Category Theory Course [Lecture notes] by John Baez

By John Baez

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Extra resources for Category Theory Course [Lecture notes]

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Left adjoints preserve colimits; right adjoints preserve limits. Proof. Sketch of proof: Let’s show that if F : C → D is a left adjoint to U : D → C, then F preserves colimits. For concreteness, let’s show F preserves pushouts - general case is analogous. So suppose we have a pushout in C: a b c x Here, x is the apex of a cocone on the diagram we’re taking a colimit of, and the universal property holds. ψ Q Choose a competitor cocone with apex Q. ψ : F ( x ) → Q making the newly formed triangle commute.

The concepts of ∪ and ∩ for subsets correspond to the operations of ∧ and ∨ on predicates. { x ∈ X | χ( x ) = T } ∪ { x ∈ X | ϕ( x ) = T } = { x ∈ X | (χ ∨ ϕ)( x ) = T } and similarly for ∩ and ∧. 5. In Set, Sub( X ) for X ∈ Set is a poset via ⊆, and thus a category where there exists a unique morphism from A to B if and only if A ⊆ B ( A, B ⊆ X ). In this category A ∩ B is the product of A and B, and A ∪ B is the coproduct. Proof. ψ ⊆ ⊆ A∩B ⊆ ⊆ A B which is true since Q ⊆ A, Q ⊆ B =⇒ Q ⊆ A ∩ B.

We get a monoidal category (C, •, e) where • is the binary product of G and e is the identity element of G. Note: In general: • If C has products, we get a monoidal category (C, ×, 1). • If C has coproducts, we get a monoidal category (C, +, 0). 3. A monoidal category (C, ⊗, I ) is symmetric if it additionally is equipped with an isomorphism s x,y : x ⊗ y → y ⊗ x for any objects x and y of C, natural in x and y, such that the following diagrams commute for all objects x, y, and z: ( x ⊗ y) ⊗ z s x,y ⊗idz (y ⊗ x ) ⊗ z a x,y,z ay,x,z x ⊗ (y ⊗ z) y ⊗ ( x ⊗ z) s x,y⊗z idy ⊗s x,z (y ⊗ z) ⊗ x ay,z,x y ⊗ (z ⊗ x ) s x,I x⊗I rx I⊗x x x x⊗y s x,y y⊗x sy,x id x×y x⊗y 52 Most of the examples of monoidal categories we have talked about are symmetric.

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