[Math] good general definition of “sheaves with values in a category”

ct.category-theoryhigher-category-theorytopos-theory

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 $[\mathcal{C}^\textrm{op}, \mathcal{A}]$.

Best Answer

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.)

Related Question