From operator categories to higher operads (original) (raw)
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From homotopy operads to infinity-operads
2014
The goal of the present paper is to compare, in a precise way, two notions of operads up to homotopy which appear in the literature. Namely, we construct a functor from the category of strict unital homotopy colored operads to the category of infinity-operads. The former notion, that we make precise, is the operadic generalization of the notion of A-infinity-categories and the latter notion was defined by Moerdijk-Weiss in order to generalize the simplicial notion of infinity-category of Joyal-Lurie. This functor extends in two directions the simplicial nerve of Faonte-Lurie for A-infinity-categories and the homotopy coherent nerve of Moerdijk-Weiss for differential graded operads; it is also shown to be equivalent to a big nerve à la Lurie for differential graded operads. We prove that it satisfies some homotopy properties with respect to weak equivalences and fibrations; for instance, it is shown to be a right Quillen functor.
Algebras of higher operads as enriched categories II
2009
One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we continue the work of [7] to adapt the machinery of globular operads [4] to this task. The resulting theory includes the Gray tensor product of 2-categories and the Crans tensor product [12] of Gray categories. Moreover much of the previous work on the globular approach to higher category theory is simplified by our new foundations, and we illustrate this by giving an expedited account of many aspects of Cheng's analysis [11] of Trimble's definition of weak n-category. By way of application we obtain an "Ekmann-Hilton" result for braided monoidal 2-categories, and give the construction of a tensor product of A-infinity algebras.
A pr 2 00 3 Axiomatic homotopy theory for operads
2003
We give sufficient conditions for the existence of a model structure on operads in an arbitrary symmetric monoidal model category. General invariance properties for homotopy algebras over operads are deduced.
An Algebraic Definition of (∞, N)-Categories
2015
In this paper we define a sequence of monads T(∞,n)(n ∈ N) on the category ∞-Gr of ∞-graphs. We conjecture that algebras for T(∞,0), which are defined in a purely algebraic setting, are models of∞-groupoids. More generally, we conjecture that T(∞,n)-algebras are models for (∞, n)-categories. We prove that our (∞, 0)-categories are bigroupoids when truncated at level 2. Introduction The notion of weak (∞, n)-category can be made precise in many ways depending on our approach to higher categories. Intuitively this is a weak∞-category such that all its cells of dimension greater than n are equivalences. Models of weak (∞, 1)-categories (case n = 1) are diverse: for example there are the quasicategories studied by Joyal and Tierney (see [24]), but also there are other models which have been studied like the Segal categories, the complete Segal spaces, the simplicial categories, the topological categories, the relative categories, and there are known to be equivalent (a survey of models ...
2003
In this paper we generalize the plus-construction given by M. Livernet for algebras over rational differential graded Koszul operads to the framework of admissible operads (the category of algebras over such operads admits a closed model category structure). We follow the modern approach of J. Berrick and C. Casacuberta defining topological plus-construction as a nullification with respect to a universal acyclic space. Similarly, we construct a universal H ∗ -acyclic algebra U and we define A −→ A+ as the U -nullification of the algebra A. This map induces an isomorphism on Quillen homology and quotients out the maximal perfect ideal of π0(A). As an application, we consider for any associative algebra R the plusconstructions of gl(R) in the categories of Lie and Leibniz algebras up to homotopy. This gives rise to two new homology theories for associative algebras, namely cyclic and Hochschild homologies up to homotopy. In particular, these theories coincide with the classical cyclic...
Algebraic structures associated to operads
arXiv: Rings and Algebras, 2017
We study different algebraic structures associated to an operad and their relations: to any operad mathbfP\mathbf{P}mathbfP is attached a bialgebra,the monoid of characters of this bialgebra, the underlying pre-Lie algebra and its enveloping algebra; all of them can be explicitely describedwith the help of the operadic composition. non-commutative versions are also given. We denote by mathbfbinfty\mathbf{b\_\infty}mathbfbinfty the operad of mathbfbinfty\mathbf{b\_\infty}mathbfbinfty algebras, describing all Hopf algebra structures on a symmetric coalgebra.If there exists an operad morphism from mathbfbinfty\mathbf{b\_\infty}mathbfbinfty to mathbfP\mathbf{P}mathbfP, a pair (A,B)(A,B)(A,B) of cointeracting bialgebras is also constructed, that it to say:$B$ is a bialgebra, and AAA is a graded Hopf algebra in the category of BBB-comodules. Most examples of such pairs (on oriented graphs, posets$\ldots$) known in the literature are shown to be obtained from an operad; colored versions of these examples andother ones, based on Feynman graphs, are introduced and compared.