Timeline for Are strict $\infty$-categories localized at weak equivalences a full subcategory of weak $\infty$-categories?
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| Mar 1, 2020 at 4:18 | comment | added | Tim Campion | One subtlety is that $L \underline X$ is a priori only connected to $X$ by a zigzag of weak equivalences passing outside the Brown-Golasinski category. But that's ok -- $L \underline X$ is cofibrant as an $\infty$-category with strict inverses, and any homotopy equivalence between products of Eilenberg-MacLane spaces lifts to an equivalence of $H \mathbb Z$-modules, so there exists a direct Brown-Golasinski equivalence $L \underline X\to X$. Thus we are indeed justified in computing the derived Brown-Golasinski homotopy classes of maps from $X$ to $Y$ using $L \underline X$. | |
| Mar 1, 2020 at 3:56 | comment | added | Tim Campion | Thus we have certain objects in the homotopy category of strict $\infty$-categories with weak inverses -- namely those objects whose inverses are strict -- which have smaller homspaces as strict $\infty$-categories than they have as weak $\infty$-categories. So the inclusion is not fully faithful. | |
| Mar 1, 2020 at 3:54 | comment | added | Tim Campion | And homotopies with respect to $I \underline X$ are in bijection with homotopies with respect to $L(I \underline X)$. Now, maps in the homotopy category of strict $\infty$-categories from $X$ to $Y$ can be described as $Hom(\underline X,Y)$ modulo $I \underline X$ homotopy. Whereas maps in the homotopy category of strict $\infty$-groupoids from $X$ to $Y$ can be described as $Hom(L\underline X, Y)$ module $L (I\underline X)$-homotopy. But these are the same set! So it appears to me that at least at the level of homotopy categories, we get the same set of maps! | |
| Mar 1, 2020 at 3:53 | comment | added | Tim Campion | @SimonHenry Hang on though -- at the 1-categorical level, strict $\infty$-categories with all strict inverses are reflective in strict $\infty$-categories; let $L$ denote the reflection functor. If $X,Y$ are strict $\infty$-categories with strict inverses and $\underline X$ is a cofibrant replacement of $X$, then any map $\underline X \to Y$ factors through some map $L \underline X \to Y$. And if $I \underline X$ is a cofibrant cylinder on $\underline X$, then because $L$ is left Quillen, we have that $L(I \underline X)$ is a cofibrant cylinder for $L\underline X$. | |
| Mar 1, 2020 at 2:50 | comment | added | Simon Henry | Exactly. Or at least, we don't know if we can. The forgetfull functor from the Brown-Golasinski model structure to the Lafont-Metayer-Worytkiewicz model structure is right Quillen, I do not known how this adjunction behave on the corresponding $(\infty,1)$-categories, ( and I have talked about this with enough people to convince me that this was not known). So as far as I know the left adjoint could be homotopy fully faithfull, in which case your argument would apply. | |
| Mar 1, 2020 at 2:35 | comment | added | Tim Campion | @SimonHenry Ah -- I completely failed to appreciate the distinction between strict $\infty$-categories with all strict inverses and strict $\infty$-categories with all weak inverses.... If I have things right, the Brown-Golasinski model structure on the former is projectively-induced from the LaFont-Metayer-Worytkiewicz model structure where the latter live. So fibrancy is not an issue, but an $\infty$-category with strict inverses won't typically be cofibrant in the LaFont-Metayer-Worytkiewicz model structure. So we can't even compute the mapping spaces in the way I was assuming... | |
| Mar 1, 2020 at 1:59 | comment | added | Simon Henry | ... more maps than in the localization of the category of strict $\infty$-groupoid. As far as I know it is an open question whether the full subcategory of the localization of the folk model structure on "reversible $\infty$-category" is equivalent to the localization of the category of strict $\infty$-groupoids. But of course I definitely agree that the answer is most likely no. | |
| Mar 1, 2020 at 1:56 | comment | added | Simon Henry | I'm not completely agreeing with your answer (though I believe you are probably right in the end). My objection is as follows: we need to distinguishes the strict $\infty$-groupoid where every arrow is strongly invertible from these where every arrows is only weekly invertible (I'll call these reversible arrows). When you compute map between two strict $\infty$-groupoids in the folk model structure you need to consider a resolution of the $\infty$-groupoid on the left by an equivalent $\infty$-category, where the arrow will only be reversible and not invertible, so there is potentially... | |
| Feb 29, 2020 at 20:28 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Feb 28, 2020 at 18:09 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Feb 28, 2020 at 7:49 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| S Feb 28, 2020 at 7:38 | history | answered | Tim Campion | CC BY-SA 4.0 | |
| S Feb 28, 2020 at 7:38 | history | made wiki | Post Made Community Wiki by Tim Campion |