Does the category of cosheaves have enough projectives? Given a general topological space $X$ does the category $\mathbf{coShv}(X,\mathbf{Mod}_R)$ have enough projectives ? I know that under some conditions this is true, for example if $X$ is a cell complex equipped with the Alexandrov topology and $R$ is a field (in the case of cellular sheaves of vector spaces). In this case one can define cosheaf homology as the left derived functor of the global section functor. I am wondering if one can always define cosheaf homology in this way ? I read that the problem with cosheaves is that cosheafification does not exist in general. Is this problem connected with the problem of defining cosheaf homology ?
 A: The answer is yes. A reference for the claim about Alexandrov spaces and cellular (co)sheaves is Justin Curry's thesis. When he proves Claim 7.1.9 he points out that the statement is true for all spaces $X$. It's also not essential that $R$ is a field. You can see directly that Justin's proof does not use any assumptions on $R$. The main point is that, given a cosheaf $F$, there is a projective cosheaf $P$ surjecting onto $F$. Once you have $P$, you can build the full projective resolution as usual. To define $P$ all you need is direct sums in the category of cosheaves of $R$-modules, and this follows from the bicompleteness of the category of $R$-modules, as observed in Theorem 2.14 of Prasolov's 2018 paper Cosheaves.
Now, just because you have projective resolutions does not mean you should expect your proof about sheaves (via injective resolutions) to immediately dualize. As Bredon writes in Cosheaves and Homology (1968) "In our opinion one cannot expect to find anything like a complete duality with sheaf theory and one must be prepared, from the start, to dispense with some basic properties."
In particular, to actually compute cosheaf homology, it's often more useful to use a resolution by flabby cosheaves, as Bredon does, but the existence of such a resolution requires X to be homologically locally connected (see page 35 of Bredon's Sheaf Theory book). The existence of cosheaf homology was never in doubt, and is discussed in this mathoverflow thread from ten years ago.
Also, without some restriction on $X$, we don't even know how to define constant cosheaves. The constant precosheaf is easy (just send everything to a fixed $R$-module $M$) but constant cosheaf is harder and traditionally required $X$ to be locally connected.
It's also true that in Bredon's time, cosheafification was not known to exist in general, as the OP wrote. However, as pointed out in this mathoverflow thread, cosheafification does exist in your setting of $R$-modules, because that's a locally presentable and accessible category. A published reference is the 2016 paper of Prasolov.
I think the reason cosheafification was considered hard in Bredon's time was the lack of cosmall objects. But this argument via local presentability gets around that. I don't think it's the same problem with the existence of resolutions. I should point out that Prasolov's 2018 paper develops quite a lot of theory that might help you, but it's about cosheaves of Pro(R)-modules. In that setting you do NOT have enough projectives but you have enough quasi-projectives; see Prasolov's appendix A to learn more.
