By Definition we just have precisely one glueing for every compatible family in $F^+$.
proof: follows from the previous results and the fact that cocontinuous functors between locally presentable categories have right adjoints. It is possible to construct the associated sheaf functor in only one step. $F^+(U)=colim_{\mathscr U}F_U(\mathscr U)$. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
proof: we already know that $\mathbf{CoShv}(\Omega, \mathcal{A})$ is cocomplete and accessible, so the result follows from [AR, corollary 2.47]. Here, the set $BC$ is part of a covering and thus has influence on $F^+(U)$. @Justin: No; no need to invoke any set-theoretical principle. as in the usual theorem follow that $L(F)$ is a separate presheaf, is a sheaf if $F$ is separate, is isomorphic to $F$ is $F$ is a sheaf, and $LL(F)$ is the sheaf associate with the canonical universal property. We show that grothendieck toposes and toposes obtained by the tripos to topos construction are instances of categories of sheaves for a suitable doctrine. Browse other questions tagged set-theory sheaf-theory or ask your own question. /H /I So I think the theorem that $X^+$ is a sheaf when $X$ is separated is a "generalization of this notion to become rigorous". (Sheafification) Let Fprebe a presheaf on the space X. Let $\mathbb{X}$ be a topological space and $\mathbf{Open}(\mathbb{X})$ the open set category of $\mathbb{X}$. You could alternatively reverse the category $\mathrm{Open}(X)$, but that destroys the open covers and presumably creates more problems. /Border [0 0 0] The tangent bundle to an infinite-dimensional manifold, + functor (used to construct sheafification)'s property. Sheafification is a popular tool in topos theory which allows to extend the internal logic of a topos with new principles. To learn more, see our tips on writing great answers. For stacks, you need to apply (a version of) the plus construction three times. What is an example of a presheaf P where P^+ is not a sheaf, only a separated presheaf? To subscribe to this RSS feed, copy and paste this URL into your RSS reader. the colimit is a cosheaf), then the inclusion functor does have a right adjoint. Exercise. It only takes a minute to sign up. rev 2020.10.9.37784, Sorry, we no longer support Internet Explorer, The best answers are voted up and rise to the top, MathOverflow works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa.
6 0 obj << /D [7 0 R /XYZ 126.672 287.016 null] Featured on Meta Creative Commons Licensing UI and Data Updates 4 0 obj But what exact component of that presheaf hindered us from getting the sheaf we wanted? /Filter /FlateDecode 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]. After stacks, there are 2-stacks (something like sheaves of 2-categories), 3-stacks, and so on. 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]. (German) Aus dem Rumänischen übersetzt von Horst Antelmann. Let the topolgy $\tau_X=${$X, U, V, A, B, A\cap B, \emptyset$} with $X=U\cup V$ and $U\cap V=A\cup B$.
contains a treatment of copresheafs and cosheafs.
given by Sherry. for if $X=U\cup V$, in general gived $s\in L(F)(U)$, $s=[(U_i, x_i)_I]$, and proof: The category of sheaves on a small site is limit-sketchable by [RP, pg. 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. Does sheafification preserve sheaves for a different topology?
Let $\mathbf{Vect}$ be the category consisting of real vector spaces as its objects and linear maps as its morphisms. cms.dm.uba.ar/academico/carreras/licenciatura/tesis/…, https://mathoverflow.net/posts/95655/edit, Responding to the Lavender Letter and commitments moving forward, $\infty$-ary tensor product on a category. In our example we have the elements $1$ and $3$ in $F(BC)$, which give the same compatible family in $F(C)$ and $F(B)$. /Length 12638 Sheaves on subsets of $X$ can be patched together, not quite uniquely, but uniquely up to canonical isomorphism. Related. %���� MathJax reference. Fix a target category $\mathcal{A}$ that is complete, cocomplete and locally presentable. 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.
Hello all. >> Use MathJax to format equations. The point now is that "twice suffices" fits into a larger pattern. Swapping out our Syntax Highlighter. 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.
Pre-cosheaf of what, and over what? Nice!
This is equivalent to asking whether the inclusion functor from the category of cosheaves into the category of pre-cosheaves has a right adjoint. R�%-��������;�����u������5�(r��q�s
are equalizers for all covering families $U_i\rightarrow U$ of all objects $U$ of $\mathscr C$. It is known that for a general presheaf on a Grothendieck Topology, we must in general apply the plus construction twice to obtain a sheaf. MathOverflow is a question and answer site for professional mathematicians. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. This is due to Eduardo Dubuc.
Cosheaves are defined dually to sheaves, that is, they send the cone induced by a sieve of the small site to a colimiting cone. Any construction and theorem valid for vector spaces can be carried over to the category of locally free sheaves.
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