nLab Joyal locus

Contents

Context

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Idea

On the one hand, \infty -groupoids form an ( , 1 ) (\infty,1) -topos (Grpd Grpd_\infty), while their stabilization – plain spectra form – form instead a stable ( , 1 ) (\infty,1) -category; on the other hand, the collection of parameterized spectra, parameterized over \infty -groupoids, forms again an ( , 1 ) (\infty,1) -topos: the “tangent ( , 1 ) (\infty,1) -toposTGrpd T Grpd_\infty.

In search for a term to capture this curious phenomenon more generally, Joyal 2015 proposed to call a (pointed) presentable ( , 1 ) (\infty,1) -category an “\infty-locus” if the collection of its Grpd Grpd_\infty-parameterized objects (namely ( , 1 ) (\infty,1) -functors Grpd 𝒞Grpd_\infty \to \mathcal{C}) forms an ( , 1 ) (\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 ( , 1 ) (\infty,1) -category, which is similarly ideosyncratic.

More generally, one may consider “loci” given by ( , 2 ) (\infty,2) -sheaves on any ( , 1 ) (\infty,1) -topos (Hoyois 2019).

Examples

In the above motivating example, 𝒞=\mathcal{C} = Spectra. In fact, every presentable styable ( , 1 ) (\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 𝒞\mathcal{C} be pointed, then every ( , 1 ) (\infty,1) -topos is a Joyal \infty-locus (cf. Hoyois 2019, Ex. 6).

For the case over over more general base ( , 1 ) (\infty,1) -toposes: sheaves of spectra over an ( , 1 ) (\infty,1) -site 𝒮\mathcal{S} may be parameterized over objects of the \infty -stack ( , 1 ) (\infinity,1) -topos Sh (𝒮)Sh_\infty(\mathcal{S}) and the collection of these parameterized sheaves of spectra forms the tangent ( , 1 ) (\infty,1) -topos TSh (𝒮)T Sh_\infty(\mathcal{S}). Analogous statements hold more generally for n n -excisive ( , 1 ) (\infty,1) -functors into any ( , 1 ) (\infty,1) -topos (see there).

References

The terminology was proposed in:

apparently motivated by the archetypical example of the tangent ( , 1 ) (\infty,1) -topos of parameterized spectra, previously noticed in

Dedicated discussion of the notion is in:

Created on July 20, 2022 at 03:18:32. See the history of this page for a list of all contributions to it.