The Tannaka Construction in General Categories,Part II

Micah Blake McCurdy

April 1, 2009

Let B be a braided autonomous category admitting certain ends orcoends which cohere with the monoidal structure. Let A be amerely monoidal category, and suppose that F : A −→ B is aseparable Frobenius functor. We have seen previously that theobjects: ∫

a∈AFa⊗ LFa

∫ a∈A

Fa⊗ LFa

can be given the structure of weak bialgebras in B. Let us todayfocus on the former, which we will today call tan F , although someof the slides refer to EF .

Recall that tan F acts canonically on F⊗n; indeed, it is theuniversal such action. Hence, there is a bijection between

f : X −→ (tan F )⊗n

andActions of X on F⊗n

We call these actions the “dicharged forms” of the mapsf : X −→ (tan F )⊗n.

The data of a weak bialgebra B are:

η : e −→ B

ε : B −→ e

µ : B ⊗ B −→ B

∆ : B −→ B ⊗ B

S : B −→ B

We recall the definitions of these maps when H = tan F .

If we assume furthermore that A is (left, say) autonomous, we canobtain an antipode

S : tan F −→ tan F

which makes tan F into a weak Hopf algebra in B, defined as:

Note that S so defined is not, in general, invertible, although it willbe if A is also right autonomous.

This is the beginning of a pattern, namely, we will consider variousstructures on A and use them to define analagous structures ontan F . We think of A as the representation category of somealgebraic object, although this is not quite true.We will not assume that F preserves these structures, but insteadmeasure the failure of F to preserve them, and use thismeasurement to define the desired structures on tan F .

We will consider three structures in particular:

I Braided Structures

I Ribbon Structures

I Cyclic (pivotal) Structures

Notice first of all that the universal action of tan F on F meansthat there are, in general, many (generalized) elements of(tan F )⊗n. For instance, given an elementg : X −→ tan F ⊗ tan F , we can define two elements g l , g r of thesame type, as having the following discharged forms:

When B is symmetric, both of these are equal to the common value

Xg−−−→ tan F ⊗ tan F

σ−−−→ tan F ⊗ tan F

, which is written as gop. However, when B is merely braided thereis no obvious relation between g l and g r and:

In the classical case, an almost cocommutative bialgebra J is oneposessing an element R ∈ J ⊗ J which satisfies

R ?∆ = ∆op ? R

That is, R measures the failure of ∆ = ∆op. In the case at hand,where B is merely braided, we will have to measure the differencebetween ∆ and ∆l as well as the difference between ∆ and ∆r .

We say that a weak hopf algebra of the form tan F is almostcocommutative if there exists a pair R,S : e −→ tan F ⊗ tan Fsuch that

R ?∆ = ∆r ? R S ?∆ = ∆l ? S

Following Etingof & Shiffman, we denote R by a semicircle and Sby a semicircle with a dash.

We write ∆r much as ∆, only decorated with a small semicircle,and ∆l decorated with a semicircle and a dash; in sum:

This semicircle notation should not be confused with the muchmore upright notation for units of duals.

An almost cocommutative weak Hopf algebra of the form tan F issaid to be braided when, furthermore:

When B is symmetric, R and S are both invertible and can beobtained from one another, namely, as

R = (S−1)op S = (R−1)op

and the many axioms above collapse to the usual ones.

Braided Structure on tan F

When A is braided, we can obtain a braided structure on tan F bymeasuring the failure of F to preserve this braid. Specifically, wedefine:

Let us now assume that B is ribbon, so that B is symmetricprecisely when this ribbon is the identity. Just as the braidedstructure (R, S) measures the differences between “∆” and “∆conjugated by the braid”, a ribbon structure on tan F will measurethe difference between “the identity on tan F” and “the identity ontan F conjugated by the ribbon”. Define ψ : tan F −→ tan F ashaving discharged form:

This is “the identity conjugated by the ribbon”.

We say that a braided weak Hopf algebra of the form tan F isribbon when there exists an invertible element v : e −→ tan F suchthat:

v ? ψ ? v−1 = tan F

Sv = v

εv = εη

R l ? (v ⊗ v) ? R = ∆v

S r ? (v−1 ⊗ v−1) ? S = ∆v−1

Note that v is not assumed to be central, as is usual. However,when B is symmetric, ψ is the identity and v ? ψ ? v−1 = tan Freduces to the centrality of v .

In the case at hand, define v as:

If A and B are cyclic, then we expect to be able to construct acyclic structure on tan F . We drop the assumption that A bebraided and ribbon and instead ask merely for A to be cyclic. Sincewe know that braided autonomous categories are ribbon if and onlyif they are cyclic, we have a canonical cyclic structure on B. Asbefore, we will obtain the cyclic structure on tan F be measuringthe failure of F to preserve the cyclic structure on A.

Let us say that a weak Hopf algebra H is cyclic if there exists aninvertible cyclic element:

c : e −→ H

such that:

∆c = (∆η) ? (c ⊗ c) εc = εη Sc = c−1

If B is symmetric, this last axiom reduces to saying that cimplements S2; furthermore, if the weak Hopf algebra has∆η = η ⊗ η, as for ordinary Hopf algebras, then the first axiomreduces to ∆c = c ⊗ c. Elements with this last property are calledgroup-like, and group-like elements of ordinary Hopf algebras whichimplement the square of the antipode have been called specialgroup-like, and have been studied by algebraists already. However,as far as I can tell, nobody has noticed that the representationcategories of Hopf algebras with special group-like elements are, asone might expect, cyclic autonomous.

In the case at hand, define c : e −→ tan F to have discharged form:

Fx ' RLFxRφFx−−−→ RRFx ' FRRx

FRφ−1x−−−−→ FRLx ' Fx

which is the same as:

As an example, let’s see the proof of:

R l ? (v ⊗ v) ? R = ∆v

As another example, let’s see the proof of:

R l ? (v ⊗ v) ? R = ∆v

R l ? (v ⊗ v) ? R = ∆v

R l ? (v ⊗ v) ? R = ∆v

R l ? (v ⊗ v) ? R = ∆v

R l ? (v ⊗ v) ? R = ∆v

R l ? (v ⊗ v) ? R = ∆v