Related papers: How many adjunctions give rise to the same monad?
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…
We establish a general method for generating reflections between categories. We then apply our technique to generate adjunctions starting from geometric morphisms between Grothendieck toposes; as particular cases, we recover various…
We present an adjunction formula for foliations on varieties and we consider applications of the adjunction formula to the cone theorem for rank one foliations and the study of foliation singularities.
We determine the dimension of every simple module for the algebra of the monoid of all relations on a finite set (i.e. Boolean matrices). This is in fact the same question as the determination of the dimension of every evaluation of a…
In this paper, motivated by the theory of operads and PROPs we reveal the combinatorial nature of tensor calculus for strict tensor categories and show that there exists a monad which is described by the coarse-graining of graphs and…
We construct a functor that inputs a retract in an $(\infty,3)$-category satisfying some adjunctibility conditions and outputs a Hopf algebra in a braided monoidal $(\infty,1)$-category. Provided the braided monoidal category is…
Let $U$ be a strong monoidal functor between monoidal categories. If it has both a left adjoint $L$ and a right adjoint $R$, we show that the pair $(R,L)$ is a linearly distributive functor and $(U,U)\dashv (R,L)$ is a linearly distributive…
The structure of the category of matroids and strong maps is investigated: it has coproducts and equalizers, but not products or coequalizers; there are functors from the categories of graphs and vector spaces, the latter being faithful;…
Categories are coreflectively embedded in multicategories via the "discrete cocone" construction, the right adjoint being given by the monoid construction. Furthermore, the adjunction lifts to the "cartesian level": preadditive categories…
We present a systematic study of join-extensions and join-completions of ordered algebras, which naturally leads to a refined and simplified treatment of fundamental results and constructions in the theory of ordered structures ranging from…
We revisit once again the connection between three notions of computation: monads, arrows and idioms (also called applicative functors). We employ monoidal categories of finitary functors and profunctors on finite sets as models of these…
A combinatorial structure, $\mathcal{F}$, with counting sequence $\{a_n\}_{n\ge 0}$ and ordinary generating function $G_\mathcal{F}=\sum_{n\ge0} a_n x^n$, is positive algebraic if $G_\mathcal{F}$ satisfies a polynomial equation…
Every principal G-bundle is classified up to equivalence by a homotopy class of maps into the classifying space of G. On the other hand, for every nice topological space Milnor constructed a strict model of loop space, that is a group.…
Containers represent a wide class of type constructions relevant for functional programming and (co)inductive reasoning. Indexed containers generalize this notion to better fit the scope of dependently typed programming. When interpreting…
Varieties of quantitative algebras are fully described by their free-algebra monads on the category Met of metric spaces. For a longer time it has been an open problem whether the resulting enriched monads are precisely the strongly…
We show that \'etale correspondences between ample groupoids induce homomorphisms of homology groups. To complement this we explore the module categories of ample groupoids. We construct an induction-restriction adjunction for subgroupoids,…
We introduce pseudoalgebras for relative pseudomonads and develop their theory. For each relative pseudomonad $T$, we construct a free--forgetful relative pseudoadjunction that exhibits the bicategory of $T$-pseudoalgebras as terminal among…
We extend the theory of distributive series of monads of \cite{EC1} by extending the definition to include an $\bN$-indexed collection of monads. Under certain conditions, distributive series of monads will have a colimit in the category of…
Lawvere's algebraic theories, or Lawvere theories, underpin a categorical approach to general algebra, and Lawvere's adjunction between semantics and algebraic structure leads to an equivalence between Lawvere theories and finitary monads…
Let $\mathbb{A}$ be a $2$-category with suitable opcomma objects and pushouts. We give a direct proof that, provided that the codensity monad of a morphism $p$ exists and is preserved by a suitable morphism, the factorization given by the…