Bousfield Localization and Algebras over Colored Operads

White, David; Yau, Donald

doi:10.1007/s10485-017-9489-8

Cited by 59 publications

(68 citation statements)

References 61 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We do not have to provide a model of the free C-colored operad functor as existence is sufficient for our work. Explicit (equivalent) models can be found in [BM07,Rez96,WY18].…”

Section: Colored Operadsmentioning

confidence: 99%

Operads for algebraic quantum field theory

Benini

Schenkel

Woike

2020

Commun. Contemp. Math.

View full text Add to dashboard Cite

We construct a colored operad whose category of algebras is the category of algebraic quantum field theories. This is achieved by a construction that depends on the choice of a category, whose objects provide the operad colors, equipped with an additional structure that we call an orthogonality relation. This allows us to describe different types of quantum field theories, including theories on a fixed Lorentzian manifold, locally covariant theories and also chiral conformal and Euclidean theories. Moreover, because the colored operad depends functorially on the orthogonal category, we obtain adjunctions between categories of different types of quantum field theories. These include novel and interesting constructions, such as timeslicification and local-to-global extensions of quantum field theories. We compare the latter to Fredenhagen's universal algebra. Report no.: ZMP-HH/17-26, Hamburger Beiträge zur Mathematik Nr. 682Keywords: algebraic quantum field theory, locally covariant quantum field theory, colored operads, change of color adjunctions, Fredenhagen's universal algebra MSC 2010: 81Txx, 18D50

show abstract

“…We do not have to provide a model of the free C-colored operad functor as existence is sufficient for our work. Explicit (equivalent) models can be found in [BM07,Rez96,WY18].…”

Section: Colored Operadsmentioning

confidence: 99%

Operads for algebraic quantum field theory

Benini

Schenkel

Woike

2020

Commun. Contemp. Math.

View full text Add to dashboard Cite

show abstract

“…Our main result, Theorem 5.14, is that the TQ A -local homotopy theory for Oalgebras (associated to the classes of maps in Definition 3.4) can be established (e.g., as a semi-model structure in the sense of Goerss-Hopkins [19] and Spitzweck [42], that is both cofibrantly generated and simplicial) by localizing with respect to a set of strong cofibrations that are TQ A -equivalences; see, for instance, Mandell [33], White [43], and White-Yau [44] where semi-model structures naturally arise in some interesting applications. A closely related (but different) notion of semi-model structure is explored in Fresse [16].…”

Section: Tq a -Local Homotopy Theorymentioning

confidence: 99%

Topological Quillen localization of structured ring spectra

Harper¹,

Zhang²

2019

Tbilisi Math. J.

View full text Add to dashboard Cite

The aim of this short paper is two-fold: (i) to construct a TQlocalization functor on algebras over a spectral operad O, in the case where no connectivity assumptions are made on the O-algebras, and (ii) more generally, to establish the associated TQ-local homotopy theory as a left Bousfield localization of the usual model structure on O-algebras, which itself is almost never left proper, in general. In the resulting TQ-local homotopy theory, the "weak equivalences" are the TQ-homology equivalences, where "TQ-homology" is short for topological Quillen homology, which is also weakly equivalent to stabilization of O-algebras. More generally, we establish these results for TQhomology with coefficients in a spectral algebra A. A key observation, that goes back to the work of Goerss-Hopkins on moduli problems, is that the usual left properness assumption may be replaced with a strong cofibration condition in the desired subcell lifting arguments: Our main result is that the TQ-local homotopy theory can be established (e.g., a semi-model structure in the sense of Goerss-Hopkins and Spitzweck, that is both cofibrantly generated and simplicial) by localizing with respect to a set of strong cofibrations that are TQ-equivalences. TQ-homology of an O-algebra with coefficients in AIf X is an O-algebra, then we may factor the map * → X * →X ≃ − − → X as a cofibration followed by an acyclic fibration; we are using the positive flat stable model structure (see, for instance, [24]). In particular,X is a cofibrant replacement of X.

show abstract

“…In light of a recent preprint ( [18]) by White and Yau investigating when analogues of Theorems 1.4, 1.5 hold for general categories, this seems likely to be a formal question.…”

Section: Directions For Future Workmentioning

confidence: 99%

“…[18]). The main goal of this paper is to prove that for the category Sp Σ of symmetric spectra, however, these questions can be answered while making minimal to no cofibrancy assumptions on O, at least provided one uses the positive S model structure as the underlying model structure on Sp Σ .…”

Section: Introductionmentioning

confidence: 99%

Cofibrancy of operadic constructions in positive symmetric spectra

Pereira¹

2016

Homology, Homotopy and Applications

View full text Add to dashboard Cite

We show that when using the underlying positive model structure on symmetric spectra one obtains cofibrancy conditions for operadic constructions under much milder hypothesis than one would need for general categories. Our main result provides such an analysis for a key operation, the "relative composition product" ○ O between right and left O-modules over a spectral operad O, and as a consequence we recover (and usually strengthen) previous results establishing the Quillen invariance of model structures on categories of algebras via weak equivalences of operads, compatibility of forgetful functors with cofibrations and Reedy cofibrancy of bar constructions.Key to the results above are novel cofibrancy results for n-fold smash powers of positive cofibrant spectra (and the relative statement for maps). Roughly speaking, we show that such n-fold powers satisfy a (new) type of Σn-cofibrancy which can be viewed as "lax Σn-free/projective cofibrancy" in that it determines a larger class of cofibrations still satisfying key technical properties of "true Σn-free/projective cofibrancy".

show abstract

Bousfield Localization and Algebras over Colored Operads

Cited by 59 publications

References 61 publications

Operads for algebraic quantum field theory

Operads for algebraic quantum field theory

Topological Quillen localization of structured ring spectra

Cofibrancy of operadic constructions in positive symmetric spectra

Contact Info

Product

Resources

About