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I am faced with a context in which the most natural notion of a sheaf $\mathcal F$ is as a functor on the category of compact subsets of a (locally compact Hausdorff) space $X$. Specifically, I say a presheaf $\mathcal F$ is a sheaf iff it satisfies the three axioms: \begin{align} \mathcal F(\varnothing)&=0\\ 0\to\mathcal F(K_1\cup K_2)\to\mathcal F(K_1)\oplus\mathcal F(K_2)\to\mathcal F(K_1\cap K_2)&\text{ is exact }\forall\text{ }K_1,K_2\subseteq X\\ \varinjlim_{\begin{smallmatrix}K\subseteq U\cr U\textrm{ open}\end{smallmatrix}}\mathcal F(\overline U)\to\mathcal F(K)&\text{ is an isomorphism }\forall\text{ }K\subseteq X \end{align}

I would very much like to not have to rewrite the entire foundations of sheaf theory in this context! I'm not sure how much the reader would appreciate such an exposition either. Granted, I would only need to write the foundations I am using, but even this turns out to require many proofs which in retrospect are mostly trivial applications of compactness and direct limit arguments.

Is there a good reference for sheaves on the category of compact subsets somewhere in the literature? I hope that there is a book which develops the relevant foundations (perhaps by comparing this notion to the more familiar notion of sheaves on open subsets) so I don't have to include too much baggage explaining the basics of such sheaves.

To be more specific about what I mean by "the basics of sheaf theory", I mean something like:

  • Sections are determined by their stalks
  • Functors $i_\ast$ and $j_!$ (for inclusions of closed and open subsets respectively)
  • A theory of Cech cohomology

I don't need anything about injective resolutions or making the category of sheaves into an abelian category.

EDIT: I withdraw this question because the category of sheaves I define above turns out to be equivalent to the (usual) category of sheaves via the functors: \begin{align} (\alpha_\ast\mathcal F)(K)&=\varinjlim_{\begin{smallmatrix}K\subseteq U\cr U\textrm{ open}\end{smallmatrix}}\mathcal F(U)\\ (\alpha^\ast\mathcal F)(U)&=\varprojlim_{\begin{smallmatrix}K\subseteq U\cr K\textrm{ compact}\end{smallmatrix}}\mathcal F(K) \end{align}

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Could you spell out what you mean by (satisfying the sheaf property for finite unions of compact sets, plus an extra "continuity" condition)? –  David Carchedi May 17 '13 at 0:18
    
@David Carchedi: sure, I edited it to give the precise definition. –  John Pardon May 17 '13 at 1:06

1 Answer 1

Let's label the three conditions you wrote as 1'), 2'), and 3)'. Combine 1') and 2') by saying for any finite cover by compact subsets... (I hope you follow what I mean). Also, in 3)', you better mean that $\bar U$ is compact, otherwise it doesn't make sense.

So now we're reduced to 1) and 2). Also, let's not talk about sheaves of abelian groups, but sheaves of sets. Sheaves of abelian groups are just abelian group objects in the category of sheaves of sets, so there is no harm to start with the latter. Now, if you want to say you are doing "sheaf theory" you better actually have a Grothendieck topology. Here is what you can do in your situation:

Given your space $X$, define the category $K\left(X\right)$ as the poset of of compact subsets of $X$ and their inclusions. You can define a Grothendieck pre-topology, by saying a covering family is a finite family of jointly surjective inclusions. Being a sheaf for this topology is equivalent to condition 1) (i.e. 1') and 2)'). Let us call this topology $J$.

Consider the category $O_c\left(X\right)$ of open subsets of $X$ which have compact closure. Since $X$ is locally compact, these form a basis for $X$. There is a Grothendieck pre-topology on this category which is the usual one (restricted to this subcategory), except we only allow finite covering families. Sheaves for the associated Grothendieck topology will in general not be sheaves on $X$ in the classical sense, unless $X$ is compact. However, given a sheaf $F$ on $K(X)$, we can define $F(U)$ for a $U$ in $O(X)_c$ by $\varinjlim F(K)$ running over all compacts containing $U,$ but note, this is the same as $F(\bar U),$ since the poset of compact subsets containing $U$ has $\bar U$ as a terminal object. Note, we may also just simply remark that there is a functor $$cl:O_c(X) \to K(X)$$ induced by taking closures and what we have done is defined $cl^*F$. Then, 1) implies that $cl^*F$ is a sheaf for finite open covers as well, since $X$ is locally compact. It would seem that this implies 2) follows automatically, however I haven't checked carefully.

EDIT: This fails in general! Condition 2) is equivalent to for all $F$ sheaves on $K(X)$, $cl_!cl^*F \cong F,$ i.e. the co-unit of the adjunction $$cl_! \dashv cl^\star$$ needs to be an iso on $Sh(K(X))$, which is if and only if $cl^{\star}$ is full and faithful when restricted to $Sh(K(X)).$ Since $cl$ is itself full and faithful, it implies that $cl_*$ is, and hence we get that $F$ must lie in the image of the pullback topos $$Sh(O_c(X)) \times_{Set^{K(X)^{op}}} Sh(K(X))$$ in $Sh(K(X))$. Concretely, this means there is a finer Grothendieck topology $J'$ on $K(X)$ which does the trick. It is possible to write down explicitly, what the covers are, but I will not attempt to do so here. To get an idea of how to do this though, look at how I construct the "compactly generated Grothendieck topology" here: http://arxiv.org/abs/0907.3925

*The following should work with either $J$ or $J'$ *

If all this is right, then all you are talking about is a sheaf on $K(X)$ with the Grothendieck topology I have mentioned. Given such a sheaf $F,$ if you restrict to a compact $C$, $F|C$ defines an ordinary sheaf on $C,$ by extending it to opens the way I described. Since every point $x$ in $X$ has a compact neighborhood, it should follow that sections are determined by their stalks, by reducing to the case of ordinary sheaves on compact spaces.

Now, if $i$ is a closed inclusion, $i:V \to X,$ then $i$ induces a functor $$i^{-1}:K(X) \to K(V),$$ by intersecting with $X.$ This functor preserves covers, so the functor $$i_*:Sh(K(V)) \to Sh(K(X)),$$ defined by $$i_*F(C)=F(C \cap V)$$ for all compact $C$ in $X,$ is well defined. It has a left adjoint $i^{\star},$ which is simply given by procompositing with $i.$

Note: The following probably fails now, with the new topology:

I am not sure how to get $j_!$ for open inclusions, since for open covers this is usually induced by the inclusion functor $O(U) \to O(X),$ but we have no inclusion functor $K(U) \to K(X).$ BUT, we do have an inclusion functor $$K(V) \to K(X);$$ it is right adjoint to the functor $i^{-1}.$ It also preserves covers, so we get an induced functor $$i_!:Sh(K(V)) \to Sh(K(X))$$ by left Kan extension. It is in fact left adjoint to $i^*$ so, from a closed inclusion we get both $i_*$ and $i_!.$

Cech cohomology is automatic, since it makes sense in any topos. This is always true

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The upshot is, your definition is equivalent to being a sheaf on $K(X)$ for a certain Grothendieck topology (which I only described implicitly), so it lies in the same world of ordinary sheaves- in topos theory. For many things, you can probably dance around without talkig about this topology, and just use the definition you gave, except, if you want to define Cech cohomology, you will have to bite the bullet and define the covers explicitly. However, you should probably use sheaf cohomology anyway, instead. –  David Carchedi May 17 '13 at 21:36
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P.S. "I don't need anything about injective resolutions or making the category of sheaves into an abelian category."- Well, you have it even if you don't need it :) –  David Carchedi May 18 '13 at 1:28

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