Spahn modus

Idea

A modus is a category with a distinguished reflective subcategory.

Definition

The category MoMo is defined by

  • An object is a pair (C,)(C,\Box) consisting of a category CC and an idempotent monad \Box on CC.

  • A morphism f:(C,)(C , )f:(C,\Box)\to (C^\prime, \Box^\prime) is defined to be a functor f:CC f:C\to C^\prime such that f(a)= f(a)f(\Box a)=\Box^\prime f(a).

The category GMoGMo of geometric modi is defined to have

  • the same objects as MoMo

  • but as morphisms only the pullback preserving morphisms of MoMo (“morphism of \Box-closed structures”).

Example

  • Admissible functor (as in Lurie DAG V, p 96)

    • In particular a “geometric envelope”, i.e. a functor f:TGf:T\to G of a pregeometry to a geometry (GG is idempotent complete an has finite limits) such that it is a geometric modus and Fun lex(G,C)GMo(T,C)Fun^{lex}(G, C)\simeq GMo(T,C) for every (,1)(\infty,1)-category CC which has finite limits.

Created on December 15, 2012 at 20:58:38. See the history of this page for a list of all contributions to it.