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.

10$\begingroup$ What is a cosheaf? $\endgroup$– Martin BrandenburgOct 23 '10 at 16:11

1$\begingroup$ Wait..."the dual of $A$ is a cosheaf"?? $\endgroup$– QfwfqOct 23 '10 at 18:09

3$\begingroup$ 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. $\endgroup$– Jonathan WiseOct 23 '10 at 23:47

3$\begingroup$ 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. $\endgroup$– thelOct 24 '10 at 17:36

1$\begingroup$ 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)... $\endgroup$– QfwfqDec 18 '10 at 12:14
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.

$\begingroup$ 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. $\endgroup$– NehaOct 23 '10 at 17:36

$\begingroup$ 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? $\endgroup$ Oct 23 '10 at 20:59

4$\begingroup$ 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. $\endgroup$ Oct 23 '10 at 23:49

$\begingroup$ Yes, I didn't suspect there was a coequalizer property defining a cosheaf  then the above source will not lead you anywhere! $\endgroup$ Oct 24 '10 at 10:44

$\begingroup$ Thanks everybody. Daniel, Can you please elaborate a bit more on localizing a coalgebra or a comodule. Where can I read about this more? $\endgroup$– NehaOct 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  http://arxiv.org/abs/1209.2995 . These are a kind of dual creatures to quasicoherent sheaves. The preprint will be updated and expanded (eventually).

2$\begingroup$ I can't decide if "Contraherent" is an ugly coification of "coherent", or if it is a really good pun. $\endgroup$ Oct 3 '12 at 11:03

6$\begingroup$ The term was chosen because contraherent cosheaves stand in the same relationship to quasicoherent 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)  en.wiktionary.org/wiki/contraherent $\endgroup$ 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

$\begingroup$ 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? $\endgroup$– NehaOct 26 '10 at 13:04

$\begingroup$ @Neha: unfortunately, cannot help you with quasicoherence. My knowledge about it resumes to knowing where I can find the definition if I ever need it. $\endgroup$ Oct 26 '10 at 23:46
Look at Bredon's 'Sheaf Theory', Chapter Six: "Cosheaves and Cech Homology"
I am not aware of any quasicoherent story.
According to Skliarienko, the BorelMoore 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.