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A monoidal category is a well-defined categorical object abstracting products to the categorical setting. The term tensor category is also used, and seems to mean a monoidal category with more structure, usually the structure of an abelian cateogry, but I can't find a precise definition. So I ask question: What is a tensor category?

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    $\begingroup$ See for instance ams.org/distribution/mmj/vol2-2-2002/deligne.pdf for the abelian category usage, or also Definition 3.29 in ncatlab.org/nlab/show/Deligne%27s+theorem+on+tensor+categories $\endgroup$ – Todd Trimble Jul 28 '18 at 18:41
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    $\begingroup$ Some authors will restrict the meaning of "tensor category" even further. For example, see Definition 2.2.4 here. $\endgroup$ – Tim Campion Jul 28 '18 at 18:53
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    $\begingroup$ @ArunDebray No less than Andre Joyal and Ross Street have used it with that meaning. So it's standard in some circles. $\endgroup$ – Todd Trimble Jul 28 '18 at 23:38
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    $\begingroup$ @NadiaSUSY Yes, the definition I'm referring to is essentially (up to some of the caveats discussed by Noah Snyder below -- incidentally one of the coauthors on the paper I linked to) the definition discussed by David White below, and in this definition a tensor category turns out to be the category of finite-dimensional modules over a finite-dimensional algebra over a field (equipped with a monoidal structure). I believe this is discussed in the paper I linked to or in the book that David White links to. $\endgroup$ – Tim Campion Jul 29 '18 at 4:53
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    $\begingroup$ Nitpick: Tim’s comment is for finite categories while David’s answer uses locally finite categories. Also it has nothing at all to do with the tensor structure. $\endgroup$ – Noah Snyder Jul 29 '18 at 11:31
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There seem to be many different definitions in the literature, based on individual papers. But, I think that might change, now that the textbook Tensor Categories, by Etingof, Gelaki, Nikshych, and Ostrik, has appeared. They define a tensor category as follows:

Let $k$ be an algebraically closed field, and $C$ a locally finite $k$-linear abelian rigid monoidal category. If the bifunctor $\otimes: C\times C\to C$ is bilinear on morphisms, then $C$ is called a multitensor category. Assume that $C$ is indecomposable (i.e. not equivalent to a direct sum of nonzero multitensor categories). If $End_C(1) \cong k$ then $C$ is called a tensor category.

Of course, I've also seen tensor category used to mean monoidal category, often in papers to do with braidings. But, generally, tensor means more than monoidal. This is also true in homotopy theory: a tensor model category has to satisfy more than a monoidal model category (it needs the functors $X\otimes -$ and $-\otimes X$ to preserve weak equivalences, for cofibrant X; see this paper of mine with Yau).

Anyway, I agree with Noah that you should try to figure it out from context, and asking questions like this is a good way to make sure people are being careful with the terminology, so that we don't end up with even more definitions! For myself, I'll only use "tensor category" for what Etingof, Gelaki, Nikshych, and Ostrik mean.

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There is no single accepted definition of “tensor category” that matches all uses. Almost always it means abelian (or a similar cocomplete condition) and k-linear. Usually it also means rigid. Often it also means the unit object is simple. Occasionally it also means symmetric. You just have to look at the definition used in each particular paper.

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