# Loci

## Disambiguation

There are other mathematical uses of the term ‘locus’. See, for example, smooth locus and derived critical locus.

## Idea

A locus is a $(\infty, 1)$-category $C$ such that the $(\infty, 1)$-category of indexed families of objects of $C$ over ∞-groupoids form a (∞,1)-topos. Since $(\infty, 1)$-toposes are closed under left exact localizations, so are loci.

## Examples

The $(\infty, 1)$-category of pointed types is a locus, since families of it are a presheaf $(\infty, 1)$-category (the $(\infty, 1)$-category of retractions or of diagrams on the walking map-equipped-with-a-section).

Similarly, the $(\infty, 1)$-category of prespectra is a locus. And the $(\infty, 1)$-category of spectra is a left exact localization of the $(\infty, 1)$-category of prespectra. Hence spectra form a locus, so parametrized spectra form an $(\infty, 1)$-topos.

## References

This page arose out of discussions at this nForum discussion, based on an idea of Andre Joyal.

Last revised on November 8, 2016 at 16:56:36. See the history of this page for a list of all contributions to it.