Timeline for Groupoidification of infinity categories and geometric realization
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Dec 1, 2021 at 20:41 | history | edited | Exit path | CC BY-SA 4.0 |
Replaced "strictification" with "groupoidification"
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| Dec 1, 2021 at 15:09 | comment | added | Theo Johnson-Freyd | I think it might stand for be "straightening", but I could be misremembering. | |
| Nov 30, 2021 at 2:31 | comment | added | Exit path | @TheoJohnson-Freyd Thanks for elaborating on this, as well as the examples. The abbreviation "str" appeared in a text I was reading, so I assumed it meant "strict". Now to choose a new name... | |
| Nov 30, 2021 at 0:58 | comment | added | Theo Johnson-Freyd | An example is a result of Mac Lane that every bicategory, and in particular every monoidal 1-category (these being the bicategories on one object), is equivalent to a strict one, and so you can "strictify" any bicategory. | |
| Nov 30, 2021 at 0:55 | comment | added | Theo Johnson-Freyd | Given this use of the word "strict", the term "strictification" is generally used to mean a procedure which replaces your not-necessarily-strict higher category by a strict one. | |
| Nov 30, 2021 at 0:54 | comment | added | Theo Johnson-Freyd | to a sub-2-category of the strict 2-category of categories (namely: to an algebra, assign its category of modules, and to a bimodule, assign the functor that tensors with that bimodule). | |
| Nov 30, 2021 at 0:54 | comment | added | Theo Johnson-Freyd | A "strict higher category" is one whose higher coherences hold not just up to (coherent) isomorphism, but actually up to equality. For example, a "strict 2-category" is a bicategory where the associativity and unitality hold strictly. The 2-category of categories and functors is a strict 2-category. The 2-category of algebras and bimodules, for example, is not a strict 2-category — the composition of bimodules is a tensor product, and if you define it as, say, some sort of quotient of a space spanned by ordered pairs, then this will not be strictly associative — but it is obviously equivalent | |
| Nov 29, 2021 at 22:42 | vote | accept | Exit path | ||
| Nov 29, 2021 at 22:37 | comment | added | Maxime Ramzi | Another common name for $Str(C)$ is "weak homotopy type of $C$" (although I don't know if that can be adapted to a name for the functor $Str$...) | |
| Nov 29, 2021 at 22:25 | comment | added | Exit path | @MikeShulman Thanks! What is the usual meaning of "strictification," if any? | |
| Nov 29, 2021 at 22:20 | comment | added | Mike Shulman | Just a note that this should not be called "strictification", but rather something like "localization" or "groupoidification" (although the latter also has other meanings). | |
| Nov 29, 2021 at 22:15 | answer | added | Maxime Ramzi | timeline score: 6 | |
| Nov 29, 2021 at 21:59 | history | asked | Exit path | CC BY-SA 4.0 |