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• First, the main difference remains: given a Lawvere theory you can only takes its models in a cartesian $\infty$-category, while you can look at the models of an operads in any monoidal $\infty$-category. So for example if you want to say something like "commutative ring spectra are commutative monoids in the category of spectra", that, as you are using the smash product of spectra which is not the cartesian product, then you really need the operads for commutative monoids and not the corresponding Lawvere theory (or to put it another way, you need to know that the Lawvere theory for $E_\infty$-algebras is "operadic").

• The discussion about modules for an $O$-algebras still applies completely unchanged to the $\infty$-world.

• Now, the way $\infty$-operads identifies with special monads/Lawvere theory become much cleaner in the $\infty$ setting: The $\infty$-category of $\infty$-operads identifies exactly with the $\infty$-category of "finitary polynomial monads" (and cartesian morphisms between them). The details of this have been worked out by Gepner, Haugseng and Kock in $\infty$-Operads as Analytic Monads.

• First, the main difference remains: given a Lawvere theory you can only takes its models in a cartesian $\infty$-category, while you can look at the models of an operads in any monoidal $\infty$-category. So for example if you want to say something like "commutative ring spectra are commutative monoids in the category of spectra", that, as you are using the smash product of spectra which is not the cartesian product, then you really need the operads for commutative monoids and not the corresponding Lawvere theory (or to put it another way, you need to know that the Lawvere theory for $E_\infty$-algebras is "operadic").

• The discussion about modules for an $O$-algebras still applies completely unchanged to the $\infty$-world.

• Now, the way $\infty$-operads identifies with special monads/Lawvere theory become much cleaner in the $\infty$ setting: The $\infty$-category of $\infty$-operads identifies exactly with the $\infty$-category of "finitary polynomial monads" (and cartesian morphisms between them, and where finitary is interpreted to mean that their arities are finite sets, not finite spaces). The details of this have been worked out by Gepner, Haugseng and Kock in $\infty$-Operads as Analytic Monads.

Edit: What about the "$\infty$-world" ?

So, that second point is really about how to go from $1$-operads to $\infty$-operads and from $1$-Lawvere theory to $\infty$-lawvere theories. It can be rephrased as the fact that the "square" that we want to draw whose corner are "$1$/$\infty$-operads/Lawvere theory" isn't a commutative square. So if one decide to focus on the $\infty$-world that problem does indeed disappear - or rather become the fact that the functor from $\infty$-operads to $\infty$-Lawvere theories doesn't preserves 1-truncated objects.

For example, the 1-Lawvere theory for commutative monoids comes from the 1-operads for commutative monoids (so the terminal symmetric operads), but when you see it as an $\infty$-Lawvere theory it is no longer the image of an operad: the operad for commutative monoids seen as an $\infty$-operads is sent to the $\infty$-Lawvere theory for $E_\infty$-algebras.

This does make the connection between $\infty$-operads and $\infty$-Lawvere theories "feel a bit different" than its 1-categorical counterparts, so I felt like it would be interesting to add a few comment about it:

• First, the main difference remains: given a Lawvere theory you can only takes its models in a cartesian $\infty$-category, while you can look at the models of an operads in any monoidal $\infty$-category. So for example if you want to say something like "commutative ring spectra are commutative monoids in the category of spectra", that, as you are using the smash product of spectra which is not the cartesian product, then you really need the operads for commutative monoids and not the corresponding Lawvere theory (or to put it another way, you need to know that the Lawvere theory for $E_\infty$-algebras is "operadic").

• The discussion about modules for an $O$-algebras still applies completely unchanged to the $\infty$-world.

• Now, the way $\infty$-operads identifies with special monads/Lawvere theory become much cleaner in the $\infty$ setting: The $\infty$-category of $\infty$-operads identifies exactly with the $\infty$-category of "finitary polynomial monads" (and cartesian morphisms between them). The details of this have been worked out by Gepner, Haugseng and Kock in $\infty$-Operads as Analytic Monads.

Homotopy models for Lawvere theories: If $C$ is a Lawvere theory (Which I see as a category with finite products) then I can define a notion of "homotopy model" or "weak model" of "C", by looking at (pseudo)-functor from $C$ to the $\infty$-category of spaces that preserves finite products (I mean send product in $C$ to homotopy products).

Homotopy models for Lawvere theories: If $C$ is a Lawvere theory (Which I see as a category with finite products) then I can define a notion of "homotopy model" or "weak model" of "C", by looking at (pseudo)-functor from $C$ to the $\infty$-category of spaces that preserves finite products (I mean send product in $C$ to homotopy products). Because Pseudo-functor to space spaces can always be strictified, this is the same as looking at actual functor $C \to Spaces$ (Where space is either topological space or simplicial sets) which sends products in $C$ to "homotopy products".