Timeline for Enrichments vs Internal homs
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 25, 2018 at 19:49 | answer | added | Giorgio Mossa | timeline score: 3 | |
| Jun 22, 2018 at 13:35 | vote | accept | Max Schattman | ||
| Jun 21, 2018 at 14:54 | answer | added | Tim Campion | timeline score: 5 | |
| Jun 21, 2018 at 14:50 | comment | added | Max Schattman | @Tim: Put this as an answer and I can accept it. | |
| Jun 21, 2018 at 13:21 | comment | added | Tim Campion | Here's an example of "wrong-way" self-enrichment: the category $Cat$ of small categories is enriched in itself in the usual way since it is cartesian closed. But it also has another self-enrichment where you take the maximal subgroupoid of each hom-category. This sort of variant enrichment is important e.g. to make $Cat$ into a simplicial model category. | |
| Jun 20, 2018 at 18:30 | comment | added | Mike Shulman | I would expect that a monoidal category $C$ could happen to be the underlying category of some $C$-enriched category in a "wrong" way so that the enriched hom-objects aren't right adjoints of the tensor product, but I don't have an example ready to hand. | |
| Jun 20, 2018 at 18:29 | comment | added | Mike Shulman | Yes, a closed monoidal category is enriched over itself. In the symmetric case this is section 1.6 of the standard reference Basic concepts of enriched category theory tac.mta.ca/tac/reprints/articles/10/tr10abs.html. I think that usually when people say "internal homs" they are referring to a monoidal structure being closed, or more generally to a closed category structure (ncatlab.org/nlab/show/closed+category) -- it doesn't make sense to talk about enrichment over a category until it has a monoidal/closed/multi/etc structure. | |
| Jun 20, 2018 at 15:51 | history | asked | Max Schattman | CC BY-SA 4.0 |