Let $P$ be a "nice" $k$-linear abelian tensor category (e.g. A tannakian category or a fusion category over a field $k$) and $F: M\to P $ an additive $k$-linear exact and faithful functor. I want to know when $F$ is equivalent to the forgetful functor from the category of co-modules over a co-algebra in $P$ to $P$. What are the known results in this direction?

A necessary condition for the existence of this equivalence is the existence of a "box product": $$\boxtimes : P\times M \to M,$$ which admits natural isomorphisms: $$F(A\boxtimes X) \simeq A \otimes F(X), \\ 1\boxtimes X\simeq X,\\ (A\otimes B)\boxtimes X\simeq A\boxtimes (B\boxtimes X).$$ Satisfying some expected coherence diagrams. Is this condition sufficient? What about the nice tensor categories mentioned above?

(For a co-module $X$ over a co-algebra $A$, with the structure map $\rho: X \to X\otimes A$,the box product $Y\boxtimes X$ is defined to be $Y\otimes X$ with co-action $$Y\otimes \rho:Y\otimes X\to Y\otimes X\otimes A.$$)

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    $\begingroup$ Have you tried applying the (co)monadicity theorem? $\endgroup$ – Qiaochu Yuan Dec 2 '15 at 4:05
  • $\begingroup$ @QiaochuYuan I know the special case $P=Vect_k $ can be proved using comonadicity. But in general case I can't find an adjoint for $F $ and can't write $F $ as a tensor product with a certain object. $\endgroup$ – Mostafa Dec 2 '15 at 5:32

If for example $P$ is a fusion category, then such a box product is what is usually referred to as a module category structure on $M$. In case $M$ is nice enough- for example semisimple with finitely many simple objects, then a result of Ostrik says that $M$ is necessarily the representation category of some algebra $A$ in $P$ (the fact that it is an algebra and not a coalgebra is not very important here). About the functor $F$: if you assume that you have such natural isomorphisms with Coherence conditions, then $F$ is necessarily given by taking tensor product over $A$ with a left $A$-module $M$. The question is whether or not this module is isomorphic with $A$ itself. This answer can be answered by applying the inner hom construction and checking it by hand. See also http://arxiv.org/pdf/math/0111139.pdf

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  • $\begingroup$ Thank you very much for the answer and the link. But in the mentioned result of Ostrik, the semisimplicity of M is very restrictive and dose not satisfied in my interested examples. There is a result of Gabber which says that every finitely generated locally finite $k$-linear abelian category is equivalent to a category of R_modules (R finite k-algebra). This result can be interpreted as a result on module categories on the category of vector spaces. Does there exist a similar result in the general case? $\endgroup$ – Mostafa Dec 16 '15 at 19:43

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