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I am struggling hard to understand the pushforwards and pullbacks of cosheaves. Are they also cosheaves? And what are quasicoherent cosheaves? Is there anything like coquasicoherent cosheaves? Please tell me a good refernce on theses topics, if there is some.

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What is a cosheaf? – Martin Brandenburg Oct 23 '10 at 16:11
Wait..."the dual of $A$ is a cosheaf"?? – Qfwfq Oct 23 '10 at 18:09
A cosheaf is a covariant functor defined on the open subsets of a space that satisfies a right exactness property (analogous to the left exactness satisfied by sheaves). If you take the dual of a sheaf "objectwise" you get a cosheaf, and vice versa. – Jonathan Wise Oct 23 '10 at 23:47
By the way, the "dual" of an equalizer is not necessarily a coequalizer, so the dual of a sheaf need not be a cosheaf. For example consider the structure sheaf of Spec(Z). If dual means Hom(-,Z) then the dual vanishes on every proper open subset. – Thel Seraphim Oct 24 '10 at 17:36
I don't understand. The dual of the tangent sheaf $T_X$ of an algebraic variety $X$ is the cotangent sheaf $T^*_X$, not the "tangent cosheaf" (whatever the latter could mean)... – Qfwfq Dec 18 '10 at 12:14
up vote 3 down vote accepted

edit: I was assuming you wanted an equalizer sheaf property, but this is not the definition of cosheaf, see comments - the following has nothing to do with cosheaves then!

If by "cosheaf" you mean a covariant functor from the opens of a space to sets/groups/etc., you could look at Moerdijk/MacLane's "Sheaves in Geometry and Logic" - there you can learn some general sheaf theory on sites, which includes the cosheaf case. In particular pushforward and pullback are transport along the the two functors comprising a "geometric morphism". The notion of quasicoherent (co)sheaf can maybe also be defined in this generality by saying that something should look locally like a pullback from the "base topos".

Sorry for the jargon, I didn't get from your question what exactly you are up to - just take a look at the book and see if it suits you.

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Thanks Peter, I want to learn all the operations that one can do with a sheaf. Like taking their external tensor product. How about the structural cosheaf? is it a cosheaf of corings? So, I have a quasicoherent sheaf $A$ of algebras over a scheme $X$. That means it is essentially $\mathcal{O}_X$-modules. The dual of $A$ is a cosheaf. Will this dual cosheaf be also quasicoherent? And will it be a quasicoherent cosheaf of coalgebras? I mean, will it be $\mathcal{O}_X$-comodules. – Neha Oct 23 '10 at 17:36
you could probably make something resembling quasicoherent if you understand what it means to localize a coalgebra or comodule(there are ways to do this) I don't know if it will be as efficient as the ordinary situation. You also have a cotensor product operation... I think for very general stuff like pushforward or pullback(not quasicoherent!) everything works fine if you just "op" the category of open sets and look at sheaves on that? – Daniel Pomerleano Oct 23 '10 at 20:59
The descent conditions for sheaves and cosheaves are different. I do not think that understanding cosheaves is simply a matter of passing to the opposite category in the manner you suggest. – Jonathan Wise Oct 23 '10 at 23:49
Yes, I didn't suspect there was a coequalizer property defining a cosheaf - then the above source will not lead you anywhere! – Peter Arndt Oct 24 '10 at 10:44
Thanks everybody. Daniel, Can you please elaborate a bit more on localizing a coalgebra or a comodule. Where can I read about this more? – Neha Oct 26 '10 at 13:07

I've written a preprint about (what I call) contraherent cosheaves of modules over the structure sheaf of rings of a scheme -- . These are a kind of dual creatures to quasi-coherent sheaves. The preprint will be updated and expanded (eventually).

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I can't decide if "Contraherent" is an ugly co-ification of "coherent", or if it is a really good pun. – Ketil Tveiten Oct 3 '12 at 11:03
The term was chosen because contraherent cosheaves stand in the same relationship to quasi-coherent sheaves as contramodules do to comodules over corings. Also, "contraherent" seems to be a legitimate Latin word (with the meaning rather close to that of "coherent", as far as I can tell) -- – Leonid Positselski Oct 3 '12 at 12:13

If you have the stomach for hard topos theory a good reference is

Singular coverings of toposes -- M. Bunge, J. Funk

The first chapter is probably enough to answer your question. The upshot is that if you have a site, then the category of cosheaves can be identified with the category of cocontinuous functors on the category of sheaves. This should give you a pretty good idea of what operations you can perform on cosheaves.

Btw, in this context, cosheaves are also called Lawvere distributions -- distributions because of the analogy with the Riesz representation theorem that identifies measures (cosheaves) with linear functionals (cocontinuous functors).

Hope it helps, regards, G. Rodrigues

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Many thanks Rodrigues. What I am still thinking is how to define quasicoherent cosheaves! It seems that it is natural to think of them as cosheaves of $\mathcal{O}^\circ_X$-comodules. and call them quasicoherent if their dual sheaf is quasicoherent. What do you say? – Neha Oct 26 '10 at 13:04
@Neha: unfortunately, cannot help you with quasi-coherence. My knowledge about it resumes to knowing where I can find the definition if I ever need it. – G. Rodrigues Oct 26 '10 at 23:46

Look at Bredon's 'Sheaf Theory', Chapter Six: "Cosheaves and Cech Homology"

I am not aware of any quasi-coherent story.

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According to Skliarienko, the Borel-Moore homology (with coefficients in a sheaf) is badly "defective"; and the "right" homology should have coefficients in a co(pre)sheaf. (He makes this point in a few of his works; but most clearly in the editor's comments to the Russian translation of Bredon's "Sheaf theory".)

In fact, there exist two papers by Beniaminov which are precisely about this. I'm a bit puzzled that Skliarienko cited them with enthusiasm in early 1970s, but later ignored on several occasions when speaking of the desired cosheaf homology. There is also a followup paper by Golovin which I haven't checked yet.

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