Sign in if you have an account, or apply for one below. the book of Bernstein/Lunts, smth around page 34 in my memory) as noted by Deligne as an elementary fact in his Hodge theory paper but in different terminology. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. The notion of stacks of groupoids on a 1-site which you discuss above is just a special case of this more general concept.

If you desperately need to know about 2-sites, you should contact Mike Shulman. The standard model structure on simplicial presheaves restricts to the standard model structure on simplicial sheaves, this restriction is a Quillen equivalence and equipped with this model structure SSh(C)SSh(C) is one of the standard models for ∞-stack (∞,1)-toposes for the site CC. @UrsSchreiber: ah, thanks for clearing that up, I had never seen it used before. A discussion forum about contributions to the.

To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. I think the entry simplicial sheaf was a little orphaned…. In category theory, a branch of mathematics, a presheaf on a category is a functor: →.If is the poset of open sets in a topological space, interpreted as a category, then one recovers the usual notion of presheaf on a topological space.. A morphism of presheaves is defined to be a natural transformation of functors. It seems that you need to read the following $X(μ_{ij})(y)∘X(p_{ij})(b_i) = X(q_{ij})(b_i)∘X(μ_{ij})(x)$ where for instance $X(μ_{ij})(y)$ is the component at $y$ of the natural transformation $X(μ_{ij})$. What's missing for F to be a stack is going in the opposite direction (existence rather than uniqueness, one usually says $F$ is effective). 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. It only takes a minute to sign up.
But how many choices of $\alpha$ do we have? I suggest Vistoli's notes on descent, around page 75, for this stuff) Please log in or leave your comment as a "guest post". To learn more, see our tips on writing great answers. Especially with the expression $μ_{ij}(y)∘X(p_{ij})(b_i)=X(q_{ij})(b_i)∘μ_{ij}(x)$ because $\mu_{ij}$ is a 2-cell while $y$ is just an element of the category $X(u)$, in fact I don't even understand any part of it . More Information: Documentation, Community Support. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Making statements based on opinion; back them up with references or personal experience. I changed it to refer to the Jardine-local model structure and say that this model structure presents only the hypercomplete (∞,1)(\infty,1)-topos, which is what the page model structure on simplicial sheaves says. MathOverflow is a question and answer site for professional mathematicians. Let $\alpha,\beta\colon a \to b$ be two morphism in the groupoid $F(U)$. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. If commenting as a "guest", please include your name in the message as a courtesy. I'm looking at the definition of 2-sheaf in the nlab http://ncatlab.org/nlab/show/2-sheaf and I get stuck with the definition of 2-separated. For example, one takes the Borel construction and simplicial sheaves over it. I doubt it, but I haven't thought about this. By the way, some geometers also say a simplicial sheaf for an appropriate concept of a sheaf over simplicial space or simplicial site. Here we need separatedness for morphisms. A presheaf of sets is called separated if we check that sections are equal on covers. (and I guess hence the nlab's distinction between 1- and 2-separatedness. Infinitely many rational nt multisection in elliptic K3 surfaces by deformation theory, Difficulties with descent data as homotopy limit of image of Čech nerve, Alternative definitions of infinite-order differential operators, Checking the functoriality of an expression involving dependent sum and product. The page simplicial sheaf said The standard model structure on simplicial presheaves restricts to the standard model structure on simplicial sheaves , this restriction is a Quillen equivalence and equipped with this model structure SSh ( C ) SSh(C) is one of the standard models for ∞-stack (∞,1)-toposes for the site C C . (as for categories fibered in groupoids and groupoid-valued functors). Asking for help, clarification, or responding to other answers. (again by separatedness $a$ is unique up to a unique isomorphism). If $a,b$ are objects of the groupoid $F(U)$ such that there exist isomorphisms $\alpha_i \colon a|U_i \to b|U_i$, whose restrictions moreover coincide on double intersections $\alpha_i|U_{ij} = \alpha_j|U_{ij}$ then there exists an isomorphism $\alpha\colon a \to b$, such that $\alpha|U_i = \alpha_i$. If F is groupoid valued then one needs to climb up a rung of the Cech ladder. In symbols, $a, b \in F(U)$ are such that for a cover $U_i \to U$ we have $a|U_i = b|U_j$ then $a = b$. So I went to record it in the References-section at simplicial sheaf, only to notice that this entry had never existed as a decent entry. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. (and separatedness ensures that it is unique), For objects you need cocycles: $a_i \in F(U_i)$, together with $\alpha_{ij} a_i|U_{ij} \to a_j|U{ij}$, satisfying the cocycle identity on $U_{ijk}$, there exists $a \in F(U)$ such that $\beta_i : a|U_i \to a_i$, and $\alpha_{ij} = \beta_j\beta_i^{-1}$ on $U_{ij}$. [[category]]. Responding to the Lavender Letter and commitments moving forward.

got any tricks to build up t-structures on derived categories?

MathJax reference. If the structure morphisms between nn-th degree sheaf component and the pullback of (n−1)(n-1)-st degree sheaf component are all iso, then we get precisely the equivariant sheaves (cf. Only a subset of the usual TeX math commands are accepted: see here for a list. I edited just a little for the moment. The nLab entry says "2-sheaf" instead of "stack" (for which there is also an entry, of course) because the 2-sheaf-entry considers the full generality of sheaves in bicategory theory, which is rarely ever considered in the literature: category valued higher sheaves on bicategorical sites. Note: only certain categories allow guest posts. Vanilla 1.1.10 is a product of Lussumo. Thanks! Thanks for contributing an answer to MathOverflow! Any explanation would be appreciated.
Of course one can go on and define $3$-sheaves (2-stacks), but you need go up the ladder and start taking quadruple intersections and so on. Out of curiosity, is there some kind of strictification/truncation to take a 2-sheaf on a 2-site to a stack on an ordinary site? I don't know about the nlab but 2-sheaves are usually called stacks and you run in the higher version of separatedness for presheaves.

For morphisms, you want them to glue like a sheaf, so if $\alpha_i,\beta_i$ are a collection of morphisms on $U_i$ such that $\alpha_i|U_{ij} = \beta_j|U_{ij}$ then you want an $\alpha$ on U such that $\alpha|U_i = \alpha_i$. $\begingroup$ The nLab entry says "2-sheaf" instead of "stack" (for which there is also an entry, of course) because the 2-sheaf-entry considers the full generality of sheaves in bicategory theory, which is rarely ever considered in the literature: category valued higher sheaves on bicategorical sites. rev 2020.10.9.37784, 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. If $\alpha|U_i = \beta|U_i$ we want $\alpha = \beta$. Want to take part in these discussions? It may happen that I just don't know the notations involved. Use MathJax to format equations.

Is this very much against the terminology here or we should take this into consideration to mention.

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