Introduction#

(under construction)

(Part of the Amsterdam-Chapman Logic Meeting, May/June 2023)

Introduction#

These notes do not cover basic notions such as category, functor, natural transformations, Yoneda, epi, mono, factorization systems, limits (products, equalisers, pullbacks), colimits, adjunctions but I am happy to explain all of this in person.

The general aim is to write these notes from the point of view of advanced category theory without assuming that reader has covered everything in basic category theory. Roughly speaking, in my view, basic category theory relies on diagram chasing while advanced category theory uses algebraic style of reasoning with equations up to isomorphism.

The particular aim of Part 1 is to explain [McKenzie, 1996] and [Porst, 2000] cited by Tommaso Moraschini [Moraschini, 2018].

Current Plan of Content#

Part 0: General Techniques

The general techniques will come up in various situations. It makes some sense to put them at the beginning, but maybe this section should better be a collection of appendices.

Part 1: Universal Algebra

Part 2: Density and Completions of Categories:

  • …

  • [Finite Limits and Filtered Colimits (Locally Finitely Presentable Categories)]

  • [Finite Products and Sifted Colimits]

  • …

Part 3: Sheaf Representation of Algebras

  • Stone Duality

  • Boolean Powers

  • Boolean Products

  • Duality for Varities Generated by Semi-Primal Algebras

  • Duality for Varieties Generated by Quasi-Primal Algebras, Keimel-Werner Duality, …

More Topics:

  • [HSP theorems]

  • [Distributive Laws]

  • …

Bibliography#

(just started)

Blo76

Stephen L. Bloom. Varieties of ordered algebras. J. Comput. Syst. Sci., 13(2):200–212, 1976. URL: https://doi.org/10.1016/S0022-0000(76)80030-X.

BW83

Stephen L. Bloom and Jesse B. Wright. P-varieties - a signature independent characterization of varieties of ordered algebras. Journal of Pure and Applied Algebra, 29(1):13–58, 1983. URL: https://doi.org/10.1016/0022-4049(83)90080-4.

KP93

G. Kelly and J. Power. Adjunctions whose counits are coequalizers and presentations of enriched monads. J.Pure Appl. Algebra, 1993. URL: https://www.sciencedirect.com/science/article/pii/0022404993900928/pdf.

KV17

Alexander Kurz and J. Velebil. Quasivarieties and varieties of ordered algebras: regularity and exactness. Math. Struct. Comput. Sci., 27(7):1153–1194, 2017. URL: https://alexhkurz.github.io/papers/Ordered-algebras.pdf.

McK96

Ralph McKenzie. An algebraic version of categorical equivalence for varieties and more general algebraic categories. Logic and Algebra, 1996.

Mor18

Tommaso Moraschini. A logical and algebraic characterization of adjunctions between generalized quasi-varieties. J. Symb. Log., 83(3):899–919, 2018. URL: https://arxiv.org/pdf/1908.00534.pdf.

Por00

H.E. Porst. Equivalence of varieties in general and of bool in particular. Algebra Universalis, 2000. URL: https://user.informatik.uni-bremen.de/porst/dvis/PorstEqV.pdf.

Further References#

  • H.-E. Porst. Generalized Morita Theories. Notices of the South African Mathematical Society, 32:4–16, 2001.

  • Porst, Hu’s Primal Algebra Theorem revisited, Commentationes Mathematicae Universitatis Carolinae, Vol. 41 (2000),

  • Porst, The Linton theorem revisited, Cahiers de topologie et géométrie différentielle catégoriques (1993).