nLab
locus

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

Examples

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

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