# tensor products NOT iterated

3-fold tensor products are usually presented in terms of the natural isomorphism of iterated tensor porducts. Where is there a treatment of 3-fold tensor products without reference to 2-fold?

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One keyword to look up here is "unbiased monoidal category." –  Qiaochu Yuan Oct 24 '12 at 19:56
And the other one is "multilinear algebra". Vote to close as too localized. –  Martin Brandenburg Oct 24 '12 at 20:15
I think it's a good reference-request question. Most linear algebra texts completely skip this definition (and I'm not talking of those who skip tensor products altogether), but papers use it (think of $\bigotimes\limits_{s\in S}V_s$ where $S$ is an infinite, or finite but not canonically ordered, set). But I think the answer should be: "Yes, you should write it up yourself and put the notes online; people will thank you for that." –  darij grinberg Oct 24 '12 at 20:47
Assuming Professor Stasheff is referring to monoidal products, and not just those pertaining to linear algebra, I wouldn't say this reduces to multilinear algebra. I think Qiaochu's answer is more appropriate; there is a circle of ideas here involving unbiased monoidal categories and multicategories, explored in Tom Leinster's book Higher Operads, Higher Categories, which would be a good reference. (By the way, you do realize, Martin, that the work of James D. Stasheff had a lot to do with the very invention of monoidal categories? Show a little respect!) –  Todd Trimble Oct 24 '12 at 23:42
@Todd: Sorry for not paying attention to the history behind the poster. I only looked at the question, which is not really precise and elaborate for MO-standards. If the poster was "NoIdeaC3PO", then this question would have already been closed. By the way, the assumption that the question refers to monoidal categories cannot be read off from the question. @Darij: Probably you mean Linear Algebra texts for students in the first semesters? Every complete treatment of Linear Algebra also contains Multilinear Algebra, and there arbitrary tensor products are studied. –  Martin Brandenburg Oct 25 '12 at 7:43