topological setting by David Roberts

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when toggle format	what		by	license	comment
Jul 24, 2020 at 14:47	vote	accept	Praphulla Koushik
Jun 15, 2020 at 7:27	history	edited	CommunityBot		Commonmark migration
Jun 1, 2020 at 7:31	comment	added	David Roberts♦		OK, thanks. Must have been a markdown formatting mistake on my part.
Jun 1, 2020 at 7:26	comment	added	Praphulla Koushik		added link for WISC. Your answer also had a link but it was not "clickable". I do not know why it is like that...
Jun 1, 2020 at 7:25	history	edited	Praphulla Koushik	CC BY-SA 4.0	deleted 25 characters in body
May 30, 2020 at 8:36	history	bounty awarded	Praphulla Koushik
May 27, 2020 at 7:23	comment	added	Praphulla Koushik		Thanks for the reference.. it is a short paper,I think I will be able to learn something from it..
May 27, 2020 at 6:33	history	edited	David Roberts♦	CC BY-SA 4.0	Expanded answer
May 27, 2020 at 6:25	comment	added	David Roberts♦		Waterhouse's paper Basically bounded functors and flat sheaves in Pacific J. Math. (projecteuclid.org/euclid.pjm/1102906018) gives an example of a presheaf on the fpqc site that does not admit a sheafification.
May 27, 2020 at 6:06	comment	added	Praphulla Koushik		I mean to say "Any specific reason for that type of choice?" For your second explanation, I think I understand what you mean.. There may not exists sheafification functor $\text{Presheaves}/\mathcal{S}\rightarrow \text{Sheaves}/\mathcal{S}$ when $\mathcal{S}$ is a large category.. I would like to read more about the precise difficulty in sheafification if you can point me to some reference.. I request you to add those lines in your previous comment to your answer.. nlab page definitio of sheafifcation starts with "Let $(C,J)$ be a site in the sense of: small category equipped with a coverage"
May 27, 2020 at 5:48	comment	added	David Roberts♦		I'm not sure what your first question is asking ("typo"?). Defining the structure of a site on a large category has no real issues, the problem is that there might not be a sheafification/stackification functor (a real size issue), even though general individual sheaves/stacks make sense. For your second question, I don't mean the 2-category of representable stacks, since they turn out to be sheaves of sets, but the 2-category of presentable stacks—this is a general term for algebraic/topological/differentiable/etc stacks over a general site. We don't need stackification for this.
May 27, 2020 at 5:07	comment	added	Praphulla Koushik		What I have understood as of now is that, even if the base category $\mathcal{S}$ is a large category, there are no size issues if I restrict my attention to the $2$-category of representable stacks over $\mathcal{S}$.. It is because this $2$-category of representable stacks over $\mathcal{S}$ is "equivalent" to the "bicategory of internal groupoids and anafunctors"..
May 27, 2020 at 4:51	comment	added	Praphulla Koushik		A side question.. David Metzler in his paper arxiv.org/abs/math/0306176 assumes a site to be defined over a small category.. nlab says similar thing "Remark 2.3. Often a site is required to be a small category. But also large sites play a role" at ncatlab.org/nlab/show/site.. It looks like even defining Grothendieck topology on a large category seems to be avoided by some people.. Any specific reason for that typo of choice?
May 27, 2020 at 0:00	comment	added	David Roberts♦		I should add that differentiable stacks can be defined using a site with countably many objects, namely the Euclidean spaces $\mathbb{R}^n$, so all this discussion is not needed for that case.
May 26, 2020 at 23:46	history	answered	David Roberts♦	CC BY-SA 4.0