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Let $C$ be a category with an initial object. Then what is the initial object of $Psh(C)$? Is it the representable functor $h_{0_C}(-):=Hom_C(-,0_C)$ or is it the empty presheaf $\emptyset(-)=\emptyset$? What if $C$ has no initial object?
Thinking about it in terms of initial objects, it seems like $\emptyset(-)$ should be the initial object, but thinking about it in terms of free cocompletions, the colimit over the empty diagram seems like it should just be the initial object.
It’s the empty presheaf. The key word in “free cocompletion” is free, which means that it ignores any colimits that might already exist in $C$, therefore generally destroying the fact that they are colimits.
Ah, is there any construction taking a category C to its minimal cocompletion (if that means anything)?
I don’t think it means very much. One thing you can do is if C has some specified collection of colimits, then you can look at the subcategory of $[C^{op},Set]$ consisting of those functors which preserve those given colimits (i.e. take them to limits in Set). That’s a reflective subcategory, hence cocomplete, and C embeds in it in a way which preserves the specified colimits. (The category of sheaves for a Grothendieck topology on C is a special case of this.) So, for example, if you look at the subcategory of presheaves preserving the initial object of C, in that category the initial object will be the presheaf represented by the initial object of C. I suppose for a “minimal” cocompletion you could look at all the colimits which C admits, assuming that C is small.
I suppose for a “minimal” cocompletion you could look at all the colimits which C admits, assuming that C is small.
Am I right in thinking that the category of presheaves preserving these is the category of sheaves for the canonical topology on $C$?
@Mike:
Yes, exactly (the last thing you said). Would such a cocompletion be idempotent, or could we not make heads or tails of it without pushing up to a higher universe (which would effectively ruin our plans for idempotence).
@FinnLawler: I don’t think so. Suppose C is not cocomplete. Then the thing Mike just mentioned will be cocomplete.
@Harry: what I mean is that you can think of a covering sieve as specifying a cocone that should become a colimit cocone in the sheaf category. The $J$-covering sieves are already colimits in $C$ iff $J$ is subcanonical. So I’m thinking that the canonical topology will be given by all of the colimit cocones in $C$, and so the category of sheaves will be that cocompletion of $C$ in which all existing colimits in $C$ are still colimits. Mac Lane and Moerdijk say (pp. 489–490) that this is true for locales (i.e the canonical topology is given by $J U = \{ \{U_i \to U\}_i \mid \bigvee_i U_i = U\}$).
But wait, isn’t the category of sheaves in the canonical topology equivalent to C?
I always thought it was, but if I’m wrong, I guess one learns something new every day.
isn’t the category of sheaves in the canonical topology equivalent to C?
Yes, for C a Grothendieck topos (see canonical topology). But I don’t think this is true for non-cocomplete C.
Coming back to the original question:
just in case there is any confusion on this point, maybe I can emphasize that checking that the presheaf constant on an initial object is initial is straightforward and elementary: just look at the naturality square
$\array{ \emptyset &\to& F(x) \\ \downarrow^{Id} && \downarrow \\ \emptyset &\to& F(y) }$@Harry: The cocompletion I described only makes sense as stated when C is small. If C is large, then you can either pass to a larger universe, or consider the category of small presheaves on it; the latter lives in the same universe as C and is its free cocompletion there. I don’t know whether you can look at “colimit-preserving small presheaves” to get a possibly-idempotent cocompletion operation; it might be possible.
@Finn: Not all colimit-preservation properties can be expressed in terms of sheaf conditions for a topology. In particular, any locally presentable category is the category of presheaves preserving some specified class of colimits on a small category, but of course not every locally presentable category is a topos.
@Mike: OK, thanks. I’ll think about this a bit more.
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