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. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. Any explanation would be appreciated. I doubt it, but I haven't thought about this. Asking for help, clarification, or responding to other answers. It may happen that I just don't know the notations involved. MathJax reference. But how many choices of $\alpha$ do we have? Please log in or leave your comment as a "guest post".
I edited just a little for the moment. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual.
Thanks for contributing an answer to MathOverflow! MathOverflow is a question and answer site for professional mathematicians. 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 $\alpha|U_i = \alpha_i$. 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). 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. Vanilla 1.1.10 is a product of Lussumo. Here we need separatedness for morphisms. 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. (and I guess hence the nlab's distinction between 1- and 2-separatedness. 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 . Want to take part in these discussions? More Information: Documentation, Community Support. 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 $a = b$. It only takes a minute to sign up. If commenting as a "guest", please include your name in the message as a courtesy. I suggest Vistoli's notes on descent, around page 75, for this stuff) To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Thanks! 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. A presheaf of sets is called separated if we check that sections are equal on covers.
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