topological setting

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Jul 24, 2020 at 14:47	vote	accept	Praphulla Koushik
S May 30, 2020 at 8:36	history	bounty ended	Praphulla Koushik
S May 30, 2020 at 8:36	history	notice removed	Praphulla Koushik
S May 28, 2020 at 15:17	history	bounty started	Praphulla Koushik
S May 28, 2020 at 15:17	history	notice added	Praphulla Koushik		Reward existing answer
May 28, 2020 at 4:18	comment	added	Praphulla Koushik		@SimonHenry Ok. Thank you :)
May 27, 2020 at 21:45	comment	added	Simon Henry		Yes. The category of topological space is not cartesian closed but the category of compactly generated is. This has nothing to do with size problem, but with properties of the compact-open topology. I'm affraid I don't really have more details to add to my comment, I think David Robert's answer pretty much cover everything.
May 27, 2020 at 18:46	comment	added	Praphulla Koushik		can you make your comment as an answer adding some details which you think might be relevant. @SimonHenry Reading some places revealed that this condition on Topological spaces has something to do with Cartesian closedness
May 27, 2020 at 15:27	comment	added	Praphulla Koushik		@RobertFurber I don’t have much experience with universes.. It would be useful if you can make your comment as an answer with some more details.
May 27, 2020 at 15:19	comment	added	Robert Furber		I was misled by such statements early in my career - universes do not allow you to ignore the difference between small and large categories. Rather, universes allow us to replace the distinction between set and proper class with the distinction between a set belonging to a fixed universe $\mathcal{U}$ and a set (still not a class) outside that universe. This means we can still apply all set-theoretic operations to "large" sets, because we never use proper classes. The definition of complete and cocomplete category must be adapted to use diagrams from $\mathcal{U}$.
May 27, 2020 at 5:30	history	edited	Praphulla Koushik	CC BY-SA 4.0	added 1306 characters in body
May 26, 2020 at 23:46	answer	added	David Roberts♦		timeline score: 7
May 26, 2020 at 16:29	comment	added	Praphulla Koushik		I wonder why there is no tag for “topological stacks”..
May 26, 2020 at 15:40	history	edited	Praphulla Koushik	CC BY-SA 4.0	edited body; edited title
May 26, 2020 at 13:20	comment	added	Praphulla Koushik		@SimonHenry oh. Sorry. I did not think if the category of compactly generated topological spaces is small or not.. I have done a rough reading of the paper by Behrang Noohi, could not find where exactly they use this restriction.. I will do another reading and see if they have done something similar to what you have mentioned. Thanks for the tip..
May 26, 2020 at 12:59	comment	added	Simon Henry		I don't think assuming that topological space are compactly generated has anything to do with a size problem. The category of compactly generated topological space is not small either. It probably has to do with wanting to consider exponential of topological space (maping space).
May 26, 2020 at 12:24	comment	added	Praphulla Koushik		@მამუკაჯიბლაძე sue, thank you :)
May 26, 2020 at 12:17	comment	added	მამუკა ჯიბლაძე		Well when you use subcanonical Grothendieck topologies, you basically do just that - make some non-representable functors representable. So any text on topos theory might be viewed relevant for that. I will come back if I figure out something more to the point.
May 26, 2020 at 12:14	comment	added	Praphulla Koushik		@მამუკაჯიბლაძე Hi, thanks for the response. I am not aware of what you have mentioned. I would like to read more if you can point me to some reference regarding making something a representable functor by adding an object representing it.. Idea is clear but some more details might clear somethings for me...
May 26, 2020 at 12:10	comment	added	მამუკა ჯიბლაძე		If I am not confusing something, "using universes" in this context just means the following: if you encounter a functor that should be representable but is not, you enlarge your category by adjoining a representing object for your functor. From this point of view, whether you distinguish small/large or not, there still remains a distinction: a functor is either representable or not.
May 26, 2020 at 11:57	history	undeleted	Praphulla Koushik
May 26, 2020 at 11:34	history	deleted	Praphulla Koushik		via Vote
May 26, 2020 at 9:29	comment	added	Praphulla Koushik		I am not asking how does one modify definitions of Angelo Vistoli's notes so that there is no issue when dealing with large categories...
May 26, 2020 at 9:26	history	asked	Praphulla Koushik	CC BY-SA 4.0

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