# A Revisionist History of (Cisinski) Model Category

Paul Lessard

## Affiliation:

Macquarie University

## Date:

Tue, 02/06/2020 - 12:00pm

## Venue:

RC-4082, The Red Centre, UNSW

## Abstract:

We'll begin with a revisionist history of model category theory (we'll imagine that, in place of homological algebra, the notion had its origin in category theory). Using this apocryphal premise, we'll describe Cisinski's theory of accessible localizers as the natural homotopification of the notion of models for a finite limit sketch in presheaf categories, a straightforward notion subsuming Grothendieck toposes, algebraic categories, and essentially algebraic categories.

We'll then use this understanding of model categories to revisit the 1963 work of Kan. While the study of categories, particularly here in Australia, has begotten the study of higher categories, e.g. $2, 3, \cdots,n, \cdots, \omega$, in strict and various weak guises, little has been made of extending the structure down. We'll present a notion of $Z$-category, with $z$-morphisms in all integer dimensions, and show that Kan's 1963 paper can be read as a proof that spectra admit presentation as locally finite pointed $Z$-groupoids.