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started entry on stabilization (in the sense of sending an (oo,1)-category to its free stable (infinity,1)-category)
I want to eventually state more properties of the effect of stabilization on objects here.
added in the References of stabilization:
Discussion of the relation between stabilization of (∞,1)-categories (to stable (∞,1)-categories) and of model categories (to stable model categories) is in section 4 of
- Marco Robalo, Noncommutative Motives I: A Universal Characterization of the Motivic Stable Homotopy Theory of Schemes, June 2012 (arxiv:1206.3645)
added also a brief paragraph under Properties – Construciton in terms of stable model categories
we had pointer to
I have added pointer to the published version, which is:
added also pointer to
for making explicit stabilization with respect to a set of objects.
Nice question! I haven’t thought about it quasi-categorically, but $(\infty, 1)$-limits in the nerve of a category should just be ordinary limits, and then the loop space functor is trivial (takes everything to the point).
One can also observe that the nerve of the category of abelian groups cannot really be stable, since its homotopy category would then be triangulated, and any such structure (if there even exists one?) will be very artificial/trivial.
I would say that to make the analogy precise, one can proceed as follows. One observes that the definition of a stable $(\infty, 1)$-category is a categorification of the notion of an abelian category. The category of abelian groups is the ’free abelian category on the category of sets’, and the $(\infty, 1)$-category of spectra is the ’free stable $(\infty, 1)$-category on the $(\infty, 1)$-category of spaces’. But the actual construction of these things is different, and only exists under circumstances which are more or less mutually exclusive (localisation of category of presheaves of sets vs $(\infty, 1)$-localisation of $(\infty, 1)$-category of presheaves of spaces).
The stabilization of the category of sets is zero. An object in Stab(Set) is a sequence of pointed sets $(E_0, E_1, \ldots)$ together with isomorphisms $\Omega(E_{i+1}) \simeq E_i$. But $\Omega(X) = \ast \times_X \ast \simeq \ast$ for every pointed set $X$. So every object is isomorphic to $(\ast, \ast, \ldots)$. The space of endomorphisms of this object is a limit of the endomorphism spaces $Map_{Set_*}(\ast, \ast) \simeq \ast$, which is again contractible.
Yes, this is what I had in mind when I wrote that the loop space functor is trivial.
Thank you for the answers. This clears up the fact that the existence of a nontrivial loop functor on the $\infty$-category of pointed spaces is something which significantly distinguishes $\infty$-category theory from ordinary category theory (where the category of pointed sets has trivial loop functor).
A question: what does stabilisation give us if we keep going up on the categorical ladder, from sets to categories, to $2$-categories, to $3$-categories, and so on?
I think the stabilisations of $\mathrm{N}_{\bullet}(\mathsf{Cats})$ and $\mathrm{N}_{\bullet}(\mathsf{Grpd})$, the nerves of the $1$-categories of categories and groupoids are trivial for the same reason that the stabilisation of $\mathrm{N}_{\bullet}(\mathsf{Sets})$ is trivial―pushouts of the form $\Omega\mathcal{C}\overset{\mathrm{def}}{=}*\times_{\mathcal{C}}*$ are isomorphic to $*$, and once again the loop space object functor will be trivial.
However, if instead we take the Duskin nerves $\mathrm{N}^{\mathrm{D}}_{\bullet}\left(\mathsf{Grpd}_{(2,1)}\right)$ and $\mathrm{N}^{\mathrm{D}}_{\bullet}\left(\mathsf{Pith}\left(\mathsf{Cats}_{(2,0)}\right)\right)$ of the $(2,1)$-category of groupoids and of the pith of the $(2,0)$-category of categories, then the based loop space object functors will no longer be trivial: $\Omega\mathcal{C}\overset{\mathrm{def}}{=}*\times^{\mathsf{h}}_{\mathcal{C}}*$ is now the pseudopullback $*\times^{\mathsf{ps}}_{\mathcal{C}}*$, which (if I’m not mistaken) is given by $\mathrm{Aut}(x_0)_\mathsf{disc}$, the discrete groupoid on the automorphism group of the point $x_0$ of a pointed category/groupoid $(\mathcal{C},x_0)$.
So $\Omega^{2}\mathcal{C}$ will once again be trivial, and hence so will the stabilisations of $\mathrm{N}^{\mathrm{D}}_{\bullet}\left(\mathsf{Grpd}_{(2,1)}\right)$ and $\mathrm{N}^{\mathrm{D}}_{\bullet}\left(\mathsf{Pith}\left(\mathsf{Cats}_{(2,0)}\right)\right)$, right?
More generally, one would expect based loop space object functors to get more and more non-trivial as $n$ goes up (e.g. for an $n$-category $\mathcal{C}$, the iterated loop space object $\Omega^{n}\mathcal{C}$ is trivial, but $\Omega^{n-1}\mathcal{C}$ isn’t, the idea being that $\Omega$ sends $n$-categories to $(n-1)$-categories (or rather to grouplike monoidal $(n-1)$-categories, or something close)), but stabilisations will nevertheless only be nontrivial for $n=\infty$, correct?
(I was thinking about adding these as examples (including $\mathrm{Aut}(x_0)_\mathsf{disc}$ as an example in the loop space object page), but I’d like to confirm that these are right with someone else before writing something possibly wrong on these pages.)
added pointer to formalization in dependent linear homotopy type theory:
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