Todd Trimble
Morphisms between tensor functors

MathOverflow User Auguste Hoang Duc asked after a proof of proposition 1.13 in Tannakian Categories by Deligne and Milne, to the effect that given two rigid tensor categories CC, DD, the category [C,D][C, D] of tensor functors and morphisms between them is a groupoid.

Thus, let F,G:CDF, G: C \to D be tensor functors and let u:FGu: F \to G be a morphism between tensor functors. For convenience, we assume associativity contraints are strict (i.e., identities). We will denote monoidal units by II (without specifying the monoidal category to which it belongs, which will be clear from context). The unit constraints are denoted λ a:Iaa\lambda_a: I \otimes a \to a and ρ a:aIa\rho_a: a \otimes I \to a (usually suppressing the subscript).

Recall that rigidity means that to each object XX there is a dual object X X^\vee together with

satisfying familiar triangular equations.

Recall that tensor functors GG come equipped with structural constraints (that are natural isomorphisms in the lower-case subscripts that appear):

subject to some coherence conditions. A morphism u:FGu: F \to G of tensor functors is a natural transformation that is moreover compatible with the structural constraints of FF and GG.

Given all this data, Deligne-Milne write down the inverse of uXu X (for an object XX of CC as

v(X):G(X)G(X ) tu(X )F(X ) F(X). v(X) : G(X) \simeq G(X^\vee)^\vee \xrightarrow{{}^t u(X^\vee)} F(X^\vee)^\vee \simeq F(X).

and the OP wishes to verify this is indeed inverse to u(X)u(X).

The proof that u(X)v(X)=1 GXu(X) \circ v(X) = 1_{G X} is displayed in the following large diagram. The proof that v(X)u(X)=1 FXv(X) \circ u(X) = 1_{F X} is similar and will be left to the reader. Starting from the occurrence of GXG X in the top row and following the string of arrows in the counterclockwise direction along the perimeter gives the composite u(X)v(X)u(X) \circ v(X). Each of the subdiagrams is readily verified to commute, using either naturality, functoriality, a coherence condition on structural constraints, a triangular equation for XX, or the compatibility of uu with structural constraints. (Note: subscripts XX, X X^\vee, etc. of components of the structural constraints α F\alpha_F, α G\alpha_G are suppressed for the sake of legibility. The symbol 11 is used to indicate identity arrows on unnamed objects; which objects these are can be inferred from context.)

IGX λ 1 GX i F1 i G1 G(λ 1) 1 FIGX uI1 GIGX α G G(IX) GX 1 GX F(η)1 G(η)1 G(η1) G(ρ) ρ uX F(XX )GX u(XX )1 G(XX )GX α G G(XX X) G(1ε) G(XI) GXI FX α F 11 α G 11 α G α G 1i G 1 uX1 ρ FXF(X )GX uXu(X )1 GXG(X )GX 1α G GXG(X X) 1G(ε) GXGI FXI 1u(X )1 uX11 uX1 uX1 1i G 1 FXG(X )GX 1α G FXG(X X) 1G(ε) FXGI\array{ & & I \otimes G X & \stackrel{\lambda^{-1}}{\leftarrow} & G X & & & & & & \\ & _\mathllap{i_F \otimes 1} \swarrow & _\mathllap{i_G \otimes 1} \downarrow & & \downarrow _\mathrlap{G(\lambda^{-1})} & \searrow _\mathrlap{1} & & & & & \\ F I \otimes G X & \stackrel{u I \otimes 1}{\to} & G I \otimes G X & \stackrel{\alpha_G}{\to} & G(I \otimes X) & & G X & \stackrel{1}{\leftarrow} & G X & & \\ _\mathllap{F(\eta) \otimes 1} \downarrow & & _\mathllap{G(\eta) \otimes 1} \downarrow & & \downarrow _\mathrlap{G(\eta \otimes 1)} & & _\mathllap{G(\rho)} \uparrow & & _\mathllap{\rho} \uparrow & \nwarrow _\mathrlap{u X} & \\ F(X \otimes X^\vee) \otimes G X & \stackrel{u(X \otimes X^\vee) \otimes 1}{\to} & G(X \otimes X^\vee) \otimes G X & \stackrel{\alpha_G}{\to} & G(X \otimes X^\vee \otimes X) & \stackrel{G(1 \otimes \varepsilon)}{\to} & G(X \otimes I) & & G X \otimes I & & F X \\ _\mathllap{\alpha_F^{-1} \otimes 1} \downarrow & & _\mathllap{\alpha_G^{-1} \otimes 1} \downarrow & & \uparrow _\mathrlap{\alpha_G} & & _\mathllap{\alpha_G} \uparrow & _\mathllap{1 \otimes i_G^{-1}} \nearrow & \uparrow _\mathrlap{u X \otimes 1} & \nearrow _\mathrlap{\rho} & \\ F X \otimes F(X^\vee) \otimes G X & \stackrel{u X \otimes u(X^\vee) \otimes 1}{\to} & G X \otimes G(X^\vee) \otimes G X & \stackrel{1 \otimes \alpha_G}{\to} & G X \otimes G(X^\vee \otimes X) & \stackrel{1 \otimes G(\varepsilon)}{\to} & G X \otimes G I & & F X \otimes I & & \\ & _\mathllap{1 \otimes u(X^\vee) \otimes 1} \searrow & _\mathllap{u X \otimes 1 \otimes 1} \uparrow & & _\mathllap{u X \otimes 1} \uparrow & & _\mathllap{u X \otimes 1} \uparrow & \nearrow _\mathrlap{1 \otimes i_G^{-1}} & & & \\ & & F X \otimes G(X^\vee) \otimes G X & \underset{1 \otimes \alpha_G}{\to} & F X \otimes G(X^\vee \otimes X) & \underset{1 \otimes G(\varepsilon)}{\to} & F X \otimes G I }

Short form

Here is the condensed, “Reader’s Digest” version of the diagram above (suppressing instances of the monoidal unit):

FXF(X )GX 1u(X )1 FXG(X )G(X) 1ε GX FX η FX1 uXu(X )1 uX11 uX GX η GX1 GXG(X )GX 1ε GX GX\array{ F X \otimes F(X^\vee) \otimes G X & \stackrel{1 \otimes u(X^\vee) \otimes 1}{\to} & F X \otimes G(X^\vee) \otimes G(X) & \stackrel{1 \otimes \varepsilon_{G X}}{\to} & F X \\ _\mathllap{\eta_{F X} \otimes 1} \uparrow & \searrow _\mathrlap{u X \otimes u(X^\vee) \otimes 1} & \downarrow _\mathrlap{u X \otimes 1 \otimes 1} & & \downarrow \mathrlap{u X} \\ G X & \underset{\eta_{G X} \otimes 1}{\to} & G X \otimes G(X^\vee) \otimes G X & \underset{1 \otimes \varepsilon_{G X}}{\to} & G X }

Here we are using the fact that a tensor functor preserves takes dual objects to dual objects, so that the bottom horizontal composite is an identity. Most of the details that are suppressed go into showing the commutativity of the triangle on the left, using compatibility of uu with the structural constraints of tensor functors.

There is a similar diagram for the other equation v(X)u(X)=1v(X) \circ u(X) = 1:

GX η FX1 FX×F(X )GX 1u(X )1 FXG(X )GX uX 11uX uXu(X )1 1ε GX FX η FX1 FXF(X )FX 1ε FX FX\array{ G X & \stackrel{\eta_{F X} \otimes 1}{\to} & F X \times F(X^\vee) \otimes G X & \stackrel{1 \otimes u(X^\vee) \otimes 1}{\to} & F X \otimes G(X^\vee) \otimes G X \\ _\mathllap{u X} \uparrow & & _\mathllap{1 \otimes 1 \otimes u X} \uparrow & _\mathllap{u X \otimes u(X^\vee) \otimes 1} \nearrow & \downarrow _\mathrlap{1 \otimes \varepsilon_{G X}} \\ F X & \underset{\eta_{F X} \otimes 1}{\to} & F X \otimes F(X^\vee) \otimes F X & \underset{1 \otimes \varepsilon_{F X}}{\to} & F X }

Generalization

Tensor categories may be regarded as 2-categories (i.e., bicategories) with one object, and there is an often-used 2-categorical generalization of the result above.

The analogy works as follows:

Here is the result:

Proposition

Let B,CB, C be 2-categories, let F,G:BCF, G: B \to C be 2-functors, and let θ:FG\theta: F \to G be an oplax natural transformation. If f:bcf: b \to c has a right adjoint f *f^\ast in BB, then the 2-cell θf\theta \cdot f is invertible.

Proof

Given an adjunction ff *f \dashv f^\ast and 2-functors FF, GG, we have FfFf *F f \dashv F f^\ast and GfGf *Gf \dashv G f^\ast. It is easily verified that the diagram

Fb Ff Fc θc Gc 1 Fb Ff * θf * Gf * 1 Gc Fb θb Gb Gf Gc\array{ F b & \overset{F f}{\to} & F c & \overset{\theta c}{\to} & G c & & \\ & _\mathllap{1_{F b}} \searrow^\mathrlap{\neArrow} & \downarrow _\mathrlap{F f^\ast} & \overset{\theta \cdot f^\ast}{\Rightarrow} & \downarrow _\mathrlap{G f^\ast \; \; \neArrow} & \searrow _\mathrlap{1_{G c}} & \\ & & F b & \underset{\theta b}{\to} & G b & \underset{G f}{\to} & G c }

pastes to give the inverse (θf) 1(\theta \cdot f)^{-1}, where the left triangle 2-cell is the unit of the adjunctions FfFf *F f \dashv F f^*, and the right triangle 2-cell is the counit of GfGf *G f \dashv G f^*.

Reference

Revised on December 13, 2012 20:09:46 by Todd Trimble