# ct.category theory – Is there a good general definition of “sheaves with values in a category”?

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.
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 $$mathcal{A}$$.

(If I’m not mistaken, assuming a sufficiently strong form of desideratum 4, desiderata 5 and 6 force the category of presheaves on $$mathcal{C}$$ with values in $$mathcal{A}$$ to be equivalent to the category of functors $$mathcal{C}^textrm{op} to mathcal{A}$$.
I am not sure whether this should be an explicit desideratum.)

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?
• … when $$mathcal{A}$$ is an abelian category, not necessarily accessible, e.g. the category of finite abelian groups?
• … when $$mathcal{A}$$ is a Grothendieck abelian category, not necessarily locally finitely presentable?

I imagine one 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 it would be nice to have a somewhat more concrete description.