-1+1971 
Introduction to Category Theory

Category theory is a branch of mathematics that emerged in the 1950s to understand how the different areas of mathematics relate to each other. Today, its influence pervades nearly all of mathematics, in addition to being an important area of study in its own right. This course will cover the basic concepts of category theory, with an emphasis on intuition and examples.

Lectures
  1. Overview

    Brief history of category theory, followed by a discussion of what it is and how it is meant to be used. Discussed syllabus and format.

  2. Background: Monoids

    Background lecture on monoids. Introduced the concepts of monoid and monoid morphism conceptually, then worked out how to implement these ideas rigorously.

  3. Categories, Functors, and Natural Transformations

    Defined categories, functors, and natural transformations, with plenty of examples.

  4. Basic Constructions and Ideas

    Defined some basic constructions on categories: duality, products, and comma categories. Also briefly discussed 2-categories, enriched categories, and large categories.

  5. Internal Features of Categories

    Following the previous lecture, which discussed operations between categories, introduced some interesting features within a single categorically. Specifically, looked at categorical generalizations / analogues of the injectivity, surjectivity, and bijectivity, as well as (co-)products and pullbacks. Hinted at Yoneda and (co-)limits, to be covered in the next few lectures.

  6. Representable Functors

    Discussion of representable functors and universality, as well as multiple perspectives on how they relate. Proved the Yoneda lemma, and explained why it's awesome. I tried to give more worked examples this time.

Prerequisites

Some familiarity with higher mathematics. Topology is a huge plus, but not necessary. I think the following as a minimum:

  • set theory: know what products, coproducts (disjoint unions), etc. are, know the paradoxes, know what posets are, know what injections/surjections/bijections are and maybe a bit about cardinals. This is pretty much just language/history and can be picked up in an afternoon or two on Wikipedia.

  • algebra: know what vector spaces/linear transformations are, ideally know what groups/monoids are. Knowledge of module theory definitely a plus. Know a few kinds of homomorphisms (linear transformations, ring homomorphisms, group homomorphisms, etc.), and understand the concept of isomorphism

  • mathematical maturity: probably the most important; basically remain calm and carry on when all the mathematics you've ever known and loved gets abstracted away into dots and arrows.

Syllabus

Dates: July 2 - .

Lectures will be posted to YouTube, assignments will probably be on Scribd

The course will cover all the major ideas in basic category theory, roughly equivalent to the first 5-7 chapters of Mac Lane.

  • categories, functors, natural transformations
  • operations on categories; duality
  • representable functors and Yoneda
  • diagrams
  • adjunctions
  • limits and colimits
  • monads
  • Kan extensions
  • applications

And some possible bonus topics:

  • topoi
  • model categories
  • generalizations of categories (higher categories, enriched categories, and so forth)

I'm very big on intuition, so expect a lot of rambling "big picture" digressions. I'll also try to mention the computer sciencey aspects.

Additional information
Teacher qualifications

Studied category theory for 2.5 years (started when I was 17)

Taken graduate courses in algebra, algebraic topology, and homological/homotopical algebra

Attended workshops/conferences on applications of category theory

Latest Update
Class subreddit (2012-10-02 18:49:43)

"Just a reminder that the class has a discussion subreddit at http://www.reddit.com/r/URIntroCategoryTheory/ !"

Roster
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957. chauhan1212
958. florian.pinault
959. foxboy
960. florianpinault
961. lupin-the-third
962. otac0n
963. soulofwren
964. Bloodrazor
965. charlie.pritscher
966. timjstewart
967. L2BSmart
968. foo
969. igbistro
970. nunofilipemargarido
971. Bennymoonbase
972. hatFolk
973. jd95
974. FuckOffGetFree
975. djhk12
976. prince_of_maths
977. mtilles69
978. mdgemr
979. crazwomanyo
980. fr0zen
981. Richo1337
982. Dardan
983. RMnumbers
984. wwalid2050
985. brettveen
986. thephotograph
987. christophermoll
988. gallabytes
989. drdude
990. bakerstreetresident
991. aeknel
992. bbmacarthur
993. zeroms
994. crichm03
995. shamwow19
996. RicoSpartan
997. sotacan
998. Zigzagh
999. buguoruci
1000. emmure91
1001. JimSunkes
1002. thecrazyboyftw
1003. HexMerck
1004. daedalus198
1005. kaneda
1006. benshats
1007. garfieldnate
1008. Threscher
1009. popotheweasel
1010. mgrollins
1011. gerardlouw
1012. OOmm
1013. kingofosraigi
1014. Suzanne2578
1015. sintrastes
1016. antone
1017. shitzfez
1018. natasakalp
1019. kingosraigi
1020. Ichikoa
1021. Theguybehindthescreen
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1024. imnotsosureyou
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1034. 5tefan0
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1036. RedCaveTree
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1038. asingh21
1039. acanete
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1044. fezspinkguitar
1045. EntropicalForest
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1047. mavinnowhere
1048. dreamfeastr
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1050. Higs
1051. martyg
1052. 20sat92
1053. Cori83