A (1-)locus is defined by Joyal to be a locally presentable category $C$ (generated by a small set of compact objects under colimits) with a zero object such that collections of objects in $C$ indexed over sets (denoted Fam($C$)) form a Grothendieck topos.

The nlab page (which briefly discusses the $\infty$-categorical analog) is here: https://ncatlab.org/nlab/show/locus

An example of a locus is the category of pointed sets (Fam($Set_{\bullet}$) is equivalent to the category of presheaves on the walking-arrow-equipped-with-a-section).

What are some other examples of loci? I couldn't find any literature on the topic (probably because it is a very new notion)

  • $\begingroup$ Your third sentence should say that the category of set-indexed families of pointed sets is equivalent to a presheaf category (not the category of pointed sets itself). $\endgroup$ – Alexander Campbell Oct 30 '16 at 9:11
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    $\begingroup$ For any locally finitely presentable symmetric monoidal closed category $\mathcal{V}$, there is a notion of $\mathcal{V}$-topos: a $\mathcal{V}$-category $\mathcal{C}$ such that there exists a small $\mathcal{V}$-category $\mathcal{A}$ and an accessible fully faithful functor $\mathcal{C} \longrightarrow [\mathcal{A}^{\text{op}},\mathcal{V}]$ with a finite-limit preserving left adjoint. I wonder if there is any relation between the notion of $\mathcal{V}$-topos, for $\mathcal{V}$ the category of pointed sets with the smash tensor product, and Joyal's notion of locus? $\endgroup$ – Alexander Campbell Oct 30 '16 at 9:36
  • $\begingroup$ Ah yes, thanks for pointing that out $\endgroup$ – user84563 Oct 30 '16 at 13:05
  • $\begingroup$ @AlexanderCampbell What sort of relation are you expecting? $\endgroup$ – user84563 Oct 30 '16 at 15:37
  • $\begingroup$ The example of pointed sets easily generalizes as follows: for any set $S$ and topos $E$, $S$-pointed objects of $E$ form a locus (well, there's no zero object if $S\neq*$, but I'm not sure what the point of that condition is). Joyal conjectured that any presentable stable ∞-category is an ∞-locus, but this looks like an ∞-only phenomenon, as such ∞-loci give rise to ∞-topoi whose underlying $n$-topos is a point for any finite $n$. $\endgroup$ – Marc Hoyois Oct 30 '16 at 17:07

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