nLab Joyal locus

Idea

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 $\mathcal{C}$ 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).

Examples

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).

Related concepts

• doubly monoidal (∞,1)-topos

• doubly closed monoidal category

References

The terminology was proposed in:

• André Joyal at 58:00, 59:52 in: New variations on the notion of topos, talk at IHES (2015) [video]

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

• André Joyal, §35 of: Notes on Logoi (2008) [pdf, pdf]

and further highlighted in

• Mathieu Anel, André Joyal, §4.2.3 in: Topo-logie, in New Spaces for Mathematics and Physics, Cambridge University Press (2021) 155-257 [doi:10.1017/9781108854429.007, pdf]

which goes back to

• Georg Biedermann, private communication with Joyal (2007)

Dedicated discussion of the notion is in:

• Marc Hoyois, Topoi of parametrized objects, Theory and Applications of Categories, 34 9 (2019) 243-248 [arXiv:1611.02267, tac:34-09]