• Corpus ID: 46657525

∞-Categories for the Working Mathematician

  title={∞-Categories for the Working Mathematician},
  author={Emily Riehl and Dominic R. Verity},
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, product, pullback, retract, and limits of towers; see Lemma C.1.1. Proof. For now see [47, 11.1.4] and dualize. On account of the dual of Lemma C.1.1, any set of maps in a cocomplete category “cellularly generates” a larger class of maps with the same left lifting property. C.1.2. Definition. The class of… 

Clifford's theorem for orbit categories

. Clifford theory relates the representation theory of finite groups to those of a fixed normal subgroup by means of induction and restriction, which is an adjoint pair of functors. We generalize this

Homotopy theory of Moore flows (II)

This paper proves that the q-model structures of Moore flows and of multipointed d-spaces are Quillen equivalent. The main step is the proof that the counit and unit maps of the Quillen adjunction

On a spectral sequence for the cohomology of infinite loop spaces

We study the mod-2 cohomology spectral sequence arising from delooping the Bousfield‐Kan cosimplicial space giving the 2‐nilpotent completion of a connective spectrum X . Under good conditions its E2

Curved L-infinity algebras and lifts of torsors

Consider an extension of finite dimensional nilpotent Lie algebras 0→ h→ g̃→ g→ 0 (over a field k of characteristic zero) corresponding to an extension of unipotent algebraic groups 1 → H → G̃ → G →

Construction of coend and the reconstruction theorem of bialgebras

Assume k is a field and let F : C → V ectk be a small k-linear functor from a k-linear abelian category C to the category of vector spaces over the field k, the purpose of this note is to use a

A short course on ∞-categories

In this short survey we give a non-technical introduction to some main ideas of the theory of ∞-categories, hopefully facilitating the digestion of the foundational work of Joyal and Lurie. Besides

The central sheaf of a Grothendieck category

. The center Z p A q of an abelian category A is the endomorphism ring of the identity functor on that category. A localizing subcategory of a Grothendieck category C is said to be stable if it is

Notes on pre-Frölicher spaces

Abstract In this work we introduce a class of Sikorski differential spaces (M, D) called pre-Fr¨olicher spaces, on which the process of yielding a Fr¨olicher structure on the underlying set M is D



Homotopy-theoretic aspects of 2-monads

We study 2-monads and their algebras using a Cat-enriched version of Quillen model categories, emphasizing the parallels between the homotopical and 2-categorical points of view. Every 2-category

Fibrations and Yoneda's lemma in an ∞-cosmos

Towards an (∞,2)-category of homotopy coherent monads in an ∞-cosmos

This thesis is part of a program initiated by Riehl and Verity to study the category theory of (infinity,1)-categories in a model-independent way. They showed that most models of


This paper is an exposition of the ideas and methods of Cisinksi, in the context of A-presheaves on a small Grothendieck site, where A is an arbitrary test category in the sense of Grothendieck. The

Vogt's theorem on categories of homotopy coherent diagrams

Let Top be the category of compactly generated topological spaces and continuous maps. The category, Top, can be given the structure of a simplicially enriched category (or S-category, S being the

Yoneda Structures from 2-toposes

  • Mark Weber
  • Mathematics
    Appl. Categorical Struct.
  • 2007
A 2-categorical generalisation of the notion of elementary topos is provided, and some of the properties of the Yoneda structure it generates are explored and results enabling one to exhibit objects as cocomplete in the sense definable within a YonedA structure are presented.

The 2-category theory of quasi-categories

Completeness results for quasi-categories of algebras, homotopy limits, and related general constructions

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 cofibrant simplicial

Calculating simplicial localizations