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O'SullivanWrite comment View comments


Could not find web page or university affiliation.

Obstruction theoryWrite comment View comments

Christensen, Dwyer, Isaksen: Obstruction theory in model cats

(de Jong blog post with many example of obstruction theories)[]

ObstructionsWrite comment View comments

arXiv:1002.1423 The Étale Homotopy Type and Obstructions to the Local-Global Principle from arXiv Front: math.AT by Yonatan Harpaz, Tomer M. Schlank In 1969 Artin and Mazur defined the étale homotopy type of an algebraic variety \cite{AMa69}. In this paper we define various obstructions to the local-global principle on a variety $X$ over a global field using the étale homotopy type of $X$ and the concept of homotopy fixed points. We investigate relations between those "homotopy obstructions" and connect them to various known obstructions such as the Brauer -Manin obstruction, the étale-Brauer obstruction and finite descent obstructions. This gives a reinterpretation of known arithmetic obstructions in terms of homotopy theory.

Oka principleWrite comment View comments
OlssonWrite comment View comments
Online notes and booksWrite comment View comments

Alex list

Lecture notes by Chen

Ferretti page

Lots of notes and surveys on arxiv.

OperadWrite comment View comments

Markl, Shnider, Stasheff book! (AMS)

Book draft used in my thesis corrections - by Fresse? Link

Smirnov: Simplicial and operad methods in algebraic topology (AMS book)

Renaissance proceedings (Loday, Stasheff, Voronov eds)

This looks interesting (Kriz and May).

Getzler talk abstract, GATA Sheffield 2008: Abstract: I give a new formulation of the axioms for operads, which covers all of the variants of the theory, such as cyclic operads, modular operads, PROPs, wheeled PROPs, as well as simplicially enriched variants, such as topological field theories of different types.

Operads in algebraic geometry, Kapranov ICM 1998

Toen and Vezzosi: Algebraic geometry over model categories. Early paper, looks very nice! Applications to interpreting DG-schemes, and to defining etale K-theory of E-infty algebras. Expectation to extend the classical work for E-infty algebras to the more general setting of AG over a model cat, for the following concepts: tangent Lie algebra, cotangent complex, Hochschild cohomology, K-theory, A-Q cohomology. Would like to do AG over a symmetric monoidal infty-cat, need strictification results. An E-infty alg should be a monoid in a SM infty-cat. Pp 34: Short nice review of operads and E-infty stuff.

\"Every operad gives rise to a triple (i.e. monad), but the converse is false.\" (Hovey p. 197)

A book by Leinstra

Operads are defined in Baez-Dolan, page 19.

Berger and Moerdijk: Axiomatic homotopy theory for operads (2003)

T. Beke -- Operads from the viewpoint of categorical algebra. In CONM227

nLab, see also algebra over and operad, and this entry on infty-one operads

Quote from Fresse: "The category of differential graded operads is a cofibrantly generated model category and as such inherits simplicial mapping spaces."

Berglund on Homological perturbation theory for algebras over operads

Loday on the operad of associative algebras with derivations, formal group laws, and more

[arXiv:1207.3467] Reedy categories which encode the notion of category actions fra arXiv Front: math.CT av Julia E. Bergner, Philip Hackney Using the structure of the categories {\Delta} and {\Omega} governing category and operad structures, respectively, we define categories which instead encode the structure of a category acting on a category, or a category acting on an operad. We prove that the former has the structure of an elegant Reedy category, whereas the latter has the structure of a generalized Reedy category.

I asked in what generality you can define a cat A object in cat B in terms of the representing functor factoring. Peter Arndt explained the following to me in July 09: There is a hierarchy as follows: PROs, PROPs, operads, Algebraic theories, sketches, gen sketches. Somehow this hierarchy explains when things commute and hence somehow answers the question. He also mentioned Joyal (maybe CRM notes) on some approach to spectra relating to this. R-mod would be an algebraic theory. See talk by Baez for some of the hierarchy.

Operad IIWrite comment View comments

Boardman Vogt W construction for modules over an operad from Top Questions - Math Overflow by Jeremy Miller The W construction of Boardman and Vogt gives a cofibrant replacement for operads. In, Salvatore describes a cofibrant replacement for algebras over an operad. Is there a similar construction which produces cofibrant resolutions of right or left modules over an operad? Does anyone have know of a reference for this?

arXiv:1205.6058 Homotopy unital Ainfinity-algebras from arXiv Front: math.AT by Volodymyr Lyubashenko It is well known that the differential graded operad of Ainfinity-algebras is a cofibrant replacement (a dg-resolution) of the operad of associative differential graded algebras without units. In this article we find a cofibrant replacement of the operad of associative differential graded algebras with units. Algebras over it are called homotopy unital Ainfinity-algebras. We prove that the operad bimodule of Ainfinity-morphisms is a cofibrant replacement of the operad bimodule of morphisms of dg-algebras without units. Similarly we show that the operad bimodule of homotopy unital A_infinity-morphisms is a cofibrant replacement of the operad bimodule of morphisms of dg-algebras with units.

OrbifoldWrite comment View comments
OrlovWrite comment View comments
Orthogonal spectraWrite comment View comments

Here is something on orthogonal spectra, which are in some sense intermediate between S-modules and symmetric spectra. See also K0408 for the equivariant version.

OstvaerWrite comment View comments