Timeline for Colimits, limits, and mapping spaces
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 14, 2018 at 9:00 | comment | added | Philippe Gaucher | @მამუკაჯიბლაძე See tac.mta.ca/tac/volumes/21/1/21-01.pdf Corollary 3.7. | |
| Jan 14, 2018 at 8:07 | comment | added | Denis Nardin | @მამუკაჯიბლაძე As I said, numerically-generated (also known as $\Delta$-generated or $[0,1]$-generated) spaces are one. They have the pleasant property that can be identified with concrete presheaves on cartesian spaces (although the identification is far from obvious) and so they are nicely analogous to diffeological spaces. | |
| Jan 14, 2018 at 5:02 | comment | added | მამუკა ჯიბლაძე | You are welcome. @PhilippeGaucher can you name it? | |
| Jan 9, 2018 at 13:15 | comment | added | Denis Nardin | @PhilippeGaucher This is possible (as I said I tend to shy away from point-set technicalities whenever possible). When I absolutely need to use topological spaces I use numerically-generated (a.k.a $\Delta$-generated) spaces, which are locally presentable. | |
| Jan 9, 2018 at 12:57 | comment | added | Philippe Gaucher | @DenisNardin None of the categories presented in ncatlab.org/nlab/show/convenient+category+of+topological+spaces are locally presentable. However, there does exist a locally presentable convenient category of topological spaces. | |
| Jan 4, 2018 at 5:07 | comment | added | მამუკა ჯიბლაძე | @Victor In a cartesian closed category this too is automatic: $\operatorname{Map}(X,-)$ preserves all existing limits, being the right adjoint to $X\times-$. | |
| Jan 4, 2018 at 1:41 | comment | added | Victor | Denis, thanks again. I was a bit slow, but I finally understood what you meant: since Map(-,Y) has a right/left adjoint it preserves colimits/limits, which is exactly my question. I still need to figure out whether in a convenient category one has $Map(X,\prod_i Y_i)\cong \prod_iMap(X,Y_i)$, but I have a feeling I am on a right track! | |
| Jan 4, 2018 at 1:35 | vote | accept | Victor | ||
| Jan 3, 2018 at 18:31 | comment | added | Denis Nardin | @Victor The nlab page I linked in the answer is a good guide for the literature on this. The starting point is probably Steenrod's classical paper titled A convenient category of spaces. I don't know the subtleties very well (I prefer to work with simplicial methods) but I think it is pretty much irrelevant which category you choose, provided it is cartesian closed, locally presentable and contains all CW complexes. | |
| Jan 3, 2018 at 17:27 | comment | added | Victor | Denis, thank you! I think that's exactly what I need. Please allow me some time to read the literature on this question. Indeed, I need a model category of topological spaces in which I would have the above property, also the exponential property $Map(X,Map(Y,Z))\cong Map(X\times Y,Z)$ and also $Map(X,\prod_i Y_i)\cong \prod_i Map(X,Y_i)$. By the way, can you suggest any good reference to read? | |
| Jan 3, 2018 at 16:37 | history | answered | Denis Nardin | CC BY-SA 3.0 |