Two-dimensional monad theory
type: Item
instance of: scholarly article
title: Two-dimensional monad theory
zbMATH Open document ID: 0675.18006
DOI: 10.1016/0022-4049(89)90160-6
author: R. Blackwell
author: Max Kelly
author: John Power
published in: Journal of Pure and Applied Algebra
publication date: 1989-01-01T00:00:00Z
review text: This is the first in a series of articles (to include and extend unpublished PhD thesis material of Kelly's students R. Blackwell and G. Bird) on the universal algebra of categories with structure. The centre- piece is the 2-category T-Alg of algebras for a ranked monad T on a suitable 2-category. Here the algebras are strict, while the morphisms merely preserve the action up to a given coherent isomorphism. Certain 2-categorical limits and colimits are shown to exist in T-Alg. These suffice to imply (with the assistance of ``flexible'' algebras) the cocompleteness of T-Alg as a bicategory. Every ``algebraic'' 2-functor T- Alg\(\to S\)-Alg is proved to admit a left adjoint in the bicategorical sense.
Mathematics Subject Classification ID: 18C15
Mathematics Subject Classification ID: 18D05
zbMATH DE Number: 4105188
zbMATH Keywords: monad
zbMATH Keywords: doctrine
zbMATH Keywords: universal algebra of categories with structure
zbMATH Keywords: 2- category
zbMATH Keywords: algebras for a ranked monad
zbMATH Keywords: bicategory
reviewed by: Ross H. Street
MaRDI profile type: Publication
full work available at URL: 0022-4049(89)90160-6
OpenAlex ID: W2081723613
cites work: Abelian categories over additive ones
cites work: Introduction to bicategories
cites work: Q4119355
cites work: Flexible limits for 2-categories
cites work: Kan extensions in enriched category theory
cites work: A presentation of topoi as algebraic relative to categories or graphs
cites work: A generalization of the functorial calculus
cites work: Q5586461
cites work: Coalgebras and cartesian categories
cites work: Braided tensor categories
cites work: Q5654206
cites work: Q5654207
cites work: Q5654208
cites work: Q4100674
cites work: Q4100657
cites work: Q4100669
cites work: Examples of non-monadic structures on categories
cites work: A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on
cites work: Q3491793
cites work: Q3324132
cites work: Elementary observations on 2-categorical limits
cites work: Coherence in closed categories
cites work: Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads
cites work: Q4100671
cites work: Closed categories generated by commutative monads
cites work: FUNCTORIAL SEMANTICS OF ALGEBRAIC THEORIES
cites work: Q5634145
cites work: Coherence for bicategories and indexed categories
cites work: The formal theory of monads
cites work: Q4092156
cites work: Q3876103
cites work: Doctrines on 2-categories
Recommended article: Two-dimensional monadicity
Recommended article: A general limit lifting theorem for 2-dimensional monad theory
Recommended article: Homotopy-theoretic aspects of 2-monads
Recommended article: The monadic theory of ω2
Recommended article: The formal theory of monads. II
Recommended article: Two-dimensional models of type theory
Recommended article: Monads in topology
Recommended article: Monads and theories
Recommended article: Monads and D-instantons
Recommended article: Monomial maps in two dimensions
description: scientific article; zbMATH DE number 4105188