Cosheafification

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 $\mathsf{F}$?

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arxiv.org/pdf/0811.2580v1.pdf see Appendix B –  David Corwin Feb 4 '13 at 10:06
@Davidac897: Isn't this as answer? –  Martin Brandenburg Feb 4 '13 at 11:10
Is it clear that David's reference answers the question for cosheaves of vector spaces? The reference only deals with cosheaves of sets, and it's remarked there that the theories of cosheaves of sets and cosheaves of abelian groups are much more different than the theories of sheaves of sets vs. abelian sheaves. –  Dan Petersen Feb 4 '13 at 11:46
I would be actually interested to see a reference discussing cosheafification of copresheaves of abelian groups or vector spaces in some detail. This being said, the OP's question as it is written at the moment is not very well-formed. What is a "pre-cosheaf in the category of vector spaces"? Pre-cosheaf of what, and over what? –  Leonid Positselski Feb 4 '13 at 14:50
If I remember right (I do not have access to the book since I am traveling), MR0275411 (43 #1168), Teleman, C. Grundzüge der Topologie und differenzierbare Manningfaltigkeiten. (German) Aus dem Rumänischen übersetzt von Horst Antelmann. Mathematische Monographien, 8 VEB Deutscher Verlag der Wissenschaften, Berlin 1968 411 pp. (loose errata). contains a treatment of copresheafs and cosheafs. –  Peter Michor Feb 5 '13 at 1:47

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

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Your easier-to-check version of the adjoint functor theorem relies on a large cardinal axiom, which is therefore not provable in ZFC unless ZFC is inconsistent. –  arsmath Mar 19 '13 at 7:00
You mean $F \colon \mathrm{Open}(X) \to \mathrm{Vect}^\mathrm{op}$, I think. You could alternatively reverse the category $\mathrm{Open}(X)$, but that destroys the open covers and presumably creates more problems. –  Ryan Reich Mar 19 '13 at 14:21
@Ryan: No, a functor $F:C\to D$ always can be used formally to define a functor $F^{op}:C^{op}\to D^{op}$. The assignment of objects remains the same, so $F^{op}(x)=F(x)$, but now a morphism $f:x\to y$ in $C$ becomes a morphism $f^{op}:y\to x$. The functor $F^{op}$ sends $f^{op}$ to $F(f)^{op}$. $F(f):F(x)\to F(y)$ defines a morphism $F(f)^{op}:F(y)\to F(x)$ in $D^{op}$, which is equal to $F^{op}(f^{op}):F^{op}(y)\to F^{op}(x)$. I don't know what you mean by "destroys covers." –  Justin Curry Mar 20 '13 at 20:48
@arsmath: Yes, you are right. It has a pretty strong consistency requirement. If I understand correctly, the strength of Vopenka's principle lies between Reinhardt cardinals and unmeasurable cardinals, but I don't have any committed opinions as to how controversial this is. I tend not to worry too much about these things, but perhaps I should. –  Justin Curry Mar 20 '13 at 21:13
It's way up there in the hierarchy. Vopenka himself thought it was false -- that's why he proposed it as an argument against the largest of large cardinal axioms -- but I think most set theorists think it's independent. Even if it's fine, it means that you're saying you have no control over the size of cosheafification: it could turn a countable set into a set of cardinality so large that it dwarfs all sets that appear in day-to-day mathematics. –  arsmath Mar 20 '13 at 22:35

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.

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Nice! Does this proof assume Vopenka's principle? –  Justin Curry Mar 20 '13 at 21:07
@Justin: No; no need to invoke any set-theoretical principle. –  G. Rodrigues Mar 21 '13 at 2:45