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.



sheaves and cosheaves
**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"](http://books.google.com/books?id=SGwwDerbEowC&printsec=frontcover&dq=%2522Sheaves+in+Geometry+and+Logic%2522&source=bl&ots=WxeQEqB_CR&sig=OTsIE4Vr1EdkGyDrLbHGDHYo5Do&hl=en&ei=VAnDTOGKJsaYOoKT1YAM&sa=X&oi=book_result&ct=result&resnum=2&ved=0CBoQ6AEwAQ#v=onepage&q&f=false) - 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.


