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March 10, 2016

Category Theory Course Notes

Posted by John Baez

Here are the notes from a basic course on category theory:

Unlike my Fall 2015 seminar, this quarter I tried to give a systematic introduction to the subject. However, many proofs (and additional theorems) were offloaded to another more informal seminar, for which notes are not available. So, many proofs here are left as ‘exercises for the reader’.

If you discover any errors in the notes please email me, and I’ll add them to the list of errors.

You can get all 10 weeks of notes in a single file here:

Or, you can look at individual weeks:

  • Week 1 (Jan. 5 and 7) - The definition of a category. Some familiar categories: Set\mathrm{Set}, Gp\mathrm{Gp}, Vect k\mathrm{Vect}_k, Ring\mathrm{Ring}, Top\mathrm{Top}, Met\mathrm{Met} and Meas\mathrm{Meas}. Various kinds of categories, including monoids, groupoids, groups, preorders, equivalence relations and posets. The definition of a functor. Doing mathematics inside a category: isomorphisms, monomorphisms and epimorphisms.
  • Week 2 (Jan. 12 and 14) - Doing mathematics inside a category: an isomorphism is a monomorphism and epimorphism, but not necessarily conversely. Products. Any object isomorphic to a product can also be a product. Products are unique up to isomorphism. Coproducts. What products and coproducts are like in various familiar categories. General limits and colimits. Examples: products and coproducts, equalizers and coequalizers, pullbacks and pushouts, terminal and initial objects.
  • Week 3 (Jan. 19 and 21) - Equalizers and coequalizers, and what they look like in Set\mathrm{Set} and other familiar categories. Pullbacks and pushouts, and what they look like in Set\mathrm{Set}. Composing pullback squares.
  • Week 4 (Jan. 26 and 28) - Doing mathematics between categories. Faithful, full, and essentially surjective functors. Forgetful functors: what it means for a functor to forget nothing, forget properties, forget structure or forget stuff. Transformations between functors. Natural transformations. Functor categories. Natural isomorphisms. In a category with binary products, the product becomes a functor, and the commutative and associative laws hold up to natural isomorphism. Cartesian categories. In a cartesian category, the left and right unit laws also hold up to natural isomorphism. A GG-set is a functor from a group GG to Set\mathrm{Set}. What is a natural transformation between such functors?
  • Week 5 (Feb. 2 and 4) - A GG-set is a functor from a group GG to Set\mathrm{Set}, and a a natural transformation between such functors is a map of G-sets. Equivalences of categories. Adjoint functors: the rough idea. The hom-functor. Adjoint functors: the definition. Examples: the left adjoint of the forgetful functor from Grp\mathrm{Grp} to Set\mathrm{Set}. The left adjoint of the forgetful functor fromVect k\mathrm{Vect}_k to Set\mathrm{Set}. The forgetful functor from Top\mathrm{Top} to Set\mathrm{Set} has both a left and right adjoint. If a category CC has binary products, the diagonal functor from CC to C×CC \times C has a right adjoint. If it has binary coproducts, the diagonal functor has a left adjoint.
  • Week 6 (Feb. 9 and 11) - Diagrams in a category as functors. Cones as natural transformations. The process of taking limits as a right adjoint. The process of taking colimits as a left adjoint. Left adjoints preserve colimits; right adjoints preserve limits. Examples: the ‘free group’ functor from sets to groups preserve coproducts, while the forgetful functor from groups to sets preserves products. The composite of left adjoints is a left adjoint; the composite of right adjoints is a right adjoint. The unit and counit of a pair of adjoint functors.
  • Week 7 (Feb. 16 and 18) - Adjunctions. The naturality of the isomorphism hom(Fc,d)hom(c,Ud) \mathrm{hom}(F c,d) \cong \mathrm{hom}(c,U d) in an adjunction. Given an adjunction, we can recover this isomorphism and its inverse from the unit and counit. Toward topos theory: cartesian closed categories and subobject classifiers. The definition of cartesian closed category, or ‘ccc’. Examples of cartesian closed categories. In a cartesian closed category with coproducts, the product distributes over the coproduct, and exponentiation distributes over the product.
  • Week 8 (Feb. 23) - Internalization. The concept of a group in a cartesian category. Any pair of objects X,YX, Y in a cartesian closed category has an ‘internal’ hom, the object Y XY^X, as well as the usual ‘external’ hom, the set hom(X,Y)\mathrm{hom}(X,Y). Evaluation and coevaluation. Internal composition. In a category with a terminal object, we can define the set of elements of any object.
    Week 8 (Feb. 25) - Guest lecture by Christina Osborne on symmetric monoidal categories.
  • Week 9 (Mar. 1 and 3) - For any category CC with a terminal object, elements define a functor elt:CSet\mathrm{elt} : C \to \mathrm{Set}. If CC is cartesian, this functor preserves finite products. If CC is cartesian closed, elt(Y X)hom(X,Y)\mathrm{elt}(Y^X) \cong \hom(X,Y), so it converts the internal hom into the external hom. The ‘name’ of a morphism. Subobjects. The subobject classifier in Set\mathrm{Set}. The general definition of subobject classifier in any category with finite limits. The definition of a topos. Examples of topoi, including the topos of graphs.
  • Week 10 (Mar. 8 and 10) - The subobject classifier in the topos of graphs. Any topos has finite colimits. Any morphism in a topos has an epi-mono factorization, which is unique up to a unique isomorphism. The image of a morphism in topos. The poset Sub(X)\mathrm{Sub}(X), whose elements are subobjects of an object XX in a topos. The correspondence between set theory and logic: given a set XX, subsets of XX correspond to predicates defined for elements of XX, intersection corresponds to ‘and’, union corresponds to ‘or’ the set XX itself corresponds to ‘true’, and the empty set corresponds to ‘false’. the intersection of subsets of XSetX \in \mathrm{Set} is their product in Sub(X)\mathrm{Sub}(X), their union is their coproduct in Sub(X)\mathrm{Sub}(X), the set XX is the terminal object in Sub(X)\mathrm{Sub}(X), and the empty set is the initial object. A lattice is a poset with finite limits and finite colimits, and a Heyting algebra is a lattice that is also cartesian closed. For any object XX in any topos, Sub(X)\mathrm{Sub}(X) is a Heyting algebra. If we think of these elements of Sub(X)\mathrm{Sub}(X) as predicates, the exponential is ‘implication’.

    Where does topos theory go from here?
Posted at March 10, 2016 9:25 PM UTC

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Re: Category Theory Course Notes

None of the links here seem to work.

list of errors ( returns

You don’t have permission to access /home/baez/qg-winter2016/errata.html on this server.

All of the other links are missing an initial ... and resolve to ....

Posted by: RodMcGuire on March 11, 2016 5:42 AM | Permalink | Reply to this

Re: Category Theory Course Notes

Sorry—it should be fixed now! The dangers of absent-mindedly copying a webpage containing relative links…

Posted by: John Baez on March 11, 2016 5:05 PM | Permalink | Reply to this

Re: Category Theory Course Notes still returns an error, even when accessed from I had tried this because it was the only link with at proper prefix (that maybe could be prepended to the pdf links) but the error made me think the whole directory was inaccessible.

Posted by: RodMcGuire on March 11, 2016 6:02 PM | Permalink | Reply to this

Re: Category Theory Course Notes

Thanks again! I forgot to set the permissions to make this particular file accessible.

One excuse is that there are, as of today, no errata. But tomorrow is another day!

Posted by: John Baez on March 17, 2016 2:23 AM | Permalink | Reply to this

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