Related papers: Operadic Fibrations and Unary Operadic 2-categorie…
We study a certain type of action of categories on categories and on operads. Using the structure of the categories {\Delta} and {\Omega} governing category and operad structures, respectively, we define categories which instead encode the…
In the sixties, Grothendieck developed the theory of pro-objects over a category. The fundamental property of the category $Pro(C)$ is that there is an embedding $C \stackrel{c}{\rightarrow} Pro(C)$, $Pro(C)$ is closed under small…
In this paper we propose and study few applications of the base structured categories $\mathcal{X} \rtimes_{\mathbf{F}} \mathbf{C}$, $\int_{\mathbf{C}} \bar{\mathbf{F}}$, $\mathcal{X} \rtimes_{\mathbb{F}} \mathbf{C}$ and ${\int_{\mathbf{C}}…
We establish, by elementary means, the existence of a cofibrantly generated monoidal model structure on the category of operads. By slicing over a suitable operad the classical Rezk model structure on the category of small categories is…
Let $\mathcal{S}$ be a small category, and suppose that we are given a full subcategory $\mathcal{U}$ such that every object of $\mathcal{S}$ can be embedded into some object of $\mathcal{U}$ in the same way as every quasi-projective…
There are two dual equivalences between the $\infty$-category of $\mathcal{O}$-monoidal $\infty$-categories with right adjoint lax $\mathcal{O}$-monoidal functors and that with left adjoint oplax $\mathcal{O}$-monoidal functors, where…
The description of algebraic structure of n-fold loop spaces can be done either using the formalism of topological operads, or using variations of Segal's $\Gamma$-spaces. The formalism of topological operads generalises well to different…
We prove that the $\infty$-category of orthogonal factorization systems embeds fully faithfully into the $\infty$-category of double $\infty$-categories. Moreover, we prove an (un)straightening equivalence for double $\infty$-categories,…
We study a functorial construction from the category of monoids to the category of set-operads and we give some combinatorial examples of applications.
Batanin defines a weak $\omega$-category as an algebra for a certain operad. Leinster refines this idea and defines the weak $\omega$-category operad as the initial object of a category of "operads with contraction". We demonstrate how a…
Given an algebraic theory which can be described by a (possibly symmetric) operad $P$, we propose a definition of the \emph{weakening} (or \emph{categorification}) of the theory, in which equations that hold strictly for $P$-algebras hold…
In this paper we study duoidal structures on $\infty$-categories of operadic modules. Let $\mathcal{O}^{\otimes}$ be a small coherent $\infty$-operad and let $\mathcal{P}^{\otimes}$ be an $\infty$-operad. If a…
Using the description of enriched $\infty$-operads as associative algebras in symmetric sequences, we define algebras for enriched $\infty$-operads as certain modules in symmetric sequences. For $\mathbf{V}$ a symmetric monoidal model…
We develop a categorical framework for reasoning about abstract properties of differentiation, based on the theory of fibrations. Our work encompasses the first-order fragments of several existing categorical structures for differentiation,…
We introduce a general definition for colored cyclic operads over a symmetric monoidal ground category, which has several appealing features. The forgetful functor from colored cyclic operads to colored operads has both adjoints, each of…
Reasoning in the 2-category Con of contexts, certain sketches for arithmetic universes (i.e. list arithmetic pretoposes; AUs), is shown to give rise to base-independent results of Grothendieck toposes, provided the base elementary topos has…
Adjunctions of two variables generalize the relationship between tensor product and the internal hom functor in a closed monoidal category. For a pair of ordinary adjunctions $(F\dashv U, F'\dashv U')$ conjugation relates natural…
We begin by explaining how any context-free grammar encodes a functor of operads from a freely generated operad into a certain "operad of spliced words". This motivates a more general notion of CFG over any category $C$, defined as a finite…
Vertex operator algebras are mathematically rigorous objects corresponding to chiral algebras in conformal field theory. Operads are mathematical devices to describe operations, that is, $n$-ary operations for all $n$ greater than or equal…
We develop theory and examples of monoidal functors on tensor categories in positive characteristic that generalise the Frobenius functor from \cite{Os, EOf, Tann}. The latter has proved to be a powerful tool in the ongoing classification…