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∞-Categories for the Working Mathematician

homotopy theory C.1. Lifting properties, weak factorization systems, and Leibniz closure C.1.1. Lemma. Any class of maps characterized by a right lifting property is closed under composition,… Expand

Categorical Homotopy Theory

- E. Riehl
- Mathematics
- 26 May 2014

This book develops abstract homotopy theory from the categorical perspective with a particular focus on examples. Part I discusses two competing perspectives by which one typically first encounters… Expand

Homotopy coherent adjunctions and the formal theory of monads

- E. Riehl, Dominic R. Verity
- Mathematics
- 30 October 2013

In this paper, we introduce a cofibrant simplicial category that we call the free homotopy coherent adjunction and characterize its n-arrows using a graphical calculus that we develop here. The… Expand

Use of physician assistants as surgery/trauma house staff at an American College of Surgeons-verified Level II trauma center.

- W. Miller, E. Riehl, M. Napier, K. Barber, H. Dabideen
- Medicine
- The Journal of trauma
- 1 February 1998

TLDR

Homotopical resolutions associated to deformable adjunctions

- A. Blumberg, E. Riehl
- Mathematics
- 14 August 2012

Given an adjunction FaG connecting reasonable categories with weak equivalences, we define a new derived bar and cobar construction associated to the adjunction. This yields homotopical models of the… Expand

The 2-category theory of quasi-categories

- E. Riehl, Dominic R. Verity
- Mathematics
- 21 June 2013

In this paper we redevelop the foundations of the category theory of quasi-categories (also called infinity-categories) using 2-category theory. We show that Joyal's strict 2-category of… Expand

Six model structures for DG-modules over DGAs: model category theory in homological action

- Tobias Barthel, Jon P. May, E. Riehl
- Mathematics
- 4 October 2013

In Part 1, we describe six projective-type model structures on the category of dierential graded modules over a dierential graded algebra A over a commutative ring R. When R is a eld, the six… Expand

HOMOTOPY (LIMITS AND) COLIMITS

- E. Riehl
- Physics
- 2009

A laser includes a laser medium, a light source for emitting light which pumps the laser medium, a pair of mirrors which are disposed on opposite sides of the laser medium and form a resonator, an… Expand

A type theory for synthetic ∞-categories

- E. Riehl, Michael Shulman
- Mathematics
- 1 February 2018

We propose foundations for a synthetic theory of $(\infty,1)$-categories within homotopy type theory. We axiomatize a directed interval type, then define higher simplices from it and use them to… Expand

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