Pseudo Algebraic Structures in CFT

Tom Fiore fioret@umich.edu

http://www.math.lsa.umich.edu/∼fioret/

Introduction

This work is a contribution to the categorical foundations of CFT.

A CFT is a morphism of stacks of pseudo commutative monoids with cancellation. (Hu, Kriz)

Bicategorical versions of limits, colimits, and adjoints are essential to this approach.

General Philosophy

• equalities are replaced by coherence isos • these coherence isos satisfy coherence dia-

grams

• bijection of sets is replaced by equivalence of categories Examples

A pseudo functor between 2-categories preserves compositions and identities up to coherence isos, which satisfy coherence diagrams.

A pseudo natural transformation α : F → G between pseudo functors F and G is an assignment A ↦ α_{A} which is natural up to a coherence iso. This coherence iso satisfies coherence diagrams.

[figure: F A; α_{A}; F f; GA; Gf; τ_{f}; F B; α_{B}; GB]

Pseudo Limits

Let F : J → C be a pseudo functor. Then L ∈ C and π ∈ P sCone(L, F) comprise a pseudo limit of F if

Mor_{C}(L′, L) → P sCone(L′, F)
f ↦ π ∘ f

is an isomorphism of categories for all L′ ∈ C.

Variants: weighted pseudolimits, bilimits, weighted bilimits Example 2-equalizer vs. Pseudo-equalizer

Suppose F : \{\xymatrix{1 \ar[r] \ar[r] & 2}\} → Cat has image

\xymatrix{A \ar[r]^{G} \ar[r]_{H} & B}.

Then Obj 2-lim F = {a ∈ A : Ga = Ha}, while the object class of the pseudo limit P sCone(1, F) is

\{(a, b, g, h) : a ∈ A, b ∈ B, \xymatrix{g : Ga \ar[r]^{≅} & b}, \xymatrix{h : Ha \ar[r]^{≅} & b}\}.

Hence 2-lim F may be empty while ps-lim F is not.

Theories and Algebras

A Lawvere theory is a category T with objects 0, 1, 2, … such that n is the product of n-copies of 1 and each n is equipped with a limiting cone.

Example The Endomorphism Theory

Let X be a set. The endomorphism theory End(X) has objects 0, 1, 2, … and morphisms Hom_{End(X)}(m, n) = Maps(X^{m}, X^{n}).

A theory is a collection of sets T (0), T (1), … equipped with maps

γ : T (k)×T (n_{1})×⋯×T (n_{k}) → T (n_{1} +⋯+n_{k})

that are associative, unital, and are equivariant with respect to subsitution.

Translation: T (n) = T (n, 1) and also T (m, n) = ∏_{i=1}^{n} T (m).

A small category X is a T -algebra or an algebra over T if it is equipped with a morphism of theories (Φ_{n} : T (n) → End(X)(n))_{n}.

Example Groups

The theory T of groups is generated by e ∈ T (0), ν ∈ T (1), μ ∈ T (2) and these satisfy the usual axioms of a group. A group is an algebra over the theory of groups.

A small category X is a pseudo T -algebra if it is equipped with

• structure maps Φ_{n} : T (n) → End(X)(n) for each n ≥ 0

• coherence isos for each operation of theories (γ, 1, and substitution) which satisfy

• coherence diagrams for each relation of theories (associativity, unitality, equivariance).

For example, for w, v, u ∈ T (1) the diagram

[figure: Φ_{1}(w) ∘ Φ_{1}(v) ∘ Φ_{1}(u); Φ_{1}(w) ∘ Φ_{1}(v ∘ u); Φ_{1}(w ∘ v) ∘ Φ_{1}(u); Φ_{1}(w ∘ v ∘ u)]

must commute.

Example Finite sets with ∐ form a pseudo commutative monoid.

A ∐ B : A × {1} ∪ B × {2} B ∐ A B × {1} ∪ A × {2}

Example Finite dimensional vector spaces with ⊕ and ⊗ form a pseudo commutative semiring.

Example Any symmetric monoidal category is a pseudo commutative monoid.

Theorem 1 (F) Let T be a theory. Let C_{T} denote the 2-category of pseudo T -algebras. Then any pseudo functor F : J → C_{T} from a 1-category J to the 2-category of pseudo T - algebras admits all weighted pseudo limits and all weighted bicolimits.

Proof: Let L := P sCone(1, U ∘ F), where U : C_{T} → Cat is the forgetful 2-functor. Then L has a pseudo T -algebra structure and it is the conical pseudo limit of F. Theorems from Street (1980) about cotensor products imply the result for conical pseudo limits. The conical bicolimits require a different construction.

This allows one to speak of stacks of pseudo T -algebras.

Theorem 2 (F) Let T be a theory. There exists a 2-monad C on Cat such that the 2-category of strict C-algebras, pseudo C-morphisms, and 2-cells is 2-equivalent to C_{T}.

Proof: The 2-monad C is constructed by taking the free theory on T and enriching it appropriately in groupoids.

Theorem 3 (Blackwell, Kelly, Power 1989) Let C be a 2-monad on a 2-category K. Then the 2-category of strict C-algebras, pseudo C-morphisms, and 2-cells admits all strictly weighted pseudo limits of strict 2-functors.

This theorem implies a special case of Theorem 1.

Bi-adjoints

Let X , A be 2-categories. A bi-adjunction ⟨F, G, ϕ⟩ : X ⇀ A consists of the following data

• Pseudo functors

\xymatrix{X \ar[r]^{F} & A \ar[l]^{G}}

• For all X ∈ ObjX and all A ∈ ObjA an equivalence of categories ϕ_{X,A} : Mor_{A}(F X, A) → Mor_{X} (X, GA) which is pseudo natural in X and A separately.

Theorem 4 (Kelly 1989) (F) Let X , A be 2categories. Let \xymatrix{X & A \ar[l]_{G}} be a pseudo functor. Then there exists a left bi-adjoint for G if and only if for every object X ∈ ObjX there exists an object R ∈ ObjA and a bi-universal arrow η_{X} : X → G(R) from X to G.

Theorem 5 (F) Let S → T be a morphism of theories and let G : C_{T} → C_{S} be the associated forgetful 2-functor. Then there exists a left bi-adjoint F : C_{S} → C_{T} which is a strict 2functor. Moreover, ϕ_{X,A} is strictly 2-natural in each variable.

2-Theories, Algebras, and Pseudo Limits

A 2-theory (Θ, T) fibered over the theory T is a natural number k, a theory T, and a contravariant functor Θ : T → Cat from the category T to the 2-category Cat of small categories such that

• ObjΘ(m) = ∐_{n≥0} Hom_{T^{k}}(m, n) for all m ≥ 0, where T^{k} is the theory with the same objects as T, but with Hom_{T^{k}}(m, n) = Hom_{T} (m, n)^{k}

• If w_{1}, …, w_{n} ∈ Hom_{T^{k}}(m, 1), then the word in Hom_{T^{k}}(m, n) with which the n-tuple is identified is the product in Θ(m) of w_{1}, …, w_{n}

• For w ∈ Hom_{T} (m, n) the functor Θ(w) : Θ(n) → Θ(m) is Θ(w)(v) = v ∘ w^{×k} on objects v ∈ Hom_{T^{k}}(n, j).

For objects w_{1}, …, w_{n}, w ∈ Mor_{T^{k}}(m, 1) ⊆ ObjΘ(m) we set

Θ(w; w_{1}, …, w_{n}) := Mor_{Θ(m)} ( n ; ∏ ; i=1 w_{i}, w).

Example Let I be a 1-category and X : I^{k} → Cat a strict 2-functor. Then there is a 2-functor End(X) : End(I)^{op} → Cat called the endomorphism 2-theory on X which is defined analogously to the endomorphism theory on a set.

A pseudo (Θ, T)-algebra consists of the following data:

• a small pseudo T -algebra I with structure maps denoted Φ : T (n) → F unctors(I^{n}, I)

• a strict 2-functor X : I^{k} → Cat

• set maps ϕ : Θ(w; w_{1}, …, w_{n}) → End(X)(Φ(w); Φ(w_{1}), …, Φ(w_{n})), where Φ(w) means to apply Φ to each component of w to make I^{k} into the product pseudo algebra of k copies of I.

• a coherence iso for each operation of a 2theory satisfying

• coherence diagrams for each relation of 2theories.

Example The 2-theory of commutative monoids with cancellation (CMC’s) is generated by 3 operations. The underlying theory T is commutative monoids and k=2. The operations are described below in terms of an algebra X : I^{2} → Cat rather than in terms of the abstract 2-theory.

+ : X_{a,b} × X_{c,d} → X_{a+c,b+d}
0 ∈ X_{0,0}
ˇ? : X_{a+c,b+c} → X_{a,b}

Axioms: + is commutative, associative, and unital ˇ? is transitive and distributive.

Example Rigged surfaces form a pseudo algebra over the 2-theory of CMC’s.

I := Finite Sets and bijections equipped with a choice of ∐ X : I^{2} → Cat X_{a,b} := the category of rigged surfaces with inbound boundary components labelled by the finite set a and outbound boundary components labelled by the finite set b

\xymatrix{+ : X_{a,b} × X_{c,d} \ar[r]^{∐} & X_{a ∐ c,b ∐ d}}
\xymatrix{ˇ? : X_{a ∐ c,b ∐ c} \ar[r]^{glue} & X_{a,b}}

Theorem 6 (F) The 2-category of pseudo (Θ, T)-algebras admits weighted pseudo limits.

This allows one to speak of stacks of pseudo (Θ, T)-algebras, in particular stacks of pseudo commutative monoids.

Definition of CFT

An abstract chiral conformal field theory on a stack of pseudo commutative monoids D is a Hilbert space H and a morphism ϕ : D → H of stacks of pseudo commutative monoids over the identity map of the underlying stack of pseudo commutative monoids.

This definition accounts for the coherence isomorphisms involved in disjoint union and gluing.

Conclusion

Certain bicategorical limits, colimits, and adjoints exist in the context of pseudo algebras over theories and 2-theories. Their existence allows one to speak of stacks in the definition of CFT formulated by Hu and Kriz.