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Naively, $m$-algebras over a field $\mathbb{k}$ are easily defined recursively: the category of 0-algebras is the symmetric monoidal category of $\mathbb{k}$-vector spaces, and for $m>0$ an $m$-algebra is an algebra object in the symmetric monoidal category of $(m-1)$-algebras.

This definition is based on the fact that algebra objects in a symmetric monoidal category should form themselves a symmetric monoidal category. Actually this is too a naive statement, and trying to make it rigorous one drops into higher categories: a semi-rigorous statement would be: "algebra objects in a symmetric monoidal $(\infty,n)$-category form a symmetric monoidal $(\infty,n+1)$-category" (this kind of statement can be found in Section 7 of Freed-Hopkins-Lurie-Teleman's Topological Quantum Field Theories from Compact Lie Groups: "Recall first that vector spaces and linear maps form a symmetric tensor category, and that algebra objects in a symmetric tensor category form in turn a symmetric tensor category (of one level higher, but who’s counting?)."

My problems here is:

Recall from where? I'm not aware of a rigorous reference for this statement, and actually even not for the notion of symmetric monoidal $(\infty,n)$-category. For the latter, I'm familiar with Lurie's On the Classification of Topological Field Theories, but there one first finds the question "(c) What is a symmetric monoidal structure on an $(\infty, n)$-category, and what does it mean for a functor to be symmetric monoidal?" but then one finds "In the interest of space, we will gloss over (b) and (c) (for an extensive discussion of (c) in the case n = 1 we refer the reader to [16])."

I'm ok with the intuitive and sketchy definitions above, but I'd like to know whether a rigorous treatment is available somewhere (and if not available yet, if there is interest in having such a rigorous treatment: I'm pondering this as a possible PhD thesis subject)

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If it's anywhere, it's in something by Lurie. I would look if either HTT or HA , probably the latter. But Jacob does respond here on MO, so I'll let him say for certain whether he's written down the details yet. – Theo Johnson-Freyd Jun 27 '11 at 2:27
It definitely not in HTT or Higher Algebra. But there was a response a while given by Jacob Lurie about the definition of symmetric monoidal $n$-category. – Thomas Nikolaus Mar 13 '12 at 20:46

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