Holmstrom Tensor category

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We consider (as motivation?) the category $Rep_F G$ of finitedimensional representations over $F$ of an affine $F$-group scheme $G$. The algebra of functions on $G$ carries the structure of a “cogebra”, actually a commutative Hopf algebra $A_G$. The abelian $F$-linear category $Rep_F G$is equivalent to the category of comodules of finite $F$-dimension, relative to the cogebra structure of $A_G$. The algebra structure of $A_G$ is hidden in the tensor structure of $Rep_F G$.

Def: A tensor category over a commutative ring $F$ is an $F$-linear cat with a tensor structure, i.e. a bilinear bifunctor, a unit object, and functorial isomorphisms expressing associativity, commutativity and unit laws; these are required to satisfy three coherence diagrams.

A rigid tensor cat over $F$ is a tensor cat $T$ over $F$ equipped with a autoduality $\vee: T \to T^{op}$ such that for every object $M$, tensoring on the right with the dual of $M$ is left adjoint to tensoring on the right with $M$, and same with left and right interchanged.

This structure gives us adjunciton morphisms $\epsilon: M \otimes M^{\vee} \to \mathbf{1}$ and $\eta: \mathbf{1} \to M^{\vee} \otimes M$, called evaluation and coevaluation, satisfying certain properties. In a rigid tensor category, every endomorphism has a trace, which is element $\epsilon \circ c \circ \eta \in End(\mathbf{1})$. We define the rank (or dimension) of an object $M$ to be the trace of $id_M$.

Example: The cat of finitedimensional linear reps of a group, over some field? The unit object is the trivial rep, and rank is the obvious thing.

Example: The cat of $\mathbb{Z}$-graded finite-dimensional $K$-vector spaces. Alternatively, $\mathbb{Z}/2$-graded. Tensor product turns this into a rigid tensor cat, in which the commutativity constraint is given be the Koszul sign rule. The rank of an object is its super-dimension $dim(V^+) - dim V^{-}$, where $V^+$ is the direct sum of the even parts etc.

Example: Combining the last two examples, one obtains the rigid tensor category of super-representations of $G$. In practice, one considers often a certain subcat of this.

Example: The cat of vector bundles on a fixed $F$-scheme. It has $End(\mathbf{1} = 1$ if proper and geometrically connected. The rank is what you think it is.

Remark: In a pseudo-abelian rigid tensor category over a field $F$ of char 0, one can define the symmetric and exterior powers of any object $M$.

Def: Tensor functor between tensor categories. (What you think it is).

If the tensor cats are rigids, any tensor functor between them is automatically compatible with taking duals, up to a natural isomorphism. Furthermore, because of duality, every morphism of tensor functors is an isomorphism.

Ezample: The free rigid tensor category on an object $M$.

nLab page on Tensor category