46 citations · 283 across the 29 of their papers we have counts for
26 papers · 1 filter
Could -category theory be taught to undergraduates?
Emily Riehl
The extension of ordinary category theory to -categories at the start of the 21st century was a spectacular achievement pioneered by Joyal and Lurie with contributions from…
On the -topos semantics of homotopy type theory
Emily Riehl
Many introductions to homotopy type theory and the univalence axiom gloss over the semantics of this new formal system in traditional set-based foundations. This expository article…
A 2-categorical proof of Frobenius for fibrations defined from a generic point
Sina Hazratpour, Emily Riehl
Consider a locally cartesian closed category with an object I and a class of trivial fibrations, which admit sections and are stable under pushforward and retract as arrows. Define…
On -cosmoi of bicategories
Emily Riehl, Mira Wattal
An -cosmos is a setting in which to develop the formal category theory of -categories. In this paper, we explore a few atypical examples of -cosmoi whos…
Cartesian exponentiation and monadicity
Emily Riehl, Dominic Verity
An important result in quasi-category theory due to Lurie is the that cocartesian fibrations are exponentiable, in the sense that pullback along a cocartesian fibration admits a ri…
On the construction of limits and colimits in -categories
Emily Riehl, Dominic Verity
In previous work, we introduce an axiomatic framework within which to prove theorems about many varieties of infinite-dimensional categories simultaneously. In this paper, we estab…
Recognizing quasi-categorical limits and colimits in homotopy coherent nerves
Emily Riehl, Dominic Verity
In this paper we prove that various quasi-categories whose objects are -categories in a very general sense are complete: admitting limits indexed by all simplicial sets. Th…
Categorical notions of fibration
Fosco Loregian, Emily Riehl
Fibrations over a category , introduced to category theory by Grothendieck, encode pseudo-functors , while the special case of discrete fibrat…
Lifting accessible model structures
Richard Garner, Magdalena Kedziorek, Emily Riehl
A Quillen model structure is presented by an interacting pair of weak factorization systems. We prove that in the world of locally presentable categories, any weak factorization sy…
Homotopy coherent structures
Emily Riehl
Naturally occurring diagrams in algebraic topology are commutative up to homotopy, but not on the nose. It was quickly realized that very little can be done with this information.…
The comprehension construction
Emily Riehl, Dominic Verity
In this paper we construct an analogue of Lurie's "unstraightening" construction that we refer to as the "comprehension construction". Its input is a cocartesian fibration $p \colo…
A type theory for synthetic -categories
Emily Riehl, Michael Shulman
We propose foundations for a synthetic theory of -categories within homotopy type theory. We axiomatize a directed interval type, then define higher simplices from it a…
Complicial sets, an overture
Emily Riehl
The aim of these notes is to introduce the intuition motivating the notion of a "complicial set", a simplicial set with certain marked "thin" simplices that witness a composition r…
Infinity category theory from scratch
Emily Riehl, Dominic Verity
We use the terms "-categories" and "-functors" to mean the objects and morphisms in an "-cosmos." Quasi-categories, Segal categories, complete Segal spaces,…
Kan extensions and the calculus of modules for -categories
Emily Riehl, Dominic Verity
Various models of -categories, including quasi-categories, complete Segal spaces, Segal categories, and naturally marked simplicial sets can be considered as the object…
Fibrations and Yoneda's lemma in an -cosmos
Emily Riehl, Dominic Verity
We use the terms -categories and -functors to mean the objects and morphisms in an -cosmos: a simplicially enriched category satisfying a few axioms, remini…
Completeness results for quasi-categories of algebras, homotopy limits, and related general constructions
Emily Riehl, Dominic Verity
Consider a diagram of quasi-categories that admit and functors that preserve limits or colimits of a fixed shape. We show that any weighted limit whose weight is a projective cofib…
Homotopy coherent adjunctions and the formal theory of monads
Emily Riehl, Dominic Verity
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 deve…
Six model structures for DG-modules over DGAs: Model category theory in homological action
Tobias Barthel, J. P. May, Emily Riehl
In Part 1, we describe six projective-type model structures on the category of differential graded modules over a differential graded algebra A over a commutative ring R. When R is…
The 2-category theory of quasi-categories
Emily Riehl, Dominic Verity
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-categ…
The theory and practice of Reedy categories
Emily Riehl, Dominic Verity
The goal of this paper is to demystify the role played by the Reedy category axioms in homotopy theory. With no assumed prerequisites beyond a healthy appetite for category theoret…
Multivariable adjunctions and mates
Eugenia Cheng, Nick Gurski, Emily Riehl
We present the notion of "cyclic double multicategory", as a structure in which to organise multivariable adjunctions and mates. The classic example of a 2-variable adjunction is t…
Monoidal algebraic model structures
Emily Riehl
Extending previous work, we define monoidal algebraic model structures and give examples. The main structural component is what we call an algebraic Quillen two-variable adjunction…
Levels in the toposes of simplicial sets and cubical sets
Carolyn Kennett, Emily Riehl, Michael Roy +1
The essential subtoposes of a fixed topos form a complete lattice, which gives rise to the notion of a level in a topos. In the familiar example of simplicial sets, levels coincide…