Cosheafification

Question

Hello all. I have a pre-cosheaf in the category of vector spaces. How do I cosheafify? I've failed to find literature on this topic.

I'll be more specific. Let $\mathbb{X}$ be a topological space and $\mathbf{Open}(\mathbb{X})$ the open set category of $\mathbb{X}$. Let $\mathbf{Vect}$ be the category consisting of real vector spaces as its objects and linear maps as its morphisms. My pre-cosheaf is a functor $\mathsf{F} : \mathbf{Open}(\mathbb{X}) \to \mathbf{Vect}$. How do I cosheafify $\mathsf{F}$?

Answer 1

Unless I have not botched things horribly, here is a general proof that cosheafification exists following the outline J. Curry gave.

Fix a target category $\mathcal{A}$ that is complete, cocomplete and locally presentable. Examples include categories of algebraic objects like linear spaces, categories of sheaves, the category of Banach spaces and linear contractions, etc.

Let $\Omega$ be a small site and $\mathbf{PCoShv}(\Omega, \mathcal{A})$ the category of precosheaves, that is, functors $\Omega\longrightarrow \mathcal{A}$. The category $\mathbf{PCoShv}(\Omega, \mathcal{A})$ is complete and cocomplete with limits and colimits computed pointwise and locally presentable by [AR, corollary 1.54].

Cosheaves are defined dually to sheaves, that is, they send the cone induced by a sieve of the small site to a colimiting cone. Denote the full subcategory of cosheaves by $\mathbf{CoShv}(\Omega, \mathcal{A})$. By the interchange of colimits theorem, $\mathbf{CoShv}(\Omega, \mathcal{A})$ is cocomplete and the inclusion functor $\mathbf{CoShv}(\Omega, \mathcal{A})\longrightarrow \mathbf{PCoShv}(\Omega, \mathcal{A})$ is cocontinuous.

proposition: $\mathbf{CoShv}(\Omega, \mathcal{A})$ is accessible. proof: The category of sheaves on a small site is limit-sketchable by [RP, pg. 331]. The same construction yields by duality that the category of cosheaves is the category of models of a colimit sketch, therefore it is accessible by [AR, corollary 2.61]. Q. E. D.

proposition: $\mathbf{CoShv}(\Omega, \mathcal{A})$ is complete and locally-presentable. proof: we already know that $\mathbf{CoShv}(\Omega, \mathcal{A})$ is cocomplete and accessible, so the result follows from [AR, corollary 2.47]. Q. E. D.

theorem: the full inclusion $\mathbf{CoShv}(\Omega, \mathcal{A})\longrightarrow \mathbf{PCoShv}(\Omega, \mathcal{A})$ has a right adjoint. proof: follows from the previous results and the fact that cocontinuous functors between locally presentable categories have right adjoints. This adjoint functor theorem can be pieced together via the concept of totality. More precisely, locally presentable categories are total by [MK, corollary 6.5 and remark 6.6] and total categories are compact, that is, every cocontinuous functor has a right adjoint -- this is [MK, theorem 5.6].

Bibliography: [AR] J. Adamek, J. Rosicky - Locally presentable and accessible categories, Cambridge University Press (1994).

[MK] Max Kelly - A survey of totality for enriched and ordinary categories, Cahiers de Top. et Géom. Diff. Catégoriques, 27 no. 2 (1986), p. 109-132

[RP] R. Pare - Some applications of categorical model theory, in Categories in Computer Science and Logic, Contemporary Mathematics, vol. 92 (1989)

edit: cleaned up, added a couple of references and made mention of the correct adjoint functor theorem.

Answer 2

I have had some conversations with Jon Woolf (the author of the paper referenced by Davidac897). He has pointed out that the forgetful functor from $for:\mathrm{Vect}\to\mathrm{Set}$ preserves limits, but not colimits. Thus a cosheaf of vector spaces need not define a cosheaf of sets and in particular cosheafifying for pre-cosheaves valued in one data category $\mathcal{D}$, may "look different" depending on what $\mathcal{D}$ is.

One can view a pre-cosheaf $\hat{F}:\mathrm{Open}(X)\to\mathrm{Vect}$ as a pre-sheaf $F:\mathrm{Open}(X)^{op}\to\mathrm{Vect}^{op}$ and try to use Grothendieck's sheafification prescription, but this will not work either. The requirements for Grothendieck's sheafification (outlined on page 24 of Schapira's notes) is that the data category be one in which filtered colimits and finite limits commute, which for $\mathrm{Vect}^{op}$ boils down to the false statement that cofiltered limits and finite colimits commute in $\mathrm{Vect}$.

The solution the following two papers get at is to work in the pro-object category, because there cofiltrant limits and finite colimits do commute, so the Grothendieck construction goes through.

http://arxiv.org/pdf/1105.3167.pdf

http://kyokan.ms.u-tokyo.ac.jp/users/preprint/pdf/2001-33.pdf

However, for some people (myself included), this is an unappealing solution. Pro-objects are diagrams in themselves, so a pre-cosheaf of pro-Vector spaces would assign to each open set a diagram of vector spaces.

So here is something one can do: One can check abstractly whether cosheafification exists. This is equivalent to asking whether the inclusion functor from the category of cosheaves into the category of pre-cosheaves has a right adjoint. Freyd's general adjoint functor theorem says that, modulo set-theoretic issues, a functor has a right adjoint (is a left adjoint) if it preserves colimits. Since the category of cosheaves is clearly closed under colimits (one just defines open-by-open it to be the colimit, and since colimits commute, the cosheaf axiom holds for this colimit pre-cosheaf, i.e. the colimit is a cosheaf), then the inclusion functor does have a right adjoint.

Of course, the devil is in the details, so I have written up the details and put them on my website here. I use an easier-to-check version of the adjoint functor theorem given by Vopenka's principle, but I think one could use the proof outlined to appeal to just Freyd's theorem.

For more on cosheaves and their possible uses, the following might be of interest:

http://arxiv.org/abs/1303.3255