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 ϕ: MS → M such that

s7→s S / MS ϕ 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 ϕ: GS → G such that

s7→s S / GS ϕ f  G commutes.

Alistair Savage (Ottawa) Universal categories January 11, 2019 4 / 37 Free algebras

Let k be a commutative ring.

Free commutative algebra

The polynomial ring k[X1,...,Xn] is the free commutative k-algebra on n generators.

Universal property: Given any commutative k-algebra R and elements r1, . . . , rn ∈ R, there exists a unique ring homomorphism

k[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) = {1X }

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 k-linear monoidal categories

Fix a commutative ground ring k. A k-linear category is a category such that each morphism space is a k-module, composition of morphisms is k-bilinear. In a strict k-linear monoidal category, we also require that tensor product of morphisms is k-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 k-linear category L(X, f) on one object X and one generating morphism f : X → X Objects: X Morphisms: Hom(X,X) = k[f].

Universal property: If C is any k-linear category, Y is an object of C, and g : Y → Y is a morphism in C, then there exists a unique k-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 k-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) = k 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 k-linear monoidal category, Y is an object of C, and g : Y → Y is a morphism in C, then there exists a unique k-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 k-linear monoidal categories, just as for monoids, groups, algebras, etc.

Objects: If the objects are generated by some collection Ai, i ∈ 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 ∈ 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 k-linear categories. Then we get an action of the group algebra

⊗n ⊗n kSn = 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 k be a field and consider the category Vectk of f.d. k-vector spaces. Unit object: k Fix a f.d. k-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 {δv : v ∈ B} be the dual basis of V . Viewing V and ↑ and V ∗ as ↓, we define

∗ P : k → V ⊗ V, 1 7→ v∈B δv ⊗ v, ∗ : V ⊗ V → k, v ⊗ f 7→ f(v), ∗ P : k → V ⊗ V , 1 7→ v∈B v ⊗ δv, ∗ : V ⊗ V → k, 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 ⊗ k −−−−−→ V ⊗ V ⊗ V −−−−−→ k ⊗ V = V, X X X w 7→ w ⊗ 1 7→ w ⊗ δv ⊗ v 7→ δv(w) ⊗ v 7→ δv(w)v = w. v∈B v∈B v∈V

The verification of the other relations is analogous.

Note: X : k → k, 1 7→ δv(v) = dim V. v∈B So the “bubble” corresponds to the dimension of the object V . Alistair Savage (Ottawa) Universal categories January 11, 2019 22 / 37 The oriented Temperley–Lieb category

The oriented Temperley-Lieb category OTL is the free k-linear monoidal category on a pair of dual objects.

Two generating objects: ↑ and ↓

Four generating morphisms:

, , , .

Relations:

= , = , = , = .

Alistair Savage (Ottawa) Universal categories January 11, 2019 23 / 37 The oriented Temperley–Lieb category

An arbitrary morphism is a k-linear combination of oriented Temperly–Lieb diagrams:

Composition is vertical “gluing”. Universal property: Any k-linear monoidal category C with a pair of dual objects V , W admits a k-linear monoidal functor OT L → C, ↑ 7→ V, ↓ 7→ W.

If we impose an extra relation that bubbles are equal to some δ (giving a category OTL(δ)), we get a universal property for objects of dimension δ. Alistair Savage (Ottawa) Universal categories January 11, 2019 24 / 37 The oriented Temperley–Lieb category

Example (Vector spaces) For any f.d. k-vector space V , we have a k-linear monoidal functor

∗ OT L → Vectk, ↑ 7→ V, ↓ 7→ V .

Example (glδ(k)-mod) δ Let V = k be the natural glδ(k)-module. We have a k-linear monoidal functor ∗ OTL(δ) → glδ(k)-mod, ↑ 7→ V, ↓ 7→ V .

Question: What if we look at self-dual objects?

Alistair Savage (Ottawa) Universal categories January 11, 2019 25 / 37 The Temperley–Lieb category

The Temperley-Lieb category TL is the free k-linear monoidal category on a self-dual object.

One generating object: |

Two generating morphisms: , .

Relations: = , = .

If we impose the extra relation = δ

we get TL(δ), the free k-linear monoidal category on a self-dual object of dimension δ. Alistair Savage (Ottawa) Universal categories January 11, 2019 26 / 37 The Temperley–Lieb category

The endomorphism algebra

⊗n TLn(δ) := EndTL(δ)(| )

is the Temperley–Lieb algebra.

Universal property: Any k-linear monoidal category C with a self-dual object V of dimension δ admits a k-linear monoidal functor

TL(δ) → C, | 7→ V.

Corollary: If C is a k-linear monoidal category with a self-dual object V of dimension δ, we have an algebra homomorphism

⊗n TLn(δ) → EndC(V ).

Alistair Savage (Ottawa) Universal categories January 11, 2019 27 / 37 The Temperley–Lieb category

Example (Inner product spaces) Consider the category Inn of real inner product spaces. For any object V of dimension δ, we have a functor

TL(δ) → Inn, | 7→ V

⊗n and an algebra homomorphism TLn(δ) → EndInn(V ).

Example (The orthogonal group) δ Let V = R be the natural representation of the orthogonal group Oδ(R). We have a functor

TL(δ) → Oδ(R)-mod, | 7→ V,

and an algebra homomorphism TL (δ) → End (V ⊗n). n Oδ(R)

Alistair Savage (Ottawa) Universal categories January 11, 2019 28 / 37 The oriented Brauer category

Idea: Let’s consider symmetric monoidal categories with duals!

The oriented Brauer category OB is the free k-linear symmetric monoidal category on a pair of dual objects.

Two generating objects: ↑ and ↓

Four generating morphisms:

, , , .

Define :=

Alistair Savage (Ottawa) Universal categories January 11, 2019 29 / 37 The oriented Brauer category

Relations:

= , = ,

= , = ,

= , = .

Then one can define = and = .

Universal property: Any k-linear symmetric monoidal category C with dual objects V,W admits a k-linear monoidal functor OB → C, ↑ 7→ V, ↓ 7→ W.

Alistair Savage (Ottawa) Universal categories January 11, 2019 30 / 37 The oriented Brauer category

Note: The oriented Temperley–Lieb category is a subcategory of the oriented Brauer category. We can enhance our previous examples.

Example (Vector spaces) For any f.d. k-vector space V , we have a k-linear monoidal functor

∗ OB → Vectk, ↑ 7→ V, ↓ 7→ V .

Example (glδ(k)-mod) δ Let V = k be the natural glδ(k)-module. We have a k-linear monoidal functor ∗ OB → glδ(k)-mod, ↑ 7→ V, ↓ 7→ V . Schur–Weyl duality: This functor is full.

Question: What if we look at self-dual objects?

Alistair Savage (Ottawa) Universal categories January 11, 2019 31 / 37 The Brauer category

The Brauer category B is the free k-linear symmetric monoidal category on a self-dual object.

One generating object: |

Three generating morphisms:

, , . Relations:

= , = ,

= , = , = , =

+ flips in the horizontal. Note: Can add relation that bubble is δ to get B(δ). Alistair Savage (Ottawa) Universal categories January 11, 2019 32 / 37 The Brauer category

An arbitrary morphism in B(δ) is a k-linear combination of Brauer diagrams:

Composition is vertical “gluing”. The endomorphism algebra

⊗n Bn(δ) := EndB(δ)(| )

is the Brauer algebra.

Alistair Savage (Ottawa) Universal categories January 11, 2019 33 / 37 The Brauer category

Universal property: Any k-linear symmetric monoidal category C with self-dual object V of dimension δ admits a k-linear monoidal functor B(δ) → C, | 7→ V.

Corollary: If C is a k-linear symmetric monoidal category with a self-dual object V of dimension δ, we have an algebra homomorphism ⊗n Bn(δ) → EndC(V ). Example (The orthogonal group) δ Let V = R be the natural representation of the orthogonal group Oδ(R). We have a functor

B(δ) → Oδ(R)-mod, | 7→ V,

and an algebra homomorphism B (δ) → End (V ⊗n). n Oδ(R)

Alistair Savage (Ottawa) Universal categories January 11, 2019 34 / 37 Braided monoidal categories

Some categories are not symmetric monoidal, but they are braided monoidal. Replace crossing generators with

and

subject to the relations

= , = , = .

Category of braids: Free braided monoidal category on one generating object. Category of framed oriented tangles: Free ribbon category (duals + twists) on one generating object (Shum–Reshetikhin–Turaev Theorem).

Alistair Savage (Ottawa) Universal categories January 11, 2019 35 / 37 Even more universal categories

Oriented skein category Start with category of framed oriented tangles. Impose a skein relation (symmetric groups Hecke algebras), twist relation, and dimension relation. Underpins the HOMFLY-PT polynomial (knot invariant).

Super versions Can work with supercategories. Now objects can be even dual or odd dual. Get odd Brauer category, etc.

Alistair Savage (Ottawa) Universal categories January 11, 2019 36 / 37 Current work (joint with J. Brundan and B. Webster)

Affine versions Allow strands to carry dots and impose some additional relations. Corresponds to symmetric groups degenerate affine Hecke algebras. Frobenius algebras Fix a Frobenius algebra F (associative algebra with a trace map). Deform categories: strands can now carry tokens labelled by elements of F . Obtain generalizations of (affine) oriented Brauer category, (affine) oriented skein category, etc.

Heisenberg categories Affine oriented Brauer category and affine oriented skein category are the charge zero case of a more general category: the (quantum) Heisenberg category.

Alistair Savage (Ottawa) Universal categories January 11, 2019 37 / 37