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Given an operad (in $Set$), there should be a Lawvere theory corresponding to it. I suppose that it’s just its category of operators.
I briefly tried to locate some standard references that talk about this explicitly, but no luck so far. For instance in May’s Uniqueness of infinite loop space machines it is curious that the phrase “preserves products (up to equivalence)” does not appear in def 1.2, def 1.5. Same for Lurie’s Commutative algebra def. 1.4.11.
It seems, notably, that Lurie’s discussion of $\infty$-operads, at least as long as algebras over them are considered just in $\infty Cat$ as around page 21, is really their description in terms of their associated $\infty$-algebraic theories. Unless I am missing something. Has anyone thought about this?
ah, stupid question. The category of operators is just semicartesian.
I just remembered Mike’s post about this… ;-)
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