Michael Barr is the Peter Redpath Emeritus Professor of Pure Mathematics at McGill University. Although his earlier work was in homological algebra, his principal research area for a number of years has been category theory.
He is on the editorial boards of Mathematical Structures in Computer Science and the electronic journal Homology, Homotopy and Applications, and is editor of the electronic journal Theory and Applications of Categories. Michael Barr has much advocated the methods of his late student Jon Beck, involving monads, especially monadicity criteria and monadic cohomology.
See also:
On Hochschild cohomology of commutative algebras:
see also
Michael Barr, A cohomology theory for commutative algebra I, Proc. Amer. Math. Soc. 16 (1965) 1379–1384 [doi:1965-016-06/S0002-9939-1965-0193119-3, pdf, pdf]
Michael Barr, A cohomology theory for commutative algebra II, Proc. Amer. Math. Soc. 16 (1965) 1385–1391 [jstor:2035937, pdf, pdf]
and in relation to monadic cohomology:
Michael Barr, Cohomology in tensored categories, Proceedings of the Conference on Categorical Algebra - La Jolla 1965, Springer (1966) 344–354 [doi:10.1007/978-3-642-99902-4_17, pdf, pdf]
Michael Barr, Shukla cohomology and triples, J. Algebra 5 2 (1967) 222–231 [doi:10.1016/0021-8693(67)90036-1, pdf, pdf]
Michael Barr, Harrison homology, Hochschild homology and triples J. Algebra 8 3 (1968) 314–323 [doi:10.1016/0021-8693(68)90062-8, pdf, pdf]
Michael Barr, Cohomology and obstructions: commutative algebras, in: Seminar on Triples and Categorical Homology Theory, Lecture Notes in Maths. 80, Springer (1969), Reprints in Theory and Applications of Categories 18 (2008) 357–374 [TAC:18, pdf, pdf]
On monads in universal algebra and (co-)homology-theory:
On cohomology of algebraic-structures (such as Lie algebra cohomology) via monads (“triples”, cf. monadic cohomology and canonical resolution):
Michael Barr, Jon Beck, Homology and Standard Constructions, in: Seminar on Triples and Categorical Homology Theory, Lecture Notes in Maths. 80, Springer (1969), Reprints in Theory and Applications of Categories 18 (2008) 186-248 [TAC:18, pdf]
Michael Barr, Cartan-Eilenberg cohomology and triples, J. Pure Applied Algebra 112 3 (1996) 219–238 [doi:10.1016/0022-4049(95)00138-7, pdf, pdf]
Michael Barr, Algebraic cohomology: the early days, in Galois Theory, Hopf Algebras, and Semiabelian Categories, Fields Institute Communications 43 (2004) 1–26 [doi:10.1090/fic/043, pdf, pdf]
On comonadicity of free monadic algebra-functors:
Introducing Barr-exact categories and regular categories:
Michael Barr, Pierre Grillet, Donovan van Osdol, Exact Categories and Categories of Sheaves, Lec. Notes in Math. 236, Springer (1971) [doi:10.1007/BFb0058579]
Michael Barr, Exact categories, in: Exact categories and categories of sheaves, Springer Lec. Notes in Math. 236 (1971) 1-120 [doi:10.1007/BFb0058580, pdf, pdf]
On coalgebras over a commutative ring (cf. CocommCoalg):
On topoi without topos points:
On the abstract construction principle behind cartesian closed convenient categories of topological spaces (such as compactly generated topological spaces):
Introducing star-autonomous categories:
On Galois theory:
Michael Barr, Abstract Galois Theory, J. Pure Appl. Algebra 19 (1980) 21–42 [pdf, pdf]
Michael Barr, Abstract Galois Theory II, J. Pure Appl. Algebra 25 3 (1982) 227–247 [doi:10.1016/0022-4049(82)90080-9, pdf, pdf]
On atomic toposes:
On toposes, monads (“triples”) and algebraic theories:
Exposition of sheaf toposes:
On models of Horn theories:
On domain theory internal to cartesian closed categories:
On star-autonomous categories and the Chu construction as categorical semantics for linear logic:
and via accessible categories:
On coalgebras for an endofunctor:
On fuzzy logic
in relation to topos theory:
and in relation to linear logic:
On the HSP theorem:
Michael Barr, HSP type theorems in the category of posets, in: Proc. 7th International Conf. Mathematical Foundation of Programming Language Semantics, Lecture Notes in Computer Science 598 (1992) 221–234 [doi:10.1007/3-540-55511-0_11, pdf]
Michael Barr, Functorial semantics and HSP type theorems, Algebra Universalis 31 (1994) 223–251 [doi:10.1007/BF01236519, pdf, pdf]
Michael Barr, HSP subcategories of Eilenberg-Moore algebras, Theory Appl. Categories 10 18 (2002), 461–468 [tac:10-18]
On category theory in computer science, via star-autonomous categories and Chu spaces:
On acyclic objects in categories of chain complexes:
Michael Barr, Acyclic models, Canadian J. Math. 48 2 (1996) 258–273 [doi:10.4153/CJM-1996-013-x, pdf, pdf]
Michael Barr, Acyclic models, CRM Monograph Series 17 (2002) [ams:crmm-17, pdf, pdf]
On the Chu construction:
On the Chu construction on Vect:
On sketches:
On the Chu construction and star-autonomous categories:
On star-autonomous categories of unit balls in Banach spaces:
More on star-autonomous categories:
On topological star-autonomous categories:
Michael Barr, Topological -autonomous categories, Theory Appl. Categories, 16 (2006) 700–708 [pdf, pdf]
Michael Barr, John Kennison, Robert Raphael, On -autonomous categories of topological modules, Theory Appl. Categories, 24 14 (2010) 278–293 [tac:24-14]
Michael Barr, Topological -autonomous categories, revisited, Tbilisi Math. J. 10 3 (2017) 51–64 [arXiv:1609.04241]
English translation of Alexander Grothendieck‘s Tohoku-article on homological algebra:
On an improved context for Pontrjagin duality:
Michael Barr, On duality of topological abelian groups [pdf, pdf]
Did you know that there is a *-autonomous category of topological abelian groups that includes all the LCA groups and whose duality extends that of Pontrjagin? The groups are characterized by the property that among all topological groups on the same underlying abelian group and with the same set of continuous homomorphisms to the circle, these have the finest topology. It is not obvious that such a finest exists, but it does and that is the key.
On the Chu construction:
On product spaces with Lindelöf topological spaces:
On computability and formal languages:
On Isbell duality:
and specifically for modules:
On discrete dynamical systems (here called “flows”) on compact Hausdorff spaces:
On coherent spaces:
On injective hulls of partially ordered monoids:
On star-autonomous categories of sup-semilattices:
On limits of integral domains among commutative rings:
On limits of integral domains among commutative rings:
On reflective and coreflective subcategories:
On contractibility of simplicial objects:
On coequalizers and free monads:
Last revised on November 12, 2023 at 11:17:48. See the history of this page for a list of all contributions to it.