Let $\mathcal{A}$ be a category. There is a common definition of "sheaves with values in $\mathcal{A}$", which is what one obtains by taking the Grothendieck-style definition of "sheaf of sets" (i.e. in terms of presheaves satisfying a certain limit condition with respect to all covering sieves) and blithely replacing $\textbf{Set}$ with $\mathcal{A}$.

In my view, this is a bad definition if we do not assume $\mathcal{A}$ is sufficiently nice – say, locally finitely presentable. When $\mathcal{A}$ is locally finitely presentable, we obtain various properties I consider to be desiderata for a "good" definition of "sheaves with values in $\mathcal{A}$", namely:

  1. The properties of limits and colimits in the category of sheaves on a general site with values in $\mathcal{A}$ are "similar" to those of $\mathcal{A}$ itself. (I am being vague here because even when $\mathcal{A}$ is locally finitely presentable, the category of sheaves with values in $\mathcal{A}$ may not be locally finitely presentable – this already happens for $\mathcal{A} = \textbf{Set}$.)
  2. The category of sheaves on a site $(\mathcal{C}, J)$ with values in $\mathcal{A}$ is (pseudo)functorial in $(\mathcal{C}, J)$ with respect to morphisms of sites. (By "morphism of sites" I mean the notion that contravariantly induces geometric morphisms.)
  3. The construction respects Morita equivalence of sites, i.e. factors through the (bi)category of Grothendieck toposes.
  4. The construction respects "good" (bi)colimits in the (bi)category of Grothendieck toposes, i.e. sends them to (bi)limits of categories. (I don't know what "good" should mean here, but at minimum it should include coproducts. When $\mathcal{A}$ is locally finitely presentable, there is a classifying topos, so in fact the construction respects all (bi)colimits.)
  5. The category of sheaves on the point with values in $\mathcal{A}$ is canonically equivalent to $\mathcal{A}$.
  6. The category of sheaves on the Sierpiński space with values in $\mathcal{A}$ is canonically equivalent to the arrow category of $\mathcal{A}$.

Question. What is a (the?) "good" definition of "sheaves with values in $\mathcal{A}$"?

  • ... when $\mathcal{A}$ is finitely accessible, not necessarily cocomplete, e.g. the category of Kan complexes, or the category of divisible abelian groups?
  • ... when $\mathcal{A}$ is an abelian category, not necessarily accessible, e.g. the category of finite abelian groups, or the category of finitely generated abelian groups?
  • ... when $\mathcal{A}$ is a Grothendieck abelian category, not necessarily locally finitely presentable?

Perhaps something like right Kan extension along the inclusion of the (bi)category of presheaf toposes into the (bi)category of Grothendieck toposes might work – but the existence of toposes with no points suggests it may not – but it would be nice to have a somewhat more concrete description.

There is a temptation to strengthen desideratum 6 to require that the category of presheaves on a (small) category $\mathcal{C}$ be equivalent to the category of functors $\mathcal{C}^\textrm{op} \to \mathcal{A}$, but this does not appear to be a good idea. As Simon Henry remarks, if $\mathcal{C}^\textrm{op}$ is filtered, then the unique functor $\mathcal{C} \to \mathbf{1}$ is a morphism of sites corresponding to a geometric morphism that has a right adjoint (i.e. the inverse image functor itself has a left adjoint that preserves finite limits), so contravariant (bi)functoriality with respect to geometric morphisms forces the induced $\mathcal{A} \to [\mathcal{C}^\textrm{op}, \mathcal{A}]$ to have a left adjoint, i.e. $\mathcal{A}$ must have colimits of shape $\mathcal{C}^\textrm{op}$. This is the same argument that shows that the category of points of a topos must have filtered colimits.

Since I am looking for a construction where $\mathcal{A}$ is not necessarily the category of models of a geometric theory, I conclude that I cannot require the category of presheaves on $\mathcal{C}$ to be $[\mathcal{C}^\textrm{op}, \mathcal{A}]$.

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    $\begingroup$ I'm not sure I agree with your claim that the "naive" way of defining sheaves in a category does nto give you these properties. As long as you restrict to category with all limits they are essentially all satisfied (excepte maybe 1 that is a very vague). Of course, trying to defines sheaves with value in a category that does not have limits is probably a bad idea as the definition involve plenty of limits. $\endgroup$ – Simon Henry Apr 5 at 13:57
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    $\begingroup$ I started writting some details in an answer because it was too long for a comment. $\endgroup$ – Simon Henry Apr 5 at 14:10
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    $\begingroup$ I have a bad feeling about this question (although I like it). It's like chasing the elephant without a picture of the elephant. You might end up framing a tiger instead. Why isn't 5 replaced, more generally, by "the category of sheaves on a discrete category $X$, with discrete topology, is canonically equivalent to an $X$-fold product of copies of $\cal A$? $\endgroup$ – fosco Apr 5 at 17:18
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    $\begingroup$ Just as an interesting example, here is a "definition" that fits all your criteria: As discussed, wanting copower force filtered colimits to exists. Assuming that $\mathcal{A}$ has filtered colimits, then you can define for a topos $T$, $Sh(T,\mathcal{A})$ as the category of functors $Pt(T) \to \mathcal{A}$ that preserve filtered colimits (where $Pt(T)$ is the category of points of $T$, which always have directed colimits). This satisfies all your requirement (with restriction on (4)). Moreover, if finite limits and filtered colimit comute in $\mathcal{A}$, then the $f^*$ are left exact. $\endgroup$ – Simon Henry Apr 6 at 12:27
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    $\begingroup$ That is certainly an interesting construction! It shows that my list doesn't capture "cohesiveness": by going via the category of points, for example Hausdorff spaces become identified with their underlying set of points, which is definitely not desirable. Hmmm... $\endgroup$ – Zhen Lin Apr 6 at 13:02

In my view, the correct notion of "sheaf of Xs" is "internal X in the topos (or $\infty$-topos) of sheaves of sets (or spaces)". (I mentioned this previously on MO here.) Since sheaves of sets are a limit theory, if X is also defined by a limit theory (i.e. the category of Xs is locally presentable), then by commutation of limits this is the same as a sheaf of Xs in the naive sense. But for other values of X it gives different answers. In fact, the answer it gives may depend on exactly how the theory of X is presented; but that's reasonable becaues sometimes there is more than one correct notion of "sheaf of Xs" (equivalently, there is more than one version of X in the internal constructive logic of a topos). For instance:

  • If X = fields, there are discrete fields, Heyting fields, and residue fields. I think discrete fields are the one that corresponds to viewing fields as models of a limit-colimit sketch (i.e. as an accessible category), but the others are often more useful (e.g. Heyting fields include the sheaf of continuous real-valued functions on a topological space).
  • The case of X = Kan complexes has already been mentioned in other answers. Although in general once you're talking about homotopy theory, it's better to incorporate the homotopy theory into the ambient $\infty$-topos and work with stacks.
  • If X = finite abelian groups, there are different notions of finite object in a topos.
  • If X = topological spaces, you can internalize that directly, but often more useful is to internalize the notion of locale -- for instance, a "sheaf of locales" on a sufficiently nice topological space $Y$ is equivalent to a space over $Y$.
  • If X = local rings, written as a geometric theory, this definition gives you the generally accepted definition of "sheaf of local rings", i.e. a sheaf of rings whose stalks are local.

This definition of "sheaf of Xs" satisfies your criteria (3) and (5). It also satisfies your criterion (1) in as strong a way as I think could be expected: the category of internal Xs in a topos behaves exactly like the ordinary category of Xs, as long as the latter is interpreted using constructive logic. And it satisfies your criteria (2), (4), and (6) if the theory of Xs is geometric, hence has a classifying topos -- which I think is the most general situation in which one can expect these properties to hold.

(Note, by the way, that your criterion (6), as well as the stronger version referring to all presheaf toposes, is a special case of your (4), since presheaves on $C$ are the Cat-enriched copower of the terminal topos by $C$ in the bicategory of toposes.)

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    $\begingroup$ It should be noted however that in the case where condition (4) & (6) are satisfied, it is for a notion of morphisms that is forced upon you by the definition of objects you used. For example, for finite sets (or finite group) it is going to be either surjection between finite set (using Kuratowski finitness) or bijection between finite sets only (using "cardinal finite") and there is no way to have finite sets/groups with all morphisms between them from a geometric theory. $\endgroup$ – Simon Henry Apr 5 at 17:26
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    $\begingroup$ I, too, am of the opinion that the correct definition is supplied by internal logic. The difficulty with this approach is that it is "intensional", in the sense that the X in "sheaves of X" needs to have an a priori definition in terms of logic (whether the traditional kind or the (co)limit sketch kind); what I am hoping for is an "extensional" approach where X can be a "black box" category. As you say, going from "intensional" to "extensional" is lossy, but perhaps there is a canonical "intension" for every "extension" that is maximal or minimal or otherwise universal in some sense. $\endgroup$ – Zhen Lin Apr 5 at 22:41
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    $\begingroup$ @SimonHenry Indeed! I believe this is an issue for local rings too -- you don't get the "local homomorphisms" from a classifying topos. $\endgroup$ – Mike Shulman Apr 5 at 23:34
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    $\begingroup$ Curiosity and, to be honest, a wish for a good general definition that could be presented to "ordinary" mathematicians who seem to be averse to "logic". (This is perhaps stretching the notion of "ordinary" a bit.) The fact that the naïve definition is repeated in many places without any caveats is quite frustrating for me but I wonder if that is because no alternative has appeared. $\endgroup$ – Zhen Lin Apr 5 at 23:51
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    $\begingroup$ @ZhenLin: Sure, calling it “intensional” makes it sound like something scary from logic. But if instead you say “this notion of $A$-valued sheaves requires extra structure on $A$, and can depend on the choice of this structure”, I don’t think most mathematicians will be scared at all — that’s a very familiar phenomenon when generalising. $\endgroup$ – Peter LeFanu Lumsdaine Apr 6 at 2:56

The naive definition of sheaves is very well behaved if you look at functoriality in the $f_*$ direction: Of course, you are going to need to assume that $\mathcal{A}$ has all limits as the definition of $\mathcal{A}$-valued sheaves involves arbitrary limit.

If you want to restrict to category that have for example finite limits, you are going to have to restrict to sites that have only have finite cover, and to geometric morphisms that satisfies some finiteness conditions.

Once you assume that $\mathcal{A}$ has finite limits, it works pretty much without any problems. In fact it has little to do with Grothendieck topologies and works well for arbitrary "limit sketches".

Definition: (maybe not completely standard terminology) By a "limit sketches" I mean a small category $\mathcal{C}$ together with $S$ a set of maps in the category $\widehat{C}$ of presheaves of sets on $\widehat{C}$.

A site is a special case with $S$ the set of covering sieves.

Given $\mathcal{E}$ a category with colimits, any functor $f:C \to \mathcal{E}$ induces an adjunction $f_! \dashv f^*$ where $f^*$ is the nerve functor $\mathcal{E} \to \widehat{C}$ and $f_!:\widehat{C} \to \mathcal{E} $ is the pointwise left kan extention of $f$.

Definition: A $(C,S)$-comodel in $\mathcal{E}$ is a functor $f:C \to \mathcal{E}$ such that $f_!$ sends all maps in $S$ to isomorphism in $\mathcal{E}$, or equivalently such that for all $X \in \mathcal{E}$, $f^* E$ is orthogonal to all maps in $S$.

Now, if $\mathcal{A}$ is a category with all limits, then a $(C,S)$-model is $\mathcal{A}$ is a a $(C,S)$-comodel in $\mathcal{A}^{op}$.

In particular, the category of $(C,S)$-model in $Set$ is the full subcategory of presheaf on $C$ that are orthogonal to all maps in $S$. I'm denoting this category by $Set(C,S)$

Propostion: For all category $\mathcal{E}$ with all colimits, the category of $(C,S)$-comodel in $\mathcal{E}$ is equivalent to the category of left adjoint functors (equivalently, colimit preserving functors) $Set(C,S) \to \mathcal{E}$.

Indeed functor $C \to \mathcal{E}$ corresponds to left adjoint functors $\widehat{C} \to \mathcal{E}$ and by definition $(C,S)$-model corresponds exactly to the ones that inverts all maps in $S$, but by classical manipulation, this is the same as left adjoints functors $Set(C,S) \to \mathcal{E}$. As $Set(C,S)$ is locally presentable, this is also the same as colimit preverving functor.

You get your functoriality requirement:

Corollary: The category of $(C,S)$-(co)model in a category with all (co)limit is functorial on all functor preserving all (co)limits.

In particular, it is functorial on site and Geometric morphisms.

Indeed, for comodel it it justs the functor represented by $Set(C,S)$ in the category of all colimits preserving functor between cocomplete category. It follows for model by duality.

  • $\begingroup$ Since desideratum 1 is vague let me ask a concrete question. Suppose $\mathcal{A}$ is an AB3 abelian category. Does your construction yield abelian categories back? Are the functors induced by geometric morphisms exact? $\endgroup$ – Zhen Lin Apr 5 at 14:16
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    $\begingroup$ So, if I is filtered then $Colim : [I,Set] \to Set$ is left exact, so is a geometric morphism. I think your requirement forces this to interpret to $Colim: [I,A] \to A$. But asking for this to be left exact is already condition $(AB5)$. So I don't think your initial requirement are compatible with this. $\endgroup$ – Simon Henry Apr 5 at 14:52
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    $\begingroup$ That is a very good point. I should have realised it myself. Functoriality + copowers already forces $\mathcal{A}$ to have filtered colimits. But it is perfectly possible to define "sheaves of finite abelian groups" for sites with enough points so that must mean the construction doesn't preserve copowers... $\endgroup$ – Zhen Lin Apr 6 at 0:22

This is a very complicated question.

Categories of Sheaves. Let me start from something that you evidently know (given the hidden references in your question). Bourceux et al. have worked on defining sheaves of something over a site $(C,J)$.

  • Borceux, Sheaves of algebras for a commutative theory, Ann. Soc. Sci. Bruxelles Sér. I 95 (1981), no. 1, 3–19
  • Borceux and Kelly, On locales of localizations, J. Pure Appl. Algebra, Volume 46, Issue 1, 1987, Pages 1-34.
  • Borceux and Veit, On the Left Exactness of Orthogonal Reflections J. Pure Appl. Algebra, 49 (1987), pp. 33-42.
  • Borceux, Subobject Classifier for Algebraic Structures. Subobject classifier for algebraic structures J. Algebra, 112 (1988), pp. 306-314.
  • Veit, Sheaves, localizations, and unstable extensions: Some counterexamples. J. Pure Appl. Algebra, Volume 140, Issue 2, July 1991, Pages 370-391.
  • Borceux and Quinteiro. A theory of enriched sheaves. Cahiers de Topologie et Géométrie Différentielle Catégoriques 37.2 (1996): 145-162.

As you mention, this theory works at its best when $\mathcal{V}$ is a regular locally finitely presentable monoidal closed category (these are the working hypotheses of the last reference in the list above). This is partially satisfactory, as in these instances we recover a nice correspondence between topologies and lexreflective localizations, and thus we can maintain the intuition for the theory of sites and lex-reflectors. On the other hand, this theory does not even recover the notion of sheaves over a topos $\mathsf{Sh}((C,J), \mathcal{E})$, as many topoi are not locally finitely presentable.

An idea. I never checked, but I always believed that most of these results can be recovered when $\mathcal{V}$ is a cocomplete precontinuous category (in the sense of Adamek, Rosicky and Vitale, see On Algebraically Exact Categories and Essential Localizations of Varieties) with a dense generator (and of course monoidal closed). This assumptions would recover the topos case and also the case of Grothendieck categories, where precontinuity is known as (AB5). (This framework would also meet all your desiderata). Of course this idea is not entirely satisfying for you, as it would not encompass those finitely accessible categories that are not cocomplete, but at least provides a good framework to study sheaves over a cocomplete precontinuous monoidal closed category with a dense generator (which, again, include topoi and Grothendieck categories).

Grothendieck topoi. The first dishonest way to answer your question is to say that you are looking for a way to describe lex-reflective $\mathcal{V}$-categories. This is the over-formalist point of view of who thinks that lex-reflectivity of categories of sheaves is not a theorem in the theory of topoi, it is a very intrinsic characterization and should be taken as a definition. I am not sure that I support this idea.

Anyway, if one wants to follow this path, there is the beautiful paper by Garner and Lack.

Garner and Lack, Lex Colimits. J. Pure Appl. Algebra. Volume 216, Issue 6, June 2012, Pages 1372-1396.

As the assumptions on $\mathcal{V}$ are very mild in this case, one cannot expect to recover a good theory of sites, and thus the very intuition of sheaf is a bit lost. Still, depending on your religious belief, this could be a starting point.

Caveat 1. I think that this tentative solution unveils the first problem of your question. If you do not choose a flavour of problems that you want to solve, or attack with this notion of sheaf, it's hard to come with a correct definition which does not rely on a specific point of view.

Caveat 2. Even the formalists that are fashinated by this approach, should be warned by the evidence. In the case of Grothendieck categories the correct notion of morphism is not that of left exact cocontinuous functors, as discussed in the very introduction of a paper of Ramos Gonzalez and myself.

Di Liberti and Ramos Gonzalez, Exponentiable Grothendieck categories in flat algebraic Geometry. arXiv:2103.07876.

So one should be very careful with getting carried by this point of view.

Elementary topoi and finite abelian groups. I was always fascinated by a very natural way to produce elementary topoi. The category of functors $\mathsf{FinSet}^C$ is an elementary topos, and when $C$ is a finite category it is even a Grothendieck topos with respect to the Grothendieck universe of finite sets. This case is very similar to that of finite abelian groups. So, are you honestly changing the notion of sheaf, or just dishonestly changing the notion of size? I think that this is a question to think about before forcing a definition that we might not need.

Kan complexes.You listed Kan complexes among finitely accessible categories that are not cocomplete, but is this the correct point of view on them? Kan compleses are indeed cocomplete with respect to the relevant notion of colimit, and if you want to see this cocompleteness, you should take sheaves over simplicial sets and study model topoi in the sense Rezk.

Rezk, Toposes and homotopy toposes.

All in all. I do not have a good answer to your question, but I have discussed a couple of remarks that I hope will thicken the debate. But I have a very informal question for you, what are the defining and conceptual features of the notion of sheaf that you want to model? Personally, I find your list of desiderata nor defining nor conceptual.

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    $\begingroup$ Sheaves of Kan complexes were considered in Ken Brown's paper introducing categories of fibrant objects. He gives (what I consider to be equivalent to) the correct definition: a sheaf of Kan complexes on a topological space is a simplicial sheaf whose stalks are Kan complexes. This generalises to internal Kan complexes in a topos. $\endgroup$ – Zhen Lin Apr 5 at 14:02
  • $\begingroup$ Yes, but isn't it the same of the theory of model topoi in the sense of Rezk? $\endgroup$ – Ivan Di Liberti Apr 5 at 14:04
  • $\begingroup$ Sure, for the purposes of actually doing homotopy theory, there are better models. But it illustrates that there is (1) a good definition of sheaves of objects that is not just the naïve one and (2) works well for categories that are not necessarily complete or cocomplete. $\endgroup$ – Zhen Lin Apr 5 at 14:07
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    $\begingroup$ To answer your question... I think categories of sheaves of <whatever> should form a stack on the category of Grothendieck toposes. For structures that are axiomatised by geometric theories, we get representable stacks, and furthermore the pullback functors have good properties in regards to (co)limits. My desiderata mostly stem from this observation. $\endgroup$ – Zhen Lin Apr 6 at 10:17
  • $\begingroup$ Thanks for this clarification, it's an interesting insight. I will think about it in the next days and hopefully come back to the question. $\endgroup$ – Ivan Di Liberti Apr 6 at 10:20

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