Related papers: Mapping Coalgebras I: Comonads
Two novel descriptions of weak {\omega}-categories have been recently proposed, using type-theoretic ideas. The first one is the dependent type theory CaTT whose models are {\omega}-categories. The second is a recursive description of a…
We establish a duality between monads and monadic morphisms in any $(\infty,2)$-category and characterize monadic morphisms in a wide class of examples. This duality unifies several dualities between algebraic structures and their…
We compare two possible ways of defining a category of 1-combs, the first intensionally as coend optics and the second extensionally as a quotient by the operational behaviour of 1-combs on lower-order maps. We show that there is a full and…
We show that the category of comodules over a coassociative coalgebra in a complete, cocomplete and well-powered category has limits and colimits under additional assumptions.
The behaviour of limits of weak morphisms in 2-dimensional universal algebra is not 2-categorical in that, to fully express the behaviour that occurs, one needs to be able to quantify over strict morphisms amongst the weaker kinds.…
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 a notion of signature whose sorts form a direct category, and study computads for such signatures. Algebras for such a signature are presheaves with an interpretation of every function symbol of the signature, and we describe…
Monads govern computational side-effects in programming semantics. They can be combined in a ''bottom-up'' way to handle several instances of such effects. Indexed monads and graded monads do this in a modular way. Here, instead, we equip…
Let CABA be the category of complete and atomic boolean algebras and complete boolean homomorphisms, and let CSL be the category of complete meet-semilattices and complete meet-homomorphisms. We show that the forgetful functor from CABA to…
We present the notion of "cyclic double multicategory", as a structure in which to organise multivariable adjunctions and mates. The classic example of a 2-variable adjunction is the hom/tensor/cotensor trio of functors; we generalise this…
In the theory of coalgebras $C$ over a ring $R$, the rational functor relates the category of modules over the algebra $C^*$ (with convolution product) with the category of comodules over $C$. It is based on the pairing of the algebra $C^*$…
Tannaka Duality describes the relationship between algebraic objects in a given category and their representations; an important case is that of Hopf algebras and their categories of representations; these have strong monoidal forgetful…
We study a coderivation from a cobimodule into a coalgebra. Vector cofields are defined by the action of a codual bicomodule on a coalgebra. This action is induced by a codifferential. A construction of a codual object in the category of…
Unstable operations in a generalized cohomology theory E give rise to a functor from the category of algebras over E to itself which is a colimit of representable functors and a comonoid with respect to composition of such functors. In this…
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 interpret tensors on a smooth manifold M as differential forms over a graded commutative algebra called the algebra of iterated differential forms over M. This allows us to put standard tensor calculus in a new differentially closed…
We provide a bicategorical generalization of Barr's landmark 1970 paper, in which he describes how to extend Set-monads to relations and uses this to characterize topological spaces as the relational algebras of the ultrafilter monad. With…
These are the notes for a minicourse held in Odessa (2016) and Belo Horizonte (2017). My aim was to provide a short introduction to basic notions of category theory and representation theory of finite-dimensional algebras. We learnt the…
We investigate the categories of weak maps associated to an algebraic weak factorisation system (AWFS) in the sense of Grandis-Tholen. For any AWFS on a category with an initial object, cofibrant replacement forms a comonad, and the…
Abstract clones serve as an algebraic presentation of the syntax of a simple type theory. From the perspective of universal algebra, they define algebraic theories like those of groups, monoids and rings. This link allows one to study the…