(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
On the one hand, -groupoids form an -topos (), while their stabilization – plain spectra – form instead a stable -category; on the other hand, the collection of (ie.: the -Grothendieck construction on) parameterized spectra, parameterized over -groupoids, forms again an -topos: the “tangent -topos” .
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 -category an “-locus” if the collection of its -parameterized objects (namely -functors ) forms an -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 -category, which is similarly ideosyncratic.
More generally, one may consider “loci” given by -sheaves on any -topos (Hoyois 2019).
In the above motivating example, Spectra. In fact, every presentable stable -category is a Joyal -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 -locus, in this sense.
If one drops the requirement that be pointed, then every -topos is a Joyal -locus (cf. Hoyois 2019, Ex. 6).
For the case over more general base -toposes: sheaves of spectra over an -site may be parameterized over objects of the -stack -topos and the collection of these parameterized sheaves of spectra forms the tangent -topos . Analogous statements hold more generally for -excisive -functors into any -topos (see there).
The terminology was proposed in:
apparently motivated by the archetypical example of the tangent -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.