Covariant & Contravariant Homotopy Theories

TL;DR

This paper develops a unified framework for covariant and contravariant homotopy theories using model structures sensitive to the directionality of cylinder objects, with applications to simplicial sets.

Contribution

It introduces a formalism that constructs direction-sensitive model structures, unifying covariant, contravariant, coCartesian, and Cartesian models within a common framework.

Findings

Models recover known structures on simplicial sets.

Natural definitions of final, initial, smooth, and proper maps.

Framework applies to various homotopy theories.

Abstract

Given a locally presentable category together with a suitable functorial cylinder object, we construct model structures which are sensitive to the `direction' of the cylinder. We show that the Covariant and Contravariant model structures on simplicial sets as well as the coCartesian and Cartesian model structures on marked simplicial sets are examples of our formalism. In this setting, notions of final and initial maps and smooth and proper maps arise very naturally and we will identify these maps in the examples.