Related papers: Dioperads, Frobenius monoidal functors and duality

We prove an adjoint functor theorem in the setting of categories enriched in a monoidal model category $\mathcal V$ admitting certain limits. When $\mathcal V$ is equipped with the trivial model structure this recaptures the enriched…

We provide a calculus of mates for functors to the $\infty$-category of $\infty$-categories and extend Lurie's unstraightening equivalences to show that (op)lax natural transformations correspond to maps of (co)cartesian fibrations that do…

We define a notion of category enriched over an oplax monoidal category $V$, extending the usual definition of category enriched over a monoidal category. Even though oplax monoidal structures involve infinitely many functors $V^n\to V$,…

For a discrete colored operad $P$, we construct an adjunction between the category of dendroidal sets over the nerve of $P$ and the category of simplicial $P$-algebras, and prove that when $P$ is $\Sigma$-free it establishes a Quillen…

Let $\mathcal{B}$ be a subcategory of a given category $\mathcal{D}$. Let $\mathcal{B}$ has monoidal structure. In this article, we discuss when can one extend the monoidal structure of $\mathcal{B}$ to $\mathcal{D}$ such that $\mathcal{B}$…

We introduce a candidate for the inner hom for $Dbl^{st}_{lx}$, the category of strict double categories and lax double functors, and characterize a lax double functor into it obtaining a lax double quasi-functor. The latter consists of a…

We study lax functors between bicategories as a generalized concept of monads and describe generalized notions and theorems of formal monad theory for lax functors. Our first approach is to use the 2-monad whose lax algebras are lax…

We develop the theory of relative monads and relative adjunctions in a virtual equipment, extending the theory of monads and adjunctions in a 2-category. The theory of relative comonads and relative coadjunctions follows by duality. While…

The tensor functor called $\alpha$-induction arises from a Frobenius algebra object, or a Q-system, in a braided unitary fusion category. In the operator algebraic language, it gives extensions of endomorphism of $N$ to $M$ arising from a…

This paper describes several cases of adjunction in the homomorphism preorder of relational structures. We say that two functors $\Lambda$ and $\Gamma$ between thin categories of relational structures are adjoint if for all structures…

We show how to "interleave" the monad for operads and the monad for contractions on the category \coll of collections, to construct the monad for the operads-with-contraction of Leinster. We first decompose the adjunction for operads and…

A pair of adjoint functors $(F,G)$ is called a Frobenius pair of the second type if $G$ is a left adjoint of $\beta F\alpha$ for some category equivalences $\alpha$ and $\beta$. Frobenius ring extensions of the second kind provide examples…

Frobenius monoidal functors preserve duals. We show that conversely, (co)monoidal functors between autonomous categories which preserve duals are Frobenius monoidal. We apply this result to linearly distributive functors between autonomous…

We show how, under certain conditions, an adjoint pair of braided monoidal functors can be lifted to an adjoint pair between categories of Hopf algebras. This leads us to an abstract version of Michaelis' theorem, stating that given a Hopf…

The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There…

Suppose that $\mathcal{A}$ is an abelian category whose derived category $\mathcal{D}(\mathcal{A})$ has $Hom$ sets and arbitrary (small) coproducts, let $T$ be a (not necessarily classical) ($n$-)tilting object of $\mathcal{A}$ and let…

This lecture series is based on joint work in progress with Shaul Barkan, as well as work in progress of the author. The five sections of these notes correspond to the five lectures, but more details have been added. $2$-dimensional…

The Eilenberg-Moore constructions and a Beck-type theorem for pairs of monads are described. More specifically, a notion of a {\em Morita context} comprising of two monads, two bialgebra functors and two connecting maps is introduced. It is…

Given a right exact functor from an abelian category into another abelian category, there is an associated abelian category called the comma category of the functor. In this paper, we characterize when left Frobenius pairs (resp. strong…

Skew-monoidal categories arise when the associator and the left and right units of a monoidal category are, in a specific way, not invertible. We prove that the closed skew-monoidal structures on the category of right R-modules are…