Universal categories Alistair Savage University of Ottawa Slides available online: alistairsavage.ca/talks Alistair Savage (Ottawa) Universal categories January 11, 2019 1 / 37 Outline Goal: Give overview of universal categories and their importance. Overview: 1 Universal constructions in algebra 2 Warm-up 3 Monoidal categories and string diagrams 4 More warm-up 5 (Oriented) Temperley{Lieb category 6 (Oriented) Brauer category 7 Current work (very brief) Alistair Savage (Ottawa) Universal categories January 11, 2019 2 / 37 Free monoid Let S be a set. The free monoid MS on S has elements given by words in S, multiplication given by concatenation: (ab2c) · (c4ba) = ab2c5ba Universal property: Given any function f : S ! M, with M a monoid, there exists a unique monoid homomorphism ': MS ! M such that s7!s S / MS ' f M commutes. Alistair Savage (Ottawa) Universal categories January 11, 2019 3 / 37 Free group Let S be a set. The free group GS on S has elements given by words in S (+ inverses), multiplication given by concatenation with reduction: (ab2c) · (c−1ba) = ab2cc−1ba = ab3a Universal property: Given any function f : S ! G, with G a group, there exists a unique group homomorphism ': GS ! G such that s7!s S / GS ' f G commutes. Alistair Savage (Ottawa) Universal categories January 11, 2019 4 / 37 Free algebras Let | be a commutative ring. Free commutative algebra The polynomial ring |[X1;:::;Xn] is the free commutative |-algebra on n generators. Universal property: Given any commutative |-algebra R and elements r1; : : : ; rn 2 R, there exists a unique ring homomorphism |[X1;:::;Xn] ! R; Xi 7! ri: Free associative algebra One can also define free associative (not necessarily commutative) algebras. Alistair Savage (Ottawa) Universal categories January 11, 2019 5 / 37 Warm-up: Some simple universal categories Free category F(X) on one object X Objects: X Morphisms: Hom(X; X) = f1X g Universal property: If C is any category and Y is an object of C, then there exists a unique functor F(X) !C;X 7! Y: Alistair Savage (Ottawa) Universal categories January 11, 2019 6 / 37 Warm-up: Some simple universal categories Free category F(X; f) on one object X and one generating morphism f : X ! X Objects: X Morphisms: Hom(X; X) is the free monoid on the generator f. Universal property: If C is any category, Y is an object of C, and g : Y ! Y is a morphism in C, then there exists a unique functor F(X; F ) !C;X 7! Y; f 7! g: Alistair Savage (Ottawa) Universal categories January 11, 2019 7 / 37 Strict monoidal categories A strict monoidal category is a category C equipped with a bifunctor (the tensor product) ⊗: C × C ! C, and a unit object 1, such that (A ⊗ B) ⊗ C = A ⊗ (B ⊗ C) for all objects A, B, C, 1 ⊗ A = A = A ⊗ 1 for all objects A, (f ⊗ g) ⊗ h = f ⊗ (g ⊗ h) for all morphisms f, g, h, 11 ⊗ f = f = f ⊗ 11 for all morphisms f. Remark: Non-strict monoidal categories In a (not necessarily strict) monoidal category, the equalities above are replaced by isomorphism, and we impose some coherence conditions. Every monoidal category is monoidally equivalent to a strict one. Alistair Savage (Ottawa) Universal categories January 11, 2019 8 / 37 |-linear monoidal categories Fix a commutative ground ring |. A |-linear category is a category such that each morphism space is a |-module, composition of morphisms is |-bilinear. In a strict |-linear monoidal category, we also require that tensor product of morphisms is |-bilinear. The interchange law The axioms of a strict monoidal category imply the interchange law: For f g A1 −! A2 and B1 −! B2, the following diagram commutes: 1⊗g A1 ⊗ B1 / A1 ⊗ B2 f⊗g f⊗1 f⊗1 & A ⊗ B / A ⊗ B 2 1 1⊗g 2 2 Alistair Savage (Ottawa) Universal categories January 11, 2019 9 / 37 Warm-up: Some simple universal categories Free |-linear category L(X; f) on one object X and one generating morphism f : X ! X Objects: X Morphisms: Hom(X; X) = |[f]. Universal property: If C is any |-linear category, Y is an object of C, and g : Y ! Y is a morphism in C, then there exists a unique |-linear functor F(X; F ) !C;X 7! Y; f 7! g: Alistair Savage (Ottawa) Universal categories January 11, 2019 10 / 37 Warm-up: Some simple universal categories Free monoidal category M(X) on one generating object X Objects: X⊗n for n ≥ 0 (where X⊗0 = 1). Morphisms: ( 1 ⊗n if n = m; Hom(X⊗n;X⊗m) = X ? if n 6= m: Universal property: If C is any monoidal category and Y is an object of C, then there exists a unique monoidal functor M(X) !C;X 7! Y: Alistair Savage (Ottawa) Universal categories January 11, 2019 11 / 37 Warm-up: Some simple universal categories Free |-linear monoidal category M(X; f) on one generating object X and one generating morphism f : X ! X Objects: X⊗n for n ≥ 0. Morphisms: ( [f ; : : : ; f ] if n = m; Hom(X⊗n;X⊗m) = | 1 n 0 if n 6= m; ⊗(i−1) ⊗(n−i) where fi = 1X ⊗ f ⊗ 1X . (The fi commute because of the interchange law!) Universal property: If C is any |-linear monoidal category, Y is an object of C, and g : Y ! Y is a morphism in C, then there exists a unique |-linear monoidal functor M(X; f) !C;X 7! Y; f 7! g: Alistair Savage (Ottawa) Universal categories January 11, 2019 12 / 37 String diagrams Fix a strict monoidal category C. We will denote a morphism f : A ! B by: B f A The identity map 1A : A ! A is a string with no label: A A We sometimes omit the object labels when they are clear or unimportant. Alistair Savage (Ottawa) Universal categories January 11, 2019 13 / 37 String diagrams Composition is vertical stacking and tensor product is horizontal juxtaposition: f = fg f ⊗ g = f g g The interchange law then becomes: f g = f g = g f A morphism f : A1 ⊗ A2 ! B1 ⊗ B2 can be depicted: B1 B2 f A1 A2 Alistair Savage (Ottawa) Universal categories January 11, 2019 14 / 37 Presentations of strict monoidal categories One can give presentations of some strict |-linear monoidal categories, just as for monoids, groups, algebras, etc. Objects: If the objects are generated by some collection Ai, i 2 I, then we have all possible tensor products of these objects: 1;Ai;Ai ⊗ Aj ⊗ Ak ⊗ A`; etc. Morphisms: If the morphisms are generated by some collection fj, j 2 J, then we have all possible compositions and tensor products of these morphisms (whenever these make sense): 1Ai ; fj ⊗ (fifk) ⊗ (f`); etc. We then often impose some relations on these morphism spaces. String diagrams: We can build complex diagrams out of our simple generating diagrams. Alistair Savage (Ottawa) Universal categories January 11, 2019 15 / 37 Symmetric monoidal categories A symmetric monoidal category is a monoidal category C such that, for each pair of objects X; Y in C, there is an isomorphism =∼ sXY : X ⊗ Y −! Y ⊗ X that is natural in both X and Y , sYX sXY = 1X⊗Y for all objects X, Y , the sXY satisfy certain (associativity and unit) coherence conditions. Examples Sets (cartesian product) Groups (cartesian product) Vector spaces over a given field (tensor product) Alistair Savage (Ottawa) Universal categories January 11, 2019 16 / 37 A universal symmetric monoidal category Define a strict monoidal category S with one generating object " and denote 1" = We have one generating morphism : " ⊗ "!" ⊗ " : We impose the relations: = ; = : Then ⊗n EndS (" ) = Sn is the symmetric group on n letters. Alistair Savage (Ottawa) Universal categories January 11, 2019 17 / 37 A universal symmetric monoidal category Universal property: If C is any symmetric monoidal category and X is an object of C, then there is a unique monoidal functor S!C; " 7! X; 7! sXX : ⊗n Immediate consequence: We have an Sn-action on EndC(X ) coming from the induced map ⊗n ⊗n Sn = EndS (" ) ! EndC(X ): Linear version: We can repeat the above with |-linear categories. Then we get an action of the group algebra ⊗n ⊗n |Sn = EndS (" ) ! EndC(X ): Alistair Savage (Ottawa) Universal categories January 11, 2019 18 / 37 Duals Suppose a strict monoidal category C has two objects " and #, with 1" = ; 1# = : A morphism 1 ! # ⊗ " would have string diagram ; where = 11: We typically omit the dotted line and draw: : 1 !# ⊗ " : Similarly, we can have : " ⊗ #! 1: Alistair Savage (Ottawa) Universal categories January 11, 2019 19 / 37 Duals We say that # is right dual to " (and " is left dual to # ) if there exist morphisms : 1 !# ⊗ " : and : " ⊗ #! 1: such that = and = : Objects " and # are both left and right dual to each other if we also have : 1 !" ⊗ # and : # ⊗ "! 1 such that = and = : Alistair Savage (Ottawa) Universal categories January 11, 2019 20 / 37 Duals: example Let | be a field and consider the category Vect| of f.d. |-vector spaces. Unit object: | Fix a f.d. |-vector space V . Claim: The dual vector space V ∗ is both left and right dual to V , in the sense mentioned above. ∗ Proof: Fix a basis B of V . Let fδv : v 2 Bg be the dual basis of V . Viewing V and " and V ∗ as #, we define ∗ P : | ! V ⊗ V; 1 7! v2B δv ⊗ v; ∗ : V ⊗ V ! |; v ⊗ f 7! f(v); ∗ P : | ! V ⊗ V ; 1 7! v2B v ⊗ δv; ∗ : V ⊗ V ! |; f ⊗ v 7! f(v): Alistair Savage (Ottawa) Universal categories January 11, 2019 21 / 37 Duals: example Let's check the relation = : The left-hand side is the composition ∼ 1V ⊗ ∗ ⊗1V ∼ V = V ⊗ | −−−−−! V ⊗ V ⊗ V −−−−−! | ⊗ V = V; X X X w 7! w ⊗ 1 7! w ⊗ δv ⊗ v 7! δv(w) ⊗ v 7! δv(w)v = w: v2B v2B v2V The verification of the other relations is analogous.