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Continued from this thread where Charles said
In the ur-$(\infty,1)$-category of topological spaces, we can consider actual commutative monoids, rather than $E_\infty$-ones. We obtain the homotopy theory of topological commutative monoids, which of course are of great significance in algebraic topology (since ordinary homology is basically the free abelian group on a space).
I remember pondering this example at some point, and concluding that strictly commutative topological monoids should be thought of as analogous to strictly-symmetric monoidal categories. I believe it is possible to describe a notion of “strictly-symmetric monoidal category” that is nevertheless “invariant” in that it transports across equivalences of categories – consider pseudo $T$-algebras where $T$ is the 2-monad for strictly-symmetric monoidal categories – but I don’t think it is an operadic structure. Is there an $(\infty,1)$-categorical (hence “invariant”) way to describe a notion corresponding to “strictly commutative topological monoid”?
You can encode “strictly commutative monoid” using algebraic theories. In particular, if you have an $(\infty,1)$-category $C$ with products, you obtain a notion of “commutative monoid” in $C$, as a product preserving functor $T\to C$, where $T$ is the theory of commutative monoids. The notion transports along equivalences of $(\infty,1)$-categories of course, since such must preserve products.
Oh, thanks for saying this, now I know what you mean (I had had the same worries as Mike voiced).
This is an important and maybe slightly subtle point, which comes up crucially in derived algebraic geometry. You all know this, but just to recall since it came up: first one may think that the homotopy theory of simplicial algebras is a good model for derived affine schemes. But then one notices that these are the $\infty$-algebras over just the 1-theory of commutative rings. Better might be the $\infty$-algebras over the $\infty$-theory of commutative rings, and these are the $E_\infty$-rings. In fact $E_\infty$-rings are already the $\infty$-algebras over the $(2,1)$-theory of commutative rings. But nevertheless, the $\infty$-alegbras over the 1-theory exist.
One can see this subtlety of passing from the 1-theory of commutative rings to the $(2,1)$-theory and hence all the way to the $(\infty,1)$-theory in much of the derived algebraic literature. Many people actually just work over the 1-theory, hence with simplicial algebras. Except JL of course…
I guess I’m confused; I don’t see how theories encode strict commutativity any more than operads do. In the theory of commutative monoids with objects labeled by the natural numbers, there is a single “addition” operation $2 \to 1$, which yields itself when composed with the “switch” isomorphism $2\to 2$. But a product-preserving $\infty$-functor out of this theory will take this commutative triangle to a pseudo-commutative triangle, so that in a model the addition is only commutative up to homotopy. What am I missing?
The theory knows not only about the “addition” operation, but also the “doubling” operation. In an $E_\infty$-space (lets think of it as a symmetric monoidal $\infty$-groupoid), the double $x\otimes x$ of a point $x$ has an automorphism given by the switch, which may be non-trivial. In a commutative monoid, the switch automorphism of $x\otimes x$ is made equivalent to the identity.
Remember that the commuative monoid theory $T$ is a category with objects=natural numbers, and morphisms $T(m,n)=\Hom(\mathbb{N}^n,\mathbb{N}^m)$; the morphism space is discrete. In the $E_\infty$-theory $T'$, the morphism spaces $T'(m,n)$ have the same $\pi_0$, but are not discrete (they are products of $B\Sigma_k$s.)
Ah, right. I should have realized that. Where is such an equivalence written down between commutative topological monoids and $\infty$-models of the theory of commutative monoids?
Do we have any conceptual understanding of why the (2,1)-theory is already the $(\infty,1)$-theory in these cases?
Where is such an equivalence written down between commutative topological monoids and ∞-models of the theory of commutative monoids?
For simplicial commutative algebras the statement goes back to Quillen’s model structure on simplicial algebras. A discussion in the context of genuine $(\infty,1)$-category theory is in section 5.2.2 of HTT. The nLab has this at
That $E_\infty$-rings are $\infty$-algebras over a $(2,1)$-algebraic theory is reviewed at (2,1)-algebraic theory of E-infinity algebras.
I suppose the conceptual reason that $E_\infty$-algebras are already a $(2,1)$-theory is that the Barratt-Eccles operad is 1-truncated. Of course one might ask “why” the $E_\infty$-operad has a cofibrant model by a 1-truncated operad. This I don’t know, I suppose.
Surely there are then analogues of the Barratt-Eccles operad where one takes not the groups $S_k$ but the automorphism $n$-groups of finite $(n-1)$-groupoids?
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