Abstract Rune Haugseng

Infinity-operads and analytic monads

Infinity-operads are a powerful formalism for encoding homotopy-coherent algebraic structures, generalizing the classical theory of operads to the setting of infinity-categories. In this lecture series I will first give a brief introduction to infinity-categories and then introduce two (equivalent) definitions of infinity-operads: that of Lurie, which encodes infinity-operads using analogues of the "categories of operators" of May-Thomason, and that of Cisinski-Moerdijk-Weiss, which encodes them as presheaves of spaces on a category of trees. I will then discuss joint work with D. Gepner and J. Kock, where we obtain a description of infinity-operads in terms of monads: we show that infinity-operads are equivalent to  analytic monads on slices of the infinity-category of spaces. This generalizes to the homotopical setting a result of Joyal, which characterizes operads in sets as analytic monads. In the one-object case, our model can also be viewed as a homotopical version of the description of operads as associative monoids in symmetric sequences.

References for infinity-categories (ordered by increasing length):

* Omar Antolín Camarena, A Whirlwind Tour of the World of (∞,1)-categories
 
http://arxiv.org/abs/1303.4669
  (An introductory survey of infinity-categories and their applications)
* Moritz Groth, A short course on ∞-categories
 
http://arxiv.org/abs/1007.2925
* Charles Rezk, Higher category theory and quasicategories
 
http://faculty.math.illinois.edu/~rezk/595-fal16/quasicats.pdf
  (Detailed lecture notes from a course on quasicategories)
* Denis-Charles Cisinski, Higher categories and homotopical algebra
 
http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf
* Emily Riehl and Dominic Verity, ∞-categories for the working mathematician
 
http://www.math.jhu.edu/~eriehl/ICWM.pdf
  (Book in progress.)
* Jacob Lurie, Higher Topos Theory
 
http://math.harvard.edu/~lurie/papers/HTT.pdf
  (Still the canonical reference - chapter 1 is a good introduction to the theory.)

 Introductions to operads:
* J. Peter May, Operads, algebras, and modules
 
http://www.math.uchicago.edu/~may/PAPERS/mayi.pdf
* John Baez, Week 191
 
http://math.ucr.edu/home/baez/week191.html
* Martin Markl, Steve Schnider and Jim Stasheff, Operads in Algebra, Topology and Physics, AMS, Providence, Rhode Island, 2002
* Jean-Louis Loday and Bruno Vallette, Algebraic operads, Grundlehren der Mathematischen Wissenschaften vol. 346, Springer, Heidelberg, 2012

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