Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References Enriched Categories Alexander Rogovskyy Proseminar \Category Theory", Summer 2020 Alexander Rogovskyy Enriched Categories 1/ 33 Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References Table of Contents Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories Alexander Rogovskyy Enriched Categories 2/ 33 Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References Example: Vec 5 f 3 R R I The morphisms between two objects are a Set I Hom(R5; ): Vec ! Set I Set is special. Can we try to get away from that? Alexander Rogovskyy Enriched Categories 3/ 33 Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References Special structure of morphisms I We often have special structure in morphisms I A notion of \combining two morphisms" into one I Concatenation: ◦ : Hom(A; B) × Hom(B; C) ! Hom(A; C) I Tensoring ⊗ : Hom(A; B) ⊗ Hom(B; C) ! Hom(A; C) I Can we generalize the notion of morphisms to reflect that? Alexander Rogovskyy Enriched Categories 4/ 33 Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References Motivation for Enriched Categories Two central questions: 1. Can we generalize over Set? 2. Can we employ special structure of morphisms () relationships) between objects Alexander Rogovskyy Enriched Categories 5/ 33 Classical categories can be seen as enriched over Set. Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References Answer: Instead of sets of morphisms (hom-sets) f g A B h we use hom-objects from another category V to represent the relationship between A and B! A C(A; B) B Alexander Rogovskyy Enriched Categories 6/ 33 Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References Answer: Instead of sets of morphisms (hom-sets) f g A B h we use hom-objects from another category V to represent the relationship between A and B! A C(A; B) B Classical categories can be seen as enriched over Set. Alexander Rogovskyy Enriched Categories 6/ 33 Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References An example: R-Vec as a Set-Category 3 Hom(R ; C) 5 3 3 ◦ 5 Hom(R ; R ) × Hom(R ; C) Hom(R ; C) 5 3 Hom(R ; R ) 5 Hom(R ; C) 5 5 3 3 3 R Hom(R ; R ) R Hom(R ; C) C Alexander Rogovskyy Enriched Categories 7/ 33 Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References An example: R-Vec as a Set-Category 3 Hom(R ; C) 5 3 3 ◦ 5 Hom(R ; R ) × Hom(R ; C) Hom(R ; C) 5 3 Hom(R ; R ) 5 Hom(R ; C) 5 5 3 3 3 R Hom(R ; R ) R Hom(R ; C) C Alexander Rogovskyy Enriched Categories 7/ 33 Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References Another point of view I Essentially, enriched categories can be thought of as directed graphs with edge labels I The edge labels come from another (regular, non-enriched) category V I We have a hom-object between any two objects =) the graph is complete B A C Important! We do not have individual morphisms anymore! Alexander Rogovskyy Enriched Categories 8/ 33 Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References Step 1: Monoidal Categories I What structure should our category of hom-objects V have? I We saw that something like a cartesian product × could be useful Alexander Rogovskyy Enriched Categories 9/ 33 =) essentially a categorical version of a monoid Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References Definition (Monoidal Category) A monoidal category consists of I A category V I A functor ⊗ : V × V ! V (\tensor product") I An element 1 2 Ob(V) (\unit") such that I ⊗ is associative (up to isomorphism) I 1 is the left and right unit w.r.t ⊗ (up to isomorphism) I certain coherence diagrams commute (see next slide) Alexander Rogovskyy Enriched Categories 10/ 33 Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References Definition (Monoidal Category) A monoidal category consists of I A category V I A functor ⊗ : V × V ! V (\tensor product") I An element 1 2 Ob(V) (\unit") such that I ⊗ is associative (up to isomorphism) I 1 is the left and right unit w.r.t ⊗ (up to isomorphism) I certain coherence diagrams commute (see next slide) =) essentially a categorical version of a monoid Alexander Rogovskyy Enriched Categories 10/ 33 Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References Some coherence Axioms ((W ⊗ X ) ⊗ Y ) ⊗ Z (W ⊗ X ) ⊗ (Y ⊗ Z) W ⊗ (X ⊗ (Y ⊗ Z)) (W ⊗ (X ⊗ Y )) ⊗ Z W ⊗ ((X ⊗ Y ) ⊗ Z) (X ⊗ 1) ⊗ Y X ⊗ (1 ⊗ Y ) X ⊗ Y Alexander Rogovskyy Enriched Categories 11/ 33 Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References Examples I Set with × and {·} I k-Vec with × (=⊕) and f0g I k-Vec with ⊗ and k I Grp with × and feg I Graph with the tensor product of graphs and I Cat with × and I R+ (with ≥ for morphisms), + and 0. Alexander Rogovskyy Enriched Categories 12/ 33 I We call this adjoint functor the internal hom I cartesian if ⊗ is the category-theoretic product and 1 is the terminal object I closed if for every X , the functor ⊗ X has a right adjoint Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References Special cases and variations A monoidal category is called... I symmetric if ⊗ is commutative up to isomorphism (and some additional coherence diagramms commute) Alexander Rogovskyy Enriched Categories 13/ 33 I We call this adjoint functor the internal hom I closed if for every X , the functor ⊗ X has a right adjoint Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References Special cases and variations A monoidal category is called... I symmetric if ⊗ is commutative up to isomorphism (and some additional coherence diagramms commute) I cartesian if ⊗ is the category-theoretic product and 1 is the terminal object Alexander Rogovskyy Enriched Categories 13/ 33 I We call this adjoint functor the internal hom Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References Special cases and variations A monoidal category is called... I symmetric if ⊗ is commutative up to isomorphism (and some additional coherence diagramms commute) I cartesian if ⊗ is the category-theoretic product and 1 is the terminal object I closed if for every X , the functor ⊗ X has a right adjoint Alexander Rogovskyy Enriched Categories 13/ 33 Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References Special cases and variations A monoidal category is called... I symmetric if ⊗ is commutative up to isomorphism (and some additional coherence diagramms commute) I cartesian if ⊗ is the category-theoretic product and 1 is the terminal object I closed if for every X , the functor ⊗ X has a right adjoint I We call this adjoint functor the internal hom Alexander Rogovskyy Enriched Categories 13/ 33 Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References Step 2: Enriched Categories Remember that we are replacing hom-sets with hom-objects. We need to have a composition morphism ◦ := ◦ABC C(A; B) ⊗ C(B; C) −!◦ C(A; C) In V = Set, this is just the usual composition of functions. Alexander Rogovskyy Enriched Categories 14/ 33 Solution Require a morphism jA : 1 !C(A; A) that represents \picking" an identity C(A; A) jA j in V 1 B C(B; B) ::: Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References I But what about the identity morphism idA? I We don't have individual morphisms anymore, only hom-objects! I We need a way to \pick" an identity Alexander Rogovskyy Enriched Categories 15/ 33 C(A; A) jA j in V 1 B C(B; B) ::: Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References I But what about the identity morphism idA? I We don't have individual morphisms anymore, only hom-objects! I We need a way to \pick" an identity Solution Require a morphism jA : 1 !C(A; A) that represents \picking" an identity Alexander Rogovskyy Enriched Categories 15/ 33 Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References I But what about the identity morphism idA? I We don't have individual morphisms anymore, only hom-objects! I We need a way to \pick" an identity Solution Require a morphism jA : 1 !C(A; A) that represents \picking" an identity C(A; A) jA j in V 1 B C(B; B) ::: Alexander Rogovskyy Enriched Categories 15/ 33 Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References Definition (Enriched Category) Let (V; ⊗; 1) be a monoidal category. A V-Category C consists of I A collection of objects Ob(C) I For A; B 2 Ob(C) a hom-object C(A; B) 2 V I For A; B; C 2 Ob(C) a composition morphism ◦ABC : C(A; B) ⊗ C(B; C) !C(A; C) I For A 2 Ob(C) an identity selector jA : 1 !C(A; A) such that the associativity and unit laws (next slides) hold. Alexander Rogovskyy Enriched Categories 16/ 33 Motivation and Intuition Constructing Enriched Categories Self-Enriched Categories References Associativity of ◦ (C(A; B) ⊗ C(B; C)) ⊗ C(C; D) C(A; B) ⊗ (C(B; C) ⊗ C(C; D)) ◦⊗id id⊗◦ C(A; C) ⊗ C(C; D) C(A; B) ⊗ C(B; D) ◦ ◦ C(A; D) Alexander Rogovskyy Enriched Categories 17/ 33 \If we pick the right identity, it will act as a unit w.r.t ◦!" Motivation