested that you thought it might not always work to construct a category of small sheaves that way. $\endgroup$ – Mike Shulman Dec 3 '17 at 17:57
  • $\begingroup$ Oh I see, I only mean't that you can fail to get a solution as a locally small category. $\endgroup$ – Simon Henry Dec 3 '17 at 19:49

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