Our seminar series might or might not be getting into category theory in the coming months. (We’re already drawing diagrams and showing that they commute; not everybody knows it yet!) To facilitate this process should we ever go in that direction, and to provide a general public service, I’m compiling a list of useful category-theory resources extant on the Wobosphere. My selection will be pedagogically oriented, rather than emphasizing the latest research; I’d like to collect reading material which could plausibly be presented to advanced undergraduate or beginning graduate students in, say, their first semester of encountering the subject. I’ll be both happy and eager to update this list with any beneficial suggestions the Gentle Readers have to offer.

Introductory:

• Mike Stay, “Monoids” (12 September 2007)
• Mike Stay, “Multiplication:composition::monoid:category” (12 September 2007)
• Bob Coecke, “Introducing Categories to the Practicing Physicist”
• Mark Chu-Carroll, “A First Glance at Category Theory” (7 June 2006)
• John Armstrong, “Categories” (22 May 2007)
• John Armstrong, “Functors” (22 May 2007)
• Mark Chu-Carroll, “Diagrams in Category Theory” (8 June 2006)
• Mark Chu-Carroll, “Some Basic Examples of Categories” (9 June 2006)
• John Armstrong, “Arrow Categories” (23 May 2007)
• John Armstrong, “Comma Categories” (26 May 2007)
• Mark Chu-Carroll, “Getting Into Functors” (12 June 2006)
• Mike Stay, “Functors as Shadows” (15 September 2007)
• Mark Chu-Carroll, “Natural Transformations and Structure” (19 June 2006)
• John Armstrong, “Natural Transformations and Functor Categories” (26 May 2007)
• Mark Chu-Carroll, “Some Definitions to Build on” (22 June 2006)
• John Armstrong, “Equivalence of Categories” (30 May 2007)
• Mark Chu-Carroll, “Products, Exponentials and Closed Cartesian Categories” (27 June 2006)
• Mike Stay, “Cartesian categories and the problem of evil” (19 September 2007)
• John Armstrong, “The Opposite Category” (31 May 2007)
• John Armstrong, “Product Categories” (1 June 2007)
• Mark Chu-Carroll, “Categories and SubThings” (4 July 2006)
• John Armstrong, “Special kinds of morphisms, subobjects, and quotient objects” (29 May 2007)
• Mark Chu-Carroll, “Arrow Equality and Pullbacks” (6 July 2006)
• Mark Chu-Carroll, “Using Natural Transformations: Recreating Closed Cartesian Categories” (10 July 2006)
• John Armstrong, “Hom Functors” (2 June 2007)
• John Armstrong, “Representable Functors” (6 June 2007)
• Mark Chu-Carroll, “Yoneda’s Lemma” (11 July 2006)
• John Armstrong, “Yoneda’s Lemma” (6 June 2007)
• John Armstrong, “What Does Yoneda’s Lemma Mean?” (7 June 2007)
• Mark Chu-Carroll, “Zeros in Category Theory” (26 July 2006)

Intermediate:

• John Armstrong, “Groupoids and Group Actions — Categorically” (8 June 2007)
• John Armstrong, “Groupoids and More Group Actions” (9 June 2007)
• John Armstrong, “Initial and Terminal Objects” (10 June 2007)
• John Armstrong, “Products and Coproducts” (11 June 2007)
• John Armstrong, “Equalizers and Coequalizers” (12 June 2007)
• John Armstrong, “Zero Objects, Kernels and Cokernels” (13 June 2007)
• John Armstrong, “Pushouts and Pullbacks” (14 June 2007)
• John Armstrong, “Multiple Products, Coproducts, Equalizers, Coequalizers, Pullbacks, Pushouts. . .” (15 June 2007)
• John Armstrong, “Diagram Categories” (16 June 2007)
• John Armstrong, “Cones and Cocones” (18 June 2007)
• John Armstrong, “Limits and Colimits” (19 June 2007)
• John Armstrong, “The Existence Theorem for Limits” (20 June 2007)
• John Armstrong, “Limits of Sets and Creation of Limits” (21 June 2007)
• John Armstrong, “Preservation of Limits” (22 June 2007)
• John Armstrong, “Limits in Functor Categories” (23 June 2007)
• John Armstrong, “Taking Limits is Functorial” (24 June 2007)
• John Armstrong, “Universal Arrows and Universal Elements” (25 June 2007)
• John Armstrong, “Representations and Universals and Colimits (oh my!)” (26 June 2007)
• John Armstrong, “Categorification” (27 July 2007)
• John Armstrong, “Monoidal Categories” (28 June 2007)
• John Armstrong, “Mac Lane’s Coherence Theorem” (29 June 2007)
• John Armstrong, “Monoidal Functors and Natural Transformations” (30 June 2007)
• John Armstrong, “The “Strictification” Theorem” (1 July 2007)

Mathematical physics:

• John Baez, “Tale of n-Categories” (1996)
• John Baez, “Quantum Quandaries: A Category-Theoretic Perspective” (7 April 2004)
• John Baez, “The Story of Nth Quantization” (8 July 2005)
• John Baez, “Categories, Quantization, and Much More” (12 April 2006)
• John Baez, Derek Wise, et al. “Quantum Gravity Seminar: Quantization and Cohomology” (2007)
• John Baez, “This Week’s Finds in Mathematical Physics (issue 252)” (27 May 2007) and n-Category Café discussion
• Bob Coeke, “Kindergarten Quantum Mechanics” (4 October 2005)
• Urs Schreiber, “Cats in MathPhys” (26 March 2006)

Number theory and games:

• Mark Chu-Carroll, “Categorical Numbers” (9 August 2006)
• John Armstrong, “Cardinals and Ordinals as Categories” (24 May 2007)
• André Joyal, “Remarks on the Theory of Two-Player Games” (1977, translated 2003)
• Robin Houston, “Categories of Games” (2003)

Topology:

• Mark Chu-Carroll, “Topological Products” (25 September 2006)
• Mark Chu-Carroll, “Big to Small, Small to Big: Topological Properties Through Sheaves” (12 December 2006), and part 2 the next day
• Mark Chu-Carroll, “Connected Topologies and Fixed Points” (2 October 2006)
• John Armstrong, “Braidings and Symmetries” (2 July 2007)
• John Armstrong, “The Category of Braids” (3 July 2007)

Logic and programming:

• Mike Stay, “Category Theory for the Java Programmer” (3 November 2007)
• Andrea Asperti and Giuseppe Longo, Categories, Types and Structures: Category Theory for the working computer scientist (1991)
• Steve Awody, “Categorical Logic” (2003)
• Mark Chu-Carroll, “Monads and Programming Languages” (12 July 2006)
• Mark Chu-Carroll, “Sequent Calculus” (17 July 2006)
• Mark Chu-Carroll, “Linear Logic” (18 July 2006)
• Mark Chu-Carroll, “Towards a Model for Linear Logic: Monoidal Categories” (19 July 2006)
• Mark Chu-Carroll, “The Category Structure for Linear Logic” (25 July 2006)
• Mark Chu-Carroll, “The Categorical Model of Linear Logic” (31 July 2006)
• Mark Chu-Carroll, “From Lambda Calculus to Cartesian Closed Categories” (10 August 2006)
• Mark Chu-Carroll, “The Theory of Monads and the Monad Laws” (31 January 2007)

Foundational questions:

• Philosophia Mathematica, Volume 13, Number 1 (February 2005)

This list is emphatically a work in progress.