Timeline for Does Grayson/Quillen's "pre group completion" have a universal property?
Current License: CC BY-SA 3.0
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| Jan 27 at 18:46 | answer | added | Georg Lehner | timeline score: 2 | |
| Oct 20, 2017 at 20:41 | history | edited | Tim Campion | CC BY-SA 3.0 |
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| Oct 20, 2017 at 20:34 | comment | added | Tim Campion | @მამუკაჯიბლაძე Right... I suppose in that sense it's pretty straightforward. And this generalizes the usual set-level construction of the group completion of a commutative monoid. I suppose what's puzzling me is the fact that this construction coincides with group completion for sets and for spaces, but not for categories. Maybe I should formulate my question as follows: What is the relationship between this construction and group completion at the level of categories? | |
| Oct 20, 2017 at 19:33 | comment | added | მამუკა ჯიბლაძე | If I understand correctly, the universal property is that of the quotient of $C\times C$ by the diagonal action of $S=Iso(C)$; generally, given an action of a monoidal category $S$ on a category $C$, Grayson defines the quotient as a universal pair $(q,\alpha)$ where $q:C\to Q$ is a functor and $\alpha$ is an isomorphism between $q\circ\text{(action)}:S\times C\to C\to Q$ and $q\circ\text{(projection)}:S\times C\to C\to Q$. | |
| Oct 20, 2017 at 18:30 | history | edited | Tim Campion | CC BY-SA 3.0 |
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| Oct 20, 2017 at 17:51 | comment | added | Neil Strickland | Thomason's paper "Beware the phony multiplication on Quillen's $\mathcal{A}^{-1}\mathcal{A}$" explains some things that look like they work, but are subtly wrong. | |
| Oct 20, 2017 at 17:33 | history | asked | Tim Campion | CC BY-SA 3.0 |