-1+1955 
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. timlovelock
958. chauhan1212
959. florian.pinault
960. foxboy
961. florianpinault
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963. otac0n
964. soulofwren
965. Bloodrazor
966. charlie.pritscher
967. timjstewart
968. L2BSmart
969. foo
970. igbistro
971. nunofilipemargarido
972. Bennymoonbase
973. hatFolk
974. jd95
975. FuckOffGetFree
976. djhk12
977. prince_of_maths
978. mtilles69
979. mdgemr
980. crazwomanyo
981. fr0zen
982. Richo1337
983. Dardan
984. RMnumbers
985. wwalid2050
986. brettveen
987. thephotograph
988. christophermoll
989. gallabytes
990. drdude
991. bakerstreetresident
992. aeknel
993. bbmacarthur
994. zeroms
995. crichm03
996. shamwow19
997. RicoSpartan
998. sotacan
999. Zigzagh
1000. buguoruci
1001. emmure91
1002. JimSunkes
1003. thecrazyboyftw
1004. HexMerck
1005. daedalus198
1006. kaneda
1007. benshats
1008. garfieldnate
1009. Threscher
1010. martyg
1011. popotheweasel
1012. mgrollins
1013. gerardlouw
1014. OOmm
1015. kingofosraigi
1016. Suzanne2578
1017. sintrastes
1018. antone
1019. shitzfez
1020. natasakalp
1021. kingosraigi
1022. Ichikoa
1023. Theguybehindthescreen
1024. arunodhaya80
1025. haraf_ijhal
1026. imnotsosureyou
1027. lemoskid
1028. esiegel
1029. NapoleonBonerpartee
1030. lazypakman
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1033. namesare4tombstones
1034. judde
1035. carmicheal99