(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
On the one hand, $\infty$-groupoids form an $(\infty,1)$-topos ($Grpd_\infty$), while their stabilization – plain spectra – form instead a stable $(\infty,1)$-category; on the other hand, the collection of (ie.: the $\infty$-Grothendieck construction on) parameterized spectra, parameterized over $\infty$-groupoids, forms again an $(\infty,1)$-topos: the “tangent $(\infty,1)$-topos” $T Grpd_\infty$.
This observation is originally due to Biedermann (2007), noted down in Joyal (2008) §35.
In search for a term to capture this curious phenomenon more generally, Joyal 2015 proposed to call a (pointed) presentable $(\infty,1)$-category an “$\infty$-locus” if the collection of its $Grpd_\infty$-parameterized objects (namely $(\infty,1)$-functors $Grpd_\infty \to \mathcal{C}$) forms an $(\infty,1)$-topos.
Notice that the terminology “locus” here is unrelated to the common use of locus in mathematics. Compare the earlier proposal in Joyal 2008 to say “logos” for $(\infty,1)$-category, which is similarly ideosyncratic.
More generally, one may consider “loci” given by $(\infty,2)$-sheaves on any $(\infty,1)$-topos (Hoyois 2019).
In the above motivating example, $\mathcal{C} =$ Spectra. In fact, every presentable stable $(\infty,1)$-category is a Joyal $\infty$-locus (essentially by the stable Giraud theorem, cf. Hoyois 2019, p. 1 and Ex. 7). But, for instance, not just spectra but already prespectra and in fact just pointed homotopy types form a Joyal $\infty$-locus, in this sense.
If one drops the requirement that $\mathcal{C}$ be pointed, then every $(\infty,1)$-topos is a Joyal $\infty$-locus (cf. Hoyois 2019, Ex. 6).
For the case over more general base $(\infty,1)$-toposes: sheaves of spectra over an $(\infty,1)$-site $\mathcal{S}$ may be parameterized over objects of the $\infty$-stack $(\infinity,1)$-topos $Sh_\infty(\mathcal{S})$ and the collection of these parameterized sheaves of spectra forms the tangent $(\infty,1)$-topos $T Sh_\infty(\mathcal{S})$. Analogous statements hold more generally for $n$-excisive $(\infty,1)$-functors into any $(\infty,1)$-topos (see there).
The terminology was proposed in:
apparently motivated by the archetypical example of the tangent $(\infty,1)$-topos of parameterized spectra, previously noticed in
and further highlighted in
which goes back to
Dedicated discussion of the notion is in:
Last revised on August 31, 2023 at 08:54:30. See the history of this page for a list of all contributions to it.