Spahn modus

Idea

A modus is a category with a distinguished reflective subcategory.

Definition

The category $Mo$ is defined by

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

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

The category $GMo$ of geometric modi is defined to have

• the same objects as $Mo$

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

Example

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

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