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Sheaves can, like all modern mathematical constructions and abstractions, be counterintuitive beasts but, like all such constructions, a few examples can allow one to visualise them simply as a generalisation of a natural object (ie. the sets of local functions on a topological space).

I have recently become rather interested in cosheaves- a sort of natural iteration of the sheaf concept- apply the sheaf functor once, and it's contravariant, twice (being terribly careful about covers being preserved in a sensible way) and one gets a covariant construction.

My problem is that this is rather hard to anchor to anything really natural as 'sheaves over sheaves' seems rather difficult to visualise without tying ones brain in knots. So my question, in its broadest terms is 'what do cosheaves look like?' (are they for example, just $\mathcal{F}^{op}$ to some sheaf $\mathcal{F}$ by analogy with various other 'co-constructions'), but more realistically (assuming that I am making assumptions) it is "Can one find a canonical example of a cosheaf that is 'visualisable' in some sense?"

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1 Answer 1

So you are probably requiring a more general sort of cosheaf -- I don't really know what you mean by "running the sheaf functor twice" -- but let me maybe provide a simpler description of cosheaves that might answer your question.

One definition of a cosheaf is simply a functor

$F:\mathrm{Open}(X)\to\mathcal{C}$

that sends colimits (unions) to colimits. By formality, we can say this is a sheaf valued in the opposite category

$F:\mathrm{Open}(X)^{op}\to\mathcal{C}^{op}.$

There are a couple of "canonical" examples of cosheaves. One is the cosheaf of compactly supported real valued functions on a space:

$U\rightsquigarrow \{f:U\to\mathbb{R}|\mathrm{supp}(f) \, \mathrm{cpt}\}$

Where the extension function is to extend by zero and partitions of unity should force the cosheaf axiom to hold.

More abstractly, using the colimit-preserving definition, any continuous map of spaces:

$f:X\to Y$

we can build a cosheaf of spaces on $Y$, by assigning to each open set $U\subset Y$

$U\rightsquigarrow f^{-1}(U) \qquad U\cup V \rightsquigarrow f^{-1}(U)\cup f^{-1}(V).$

Another, very closely related canonical example of a cosheaf, is to take $\pi_0(f^{-1}(U))$ and as long as $Y$ is locally connected, this will be a cosheaf. See Jon Woolf's paper The Fundamental Category of a Stratified Space and appendix B in there.

Finally, due to an observation of Bob MacPherson, if we think of a cell complex $X$ as a category with objects the cells $\sigma\in X$ with morphisms given by the face relation $\sigma\subseteq \bar{\tau}$, then a constructible sheaf is equivalent to a functor $F:X\to\mathrm{Vect}$ and a constructible cosheaf $F:X^{op}\to\mathrm{Vect}$. These gadgets are bona fide sheaves and cosheaves in the Alexandrov topology on the associated face relation poset, i.e. open sets are $U\subset X$ such that $x\in U$ $x\leq y$ implies $y\in U$.

Finally, I should say that some of my thesis work applies cosheaves to Morse theory, persistent homology and sensor networks, which should provide some more intuitive examples.

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