Denis-Charles Cisinski

Wednesday, July 22, 3:00 - 5:00, MIT Room 2-131

Homotopy weakening of algebras

Abstract:
Given a suitable monad T on a presheaf
category, one can define the homotopy theory of
`quasi-T-algebras' and the Quillen equivalent
homotopy theory of `complete Segal T-spaces'.
In the case where T is the free category monad
on the category of graphs, we recover respectively
the Joyal model structure for quasi-categories, and
the Rezk model structure of complete Segal spaces.
In the case where T is the free operad monad on the
category of multigraphs, we recover the
model structures on (simplicial) dendroidal sets
corresponding to quasi-operads in the sense of
Moerdijk-Weiss.
The case of the free n-category monad on n-graphs
defines suitable notions of quasi-n-categories for n>0.

If there is time we will go on to discuss some continuations.
Given a model structure on a presheaf category C
which is compatible with cartesian product,
I can explain how to produce the homotopy
theory of quasi-categories enriched in Ho(C)
(with respect to cartesian product). Here again,
there is a Joyal version and a Rezk version
(the latter has been worked out by Rezk himself
in a recent preprint). This allows an inductive description of
quasi-n-categories.