Related papers: Dwyer--Kan homotopy theory for cyclic operads
The category of (colored) props is an enhancement of the category of colored operads, and thus of the category of small categories. In this paper, the second in a series on "higher props," we show that the category of all small colored…
Generalizing the approach to pseudo monoidal DG-categories as certain colored non-symmetric DG-operads, we introduce a certain relaxed notion of a category enriched in DG-categories. We construct model structures on the category of colored…
We define a generalization of (coloured) operads based on double lax functors and we construct a model structure on the associated category of generalized simplicial (coloured) operads. In particular, we obtain a model structure on the…
Over suitable monoidal model categories, we construct a Dwyer-Kan model category structure on the category of algebras over an augmented operadic collection. As examples we obtain Dwyer-Kan model category structure on the categories of…
We recall several categories of graphs which are useful for describing homotopy-coherent versions of generalized operads (e.g. cyclic operads, modular operads, properads, and so on), and give new, uniform definitions for their morphisms.…
Coloured PROPs are a generalisation of coloured operads. In this article, we prove the existence of a Dwyer-Kan model structure on the category of small coloured PROPs enriched in a (sufficiently nice) monoidal model category V. This model…
In this work we provide a definition of a coloured operad as a monoid in some monoidal category, and develop the machinery of Gr\"obner bases for coloured operads. Among the examples for which we show the existance of a quadratic Gr\"obner…
The category of (colored) props is an enhancement of the category of colored operads, and thus of the category of small categories. The titular category has nice formal properties: it is bicomplete and is a symmetric monoidal category, with…
Operads may be represented as symmetric monoidal functors on a small symmetric monoidal category. We discuss the axioms which must be imposed on a symmetric monoidal functor in order that it give rise to a theory similar to the theory of…
We prove that, under certain conditions, the model structure on a monoidal model category $\mathcal{V}$ can be transferred to a model structure on the category of $\mathcal{V}$-enriched coloured (symmetric) operads. As a particular case we…
We introduce an explicit combinatorial characterization of the minimal model ${\cal O}_{\infty}$ of the coloured operad ${\cal O}$ encoding non-symmetric operads. In our description of ${\cal O}_{\infty}$, the spaces of operations are…
We construct a model structure on the category of small categories enriched over a combinatorial closed symmetric monoidal model category satisfying the monoid axiom. Weak equivalences are Dwyer-Kan equivalences, i.e. enriched functors…
We present a homotopy theory for a weak version of modular operads whose compositions and contractions are only defined up to homotopy. This homotopy theory takes the form of a Quillen model structure on the collection of simplicial…
Constructions of spectra from symmetric monoidal categories are typically functorial with respect to strict structure-preserving maps, but often the maps of interest are merely lax monoidal. We describe conditions under which one can…
We establish a highly flexible condition that guarantees that all colored symmetric operads in a symmetric monoidal model category are admissible, i.e., the category of algebras over any operad admits a model structure transferred from the…
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…
We present a Markl-style definition of operads colored by a small category. In the presence of a unit these are equivalent to substitudes of Day and Street. We show that operads colored by a category are internal algebras of a certain…
It is shown how double categories provide a direct abstract approach to coloured operads; namely, product-preserving normal lax functors from (Pb C)^op (the opposite of the double category of pullback squares in C) to Cat (the double…
The homotopy theory of infinity-operads is defined by extending Joyal's homotopy theory of infinity-categories to the category of dendroidal sets. We prove that the category of dendroidal sets is endowed with a model category structure…
We introduce a functorial construction which, from a monoid, produces a set-operad. We obtain new (symmetric or not) operads as suboperads or quotients of the operad obtained from the additive monoid. These involve various familiar…