Related papers: Coherence for Pseudo Commutative Monads

We show that every braiding on a monoidal bicategory induces a monoidal structure on its bicategory of monoids, such that if the former is sylleptic or symmetric then the latter is braided or symmetric, respectively. This extends a classic…

This paper is a fundamental study of comodules and contramodules over a comonoid in a symmetric closed monoidal category. We study both algebraic and homotopical aspects of them. Algebraically, we enrich the comodule and contramodule…

We introduce rigid algebras, a generalization of rigid categories to arbitrary symmetric monoidal $(\infty,2)$-categories. We develop their general theory, showing in particular that the a priori $(\infty,2)$-category of rigid algebras is…

Many recursive functions can be defined elegantly as the unique homomorphisms, between two algebras, two coalgebras, or one each, that are induced by some universal property of a distinguished structure. Besides the well-known applications…

We construct families of quilted surfaces parametrized by the multiplihedra, and define moduli spaces of pseudoholomorphic quilted disks using the theory of pseudoholomorphic quilts of Wehrheim and Woodward. We prove a gluing theorem for…

We study equivariant vector bundles over configuration spaces with diagonals included, viewed as orbifold quotients $M^n/\mathfrak{S}_n$ by permutation groups. Working in the equivalent language of equivariant vector bundles, we construct…

Delta lenses are a kind of morphism between categories which are used to model bidirectional transformations between systems. Classical state-based lenses, also known as very well-behaved lenses, are both algebras for a monad and coalgebras…

An operad describes a category of algebras and a (co)homology theory for these algebras may be formulated using the homological algebra of operads. A morphism of operads $f:\mathcal{O}\rightarrow\mathcal{P}$ describes a functor allowing a…

In this paper we extend some previous results on weak pseudo-bosons and on their related bi-coherent states. The role of {\em compatible} functions is discussed in details, and some examples are considered. The pseudo-bosonic ladder…

We show that, for a right exact functor from an abelian category to abelian groups, Yoneda's isomorphism commutes with homology and, hence, with functor derivation. Then we extend this result to semiabelian domains. An interpretation in…

In this paper, we address the construction of homotopy bicategories of $(\infty,2)$-categories, which we take as being modeled by 2-fold Segal spaces. Our main result is the concrete construction of a functor $h_2$ from the category of…

We define a version of Hochschild homology and cohomology suitable for a class of algebras admitting compatible actions of bialgebras, called module algebras. We show this (co)homology, called Hopf--Hochschild (co)homology, can also be…

We define a cohomology for an arbitrary $K$-linear semistrict semigroupal 2-category $(\mathfrak{C},\otimes)$ (called in the paper a Gray semigroup) and show that its first order (unitary) deformations, up to the suitable notion of…

The category of involutive non-commutative sets encodes the structure of an involution compatible with a (co)associative (co)multiplication. We prove that the category of involutive bimonoids in a symmetric monoidal category is equivalent…

A theory of higher colimits over categories of free presentations is developed. It is shown that different homology functors such as Hoshcshild and cyclic homology of algebras over a field of characteristic zero, simplicial derived…

For a 2-category $\mathcal{K}$, we consider Street's 2-category Mnd($\mathcal{K}$) of monads in $\mathcal{K}$, along with Lack and Street's 2-category EM($\mathcal{K}$) and the identity-on-objects-and-1-cells 2-functor Mnd($\mathcal{K}$)…

In the first part of this article, we give an analysis of the free monad sequence in non-cocomplete categories, with the needed colimits explicitly parametrized. This enables us to state a more finely grained functoriality principle for…

In The factorization of the Giry monad (arXiv:1707.00488v2) the author asserts that the category of convex spaces is equivalent to the category of Eilenberg-Moore algebras over the Giry monad. Some of the statements employed in the proof of…

2-Theories are a canonical way of describing categories with extra structure. 2-theory-morphisms are used when discussing how one structure can be replaced with another structure. This is central to categorical coherence theory. We place a…

The notion of multiplier Hopf monoid in any braided monoidal category is introduced as a multiplier bimonoid whose constituent fusion morphisms are isomorphisms. In the category of vector spaces over the complex numbers, Van Daele's…