Tannaka duality

Given a compact group G, the set $\Rep G$ of isomorphism classes of appropriate representations admits various operations; for example direct sum and tensor product. Tannaka's duality theorem (1939) provided a recipe for recovering a compact group $\Gp R$ from a structure R such as $\Rep G$ whereby $\Gp\Rep G\cong G$.

For algebraic groups, Saavedra Rivano [Riv72] considered the category of appropriate representations together with the tensor structure and the underlying functor into vector spaces. He gave criteria on a tensor functor into vector spaces under which it should be equivalent to such an underlying functor. A non-commutative generalization of this was given by Ulbrich [Ulb89]. We shall lead into this Hopf algebra version by examining the 2-functor $\Mod_\scrV$ of the previous section.

For simplicity of exposition we suppose our tensor category $\scrV\,$ is strict. This loses no generality in fact since every tensor category is equivalent to a strict one (MacLane's coherence theorem). We also suppose that $\scrV$ is symmetric (but we cannot suppose the tensor product is strictly commutative). A consequence of this simplification is that we really do have a weak tensor functor

$$\map \Mod_\scrV \from \Mon_\scrV^\op \to {\CatV}$$

We are now interested in going back from $\CatV$ to $\MonV^\op\!$. A possible way to do this is via a left adjoint to $\ModV$, if it exists. Under reasonable conditions, a left adjoint $\EsubF\in \MonV)$ does exist at an object $\bigl(\,\scrC\,,F\,\bigr)$ of $\CatV$. It is constructed as follows.

If each internal hom $[\,\FX\,,\FX\,]$ exists in $\scrV$ and if $\scrV$ is suitably complete, we put

$$\EsubF \; = \; \int_{X\in \scrC}[\,\FX\,,\FX\,]$$

where the integral sign denotes the end (see MacLane [Mac71]) of the functor $\functor \from \,\scrC^\op \times \scrC \to \scrV$ taking (X ,Y) to $[\,\FX,FY\,]$; it is the equalizer of the obvious pair of arrows

$$\diagram \relax\prod\limits_X\, [\,\FX\,,\FX\,] \rto<1.1ex... ...ptstyle f:X \rightarrow Y} [\,\FX\,,FY\,]\rlap{~~.}\enddiagram$$

There are projection arrows

$$\map \pi_{\!X}^{\phj} \from \EsubF \to {[\,\FX\,,\FX\,]}$$

for each object $X\!\in\scrC$. These correspond, using the definition of internal hom, to arrows

$$\map \muX \from \EsubF \XO \FX \to {\FX}\;.$$

The univeral properties of end and internal hom show that there exists a bijection between the arrows $\map f \from A \to \EsubF$ in $\scrV$ and natural families of arrows $\map \thetaX \from A\XO \FX \to {\FX}$, given by

$$\thetaX \; = \; \muX \circ (f \xO 1_{\!X}^{\phj} )\;.$$

The natural families

$$\spreaddiagramcolumns{1pc}\diagram \relax\EsubF \XO\EsubF\X... ...m \relax\FX \rto^-{1_{\!\FX}^{\phj}\;} & \relax\FX \enddiagram$$

induce, under such bijections, the monoid structure on $\EsubF\,$:

$$\map \mu \from \EsubF\XO\EsubF \to \EsubF \quad, \quad \map \eta \from I \to \EsubF \;\;.$$

Example 17.1  

Take $\scrV=\Mod_R$ for some commutative ring R . Then we have that $\MonV=\Alg_R$. Now for any functor $\functor F \from \,\scrC \to \scrV$, the algebra $\EsubF$ has as elements the natural families $\theta=(\thetaX)_{X\in G}^{\phj}$ of R-linear morphisms $\map \theta_{\!X}^{\phj} \from \FX \to {\FX}$; addition and multiplication by scalars are done componentwise, while multiplication is componentwise composition. In particular, for any R-algebra A , if we have that

$$F \; = \; \map \UsubA \from \Mod_R(A) \to {\Mod_R} \quad,\quad\map \eta \from I \to \EsubF$$

then there is a natural isomorphism of algebras

$$\EsubF \;\isom\; A\;.$$

To see this, notice that each element m of an A-module M determines a unique $\map \hat m \from A \to M$ in $\Mod_R(A)$ with $\hat m(1)=m$; so for a natural $\map \theta \from \UsubA \to {\UsubA}$, we have

$$\spreaddiagramrows{1pc}\spreaddiagramcolumns{1pc}\diagram A... ...-{\hat m} \\ M \rto^-{\theta_{\!\!M}^{\phj}\;}& M \enddiagram$$

which implies $\theta_{\!M}^{\phj}(m)=\thetA(1)\,m$; so $\theta$ is determined by $\thetA(1)\in A$.

Suppose A is a monoid in $\scrV\,$. The two axioms which are required for an arrow $\map f \from A \to \EsubF$ to be in $\MonV$ translate to the two conditions on the corresponding family of arrows $\map \thetaX \from A\XO \FX \to {\FX}$ which say that each $\thetaX$ is an action of A on $\FX$. This is precisely what is needed to lift F to a functor $\functor T \from \,\scrC \to {\ModV(A)}$ such that $\UsubA T = F\,$; just put $TX=\bigl(\,\FX\,,\thetaX\,\bigr)\,$. This gives a natural bijection

$$\MonV(A\,,\EsubF)\;\cong\; \bigl(\CatV\bigr)\Bigl(\bigl(\,\scrC\,,F\,\bigr)\,, \bigl(\,\ModV(A)\,,\UsubA\,\bigr)\Bigr)$$

between hom sets, which means that $\bigl(\,\scrC\,,F\,\bigr)\xymapsto \EsubF\,$ is left adjoint to $\map \ModV \from \MonV^\op \to {\CatV}$. In fact, the above bijection becomes an isomorphism of categories, since it extends to 2-cells:

$$\spreaddiagramcolumns{2pc}\diagram A \rtwocell^f_{f'}{\;\xi... ...\ModV\rlap{$(A)$}\\ & \;\;\scrV\; \urto_-{\UsubA} \enddiagram$$

This is expressed by saying that $\bigl(\,\scrC\,,F\,\bigr) \xymapsto \EsubF$ is left 2-adjoint to $\ModV\,$.

Taking $A=\EsubF$ in the above bijection and looking at the image of the identity arrow, we obtain

$$\spreaddiagramcolumns{1pc}\spreaddiagramrows{.5pc}\diagram ... ...\dlto^-{\quad U_{\!\EsubF}^{}} \\ & \relax\scrV & \enddiagram$$

where $NX=\bigl(\,FX\,,\muX\,\bigr)\,$. We obtain a (partial) 2-functor

$$\map \Esub \from \CatV \to {(\MonV)^\op}$$

by taking the 2-cell $\twomap\alpha\from T\to\to T'\from \bigl(\,\scrC\,,F\,\bigr)\to {\bigl(\,\scrD\,,G\,\bigr)}$ in $\CatV\,$ into the 2-cell $\twomap E_{\!\alpha}^{\phj} \from E_T^{\phj} \to\to E_{T'}^{\phj} \from \EsubG \to {\EsubF}$ in $\MonV$ corresponding (under the 2-adjunction) to the 2-cell in $\CatV\,$:

$$\twomap N \alpha \from N T \to\to N T' \from \bigl(\,\scrC\,,F\,\bigr) \to {\bigl(\,\ModV(\EsubG)\,, U_{\EsubG}\,\bigr)}\;.$$

Remark 17.2   Formal Tannaka Duality criteria on $\functor F \from \,\scrC \to \scrV$ are that $\functor N \from \,\scrC \to {\ModV(\EsubF)}$ should be faithful and also that every ``appropriate'' $\EsubF$-module should be isomorphic to some NX .

We can equally well regard $\Esub$ as a 2-functor

$$\map \Esub \from (\CatV)^\op \to {\MonV}$$

whereupon (for general reasons as an adjoint to $\ModV$) it is a weak tensor functor. It preserves the unit in the sense that $E_{\!I}^{\phj}\cong I$, while we have a canonical arrow $\phi$ such that

$$\spreaddiagramcolumns{1.5pc}\spreaddiagramrows{1pc}\diagram ... ...,] \rto^{\_\,\xo\_\;} & [\,\FX \XO GY,\FX\XO GY\,] \enddiagram$$

where the bottom arrow corresponds to the composite

$$\spreaddiagramcolumns{1pc}\spreaddiagramrows{-1pc}\diagram ... ...{e\xo e}&\FX\XO GY\;.\qquad\qquad\qquad\qquad\quad \enddiagram$$

So $\Esub$ takes monoids in $(\CatV)^\op\,$ to monoids in $\MonV$, the latter being the commutative monoids in $\scrV\,$, but this is of no interest to us here.

Our real interest is in to what extent

$$\map \Esub \from (\CatV) \to {\MonV^\op}$$

takes monoids to monoids. This will be true of those monoids $\bigl(\,\scrC\,,F\,\bigr)$ in $\CatV$ for which $\map \phisub{F\,F}\from \EsubF\XO\EsubF \to {E_{\!F\xO F}^{\phj}}$ is invertible.

Is this a reasonable condition? At first glance, invertibility of

$$\map {\phisub{F\,G}} \from \int_{\!X}^{\phj}\,[\,\FX\,,\F... ...Y,GY\,] \to {\int_{\!X,Y}^{\phj}\,[\,\FX\XO GY,\FX\XO GY\,]}$$

looks unlikely. It would be implied by the two conditions:

(a)

each $\map A\XO\slot\, \from \scrV \to \scrV$ preserves ends; and

(b)

each $\raise.5ex\hbox{$\spreaddiagramcolumns{1pc}\diagram [A\,,B\,] \XO [\,C\,,D\,] \rto^{\_\,\xo\_\;}& [\,A\XO C\,,B\XO D\,] \enddiagram $}$ is invertible, for every $A=\FX$ and every C=GY .

However these look unlikely too, if we think in terms of the example.

We shall look at the conditions (a) and (b) more closely. If $\scrV$ is a closed tensor category then $A\XO\!\slot$ preserves colimits (since it has a right adjoint $[A\,,\slot\,]\,$; see Mac Lane ([Mac71, Chapter V §5]). But end is a limit, not a colimit. So (a) can be ensured by taking $\scrV\,$ to be the opposite of a complete closed tensor category. We need to be careful here since we still need the internal homs of the form $[\FX,\FX\,]$ in $\scrV\,$, not in $\scrV^\op$.

Condition (b) is true, for example for finite-dimensional vector spaces. What is needed is that A and C should have duals; then we have canonical isomorphisms

$$\begin{eqnarray*} &[\,A\,,B\,] \XO [\,C\,,D\,] &\isom\; \Astar\XO\,B\,\XO\,C^*\X... ...m\; (A \XO C)^*_{}\XO\,(B\XO D)\;\isom\;[\,A\XO C\,,B\XO D\,]\;. \end{eqnarray*}$$

Hence, conditions (a) and (b) are not unreasonable after all. They are satisfied when $\scrV$ is the opposite of a closed symmetric tensor category which is cocomplete enough for co-ends over $\scrC$ to exist, and when each $\FX$ and GY has a dual.

Suppose then that $\scrV^\op$ is a closed symmetric (strict) tensor category which is (small) cocomplete. Suppose $\,\scrC\,$ is a left autonomous small (strict) tensor category and $\functor F \from \,\scrC \to \scrV$ is a (strict) tensor functor. Then each $\FX$ has a dual $\FX^*_{}$. Since $\bigl(\,\scrC\,,F\,\bigr)$ is a monoid in $\CatV$, we obtain a monoid $\EsubF$ in $\smash{\MonV}^\op$; that is, a bi-monoid $\EsubF$ in $\scrV$. This gives a factorization

$$\spreaddiagramcolumns{1pc}\spreaddiagramrows{.5pc}\diagram ... ...F)$} \dlto^{\quad U_{\!\EsubF}} \\ & \relax\scrV \enddiagram$$

of our tensor functor F into tensor functors N and $U_{\!\EsubF}$.

In fact, $\EsubF$ is a Hopf monoid. To see this, define $\functor F^* \from \;\scrC^\op \to \scrV$ by $F^*X=\FX^*$. We obtain a monoid arrow

$$\functor F^* \from \,\scrC^\op \to \scrV\$$

in $\CatV$. This induces a monoid arrow $\nu$ with

$$\spreaddiagramrows{.5pc}\diagram E_{\!F^*_{}}^{\phj}\drto_{... ...\quad\;\;\pi_{\!X^*}^{}} \\ & \relax\FX \XO \FX^* \enddiagram$$

It is easy to see that $E_{F^*_{}}=\EsubF^\op$ as bi-monoids in $\scrV$ (that is, E_F^* is just $\EsubF$ with switched multiplication and switched comultiplication), and $\nu$ is an antipode for the bi-monoid $\EsubF$.

Now suppose $\,\scrC\,$ is braided. The braiding can be regarded as an invertible 2-cell:

$$\spreaddiagramcolumns{-.5pc}\spreaddiagramrows{.5pc}\diagram... ...,F\,\bigr) \xtwocell[-1,0]{+(-4,0)}\omit{<.5>c\;\;} \enddiagram$$

in $\CatV$. Applying $\Esub\,$, we obtain an invertible 2-cell in $\MonV$:

$$\spreaddiagramrows{.5pc}\diagram \relax\EsubF\XO\EsubF & \r... ...elta} \xtwocell[-1,0]{+(-4,0)}<\omit>{^<2>\gamma} \enddiagram$$

The braiding arrows for c on $\scrC$ carry over precisely to those for $\gamma$ on $\EsubF$. Moreover, $\map N \from \,\scrC \to {\ModV(\EsubF)}$ becomes a braided tensor functor.

Next suppose $\,\scrC\,$ is balanced. The twist on $\,\scrC\,$ can be regarded as being an invertible 2-cell:

$$\spreaddiagramcolumns{2pc}\spreaddiagramrows{-1.65pc}\diagra... ..._{1_\scrC}{\theta} & \relax\bigl(\,\scrC\,,F\,\bigr)\enddiagram$$

in $\CatV$, and, applying $\Esub\,$, we obtain a twist

$$\spreaddiagramcolumns{2pc}\spreaddiagramrows{-1.75pc}\diagram \relax\EsubF \rtwocell^1_1{\tau}& \relax\EsubF \enddiagram$$

for the braided bi-monoid $\EsubF$. So $\EsubF$ becomes a balanced Hopf monoid and $\map N \from \,\scrC \to {\ModV(\EsubF)}$ becomes a balanced tensor functor.

Finally, if $\,\scrC\,$ is a tortile tensor category, $\EsubF$ is a tortile bimonoid in $\scrV$.

To obtain Ulbrich's [Ulb89] setting, we take $\scrV=\Mod_R^\op$ for a commutative ring R . For each R-coalgebra C, we have

$$\Mod_\scrV(\scrC)^\op \; = \; \Comod_R(\scrC)\;.$$

We use the notation $\Comod_R(\scrC)\cauchy$ to denote the full subcategory consisting of C-comodules M for which the underlying R-module $U_{\!C}^{\phj} M$ is cauchy.

Consider a small category $\scrC$ and a functor $\functor F \from \;\scrC \to {\Mod_R}$ whose values $\FX$ are cauchy R-modules. The coend

$$E'_{\!F} \; =\; \int^X \!\FX \,\xR\,(\FX)^*$$

becomes an R-coalgebra and we have

$$\spreaddiagramcolumns{-.5pc}\spreaddiagramrows{.5pc}\diagram... ...R\rlap{$(E'_{\!F})$} \dlto^-{\quad U}\\ & \;\Mod_R \enddiagram$$

(since we can apply our previous theory to F regarded as going from $\scrC^\op$ to $\scrV=\Mod_R^\op\;$). Notice that N actually lands in $\Comod_R(E'_{\!F})\cauchy$.

If $\,\scrC\,$ is a tensor category and F is a tensor functor then E'_!F becomes an R-bialgebra and N becomes a tensor functor. If $\,\scrC\,$ is left autonomous then E'_!F becomes a Hopf algebra with invertible antipode. If $\,\scrC\,$ is a tortile tensor category then E'_!F becomes a cotortile R-bialgebra (quantum group!) and N becomes a balanced tensor functor.

An important case of Tannaka duality is the characterization of those $\functor F \from \,\scrC \to {\Mod_R\;}$ equivalent to $\map \;U_{\!H}^{\phj} \from \Comod_R(H)\cauchy \to {\Mod_R}$ for some Hopf algebra H. This can be investigated by looking at when the functor $\map N \from \,\scrC \to {\Comod_R(E'_{\!F})\cauchy}$ is an equivalence.

The question arises here as to whether $E'_{\!F}\isom C$ when the equality $F = \map U_{\mkern-2mu C}^{\phj} \from \Comod\!(C)\cauchy \to {\Mod_R}$ holds for a coalgebra C . We cannot use the technique of the example since, although C is a C-comodule, it is generally not cauchy as an R-module.

Proposition 17.3   If C is a coalgebra over a field R and U denotes the forgetful functor $\functor U \from \,\Comod\!(C)\cauchy \to {\Mod_R}$, then there is a coalgebra isomorphism

$$E'_U\;\isom\;C\;.$$