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A good question! I’m a little confused though — if by Δmaclane you mean the augmented simplex category Δa then it’s not a Lawvere finite-product theory either, but a monoidal theory: the “addition” operation on Δa is not the cartesian product.
I believe the answer, though, is that Γ is the semicartesian symmetric monoidal theory of commutative monoids. See this blog post. This would be great to add to the Γ-spaces page.
I’d start from the following fact. Write F for the category of finite sets (or a skeleton thereof), viewed as a symmetric monoidal category via coproduct. Then for any cartesian monoidal category E, a functor Γop→E is the same thing as a colax symmetric monoidal functor F→E.
This equivalence can be refined. The functors Γop→E for which the Segal maps are all isomorphisms correspond to the strong symmetric monoidal functors F→E. That is, they’re the commutative monoids in E.
The same thing works in the non-symmetric world, changing Γ to Δ (the category of nonempty totally ordered sets) and F to the category D of possibly empty totally ordered sets (which is monoidal). So, for example, a simplicial set is the same thing as a colax monoidal functor D→Set.
So we’re very much in the world of (symmetric or not) monoidal theories. It’s an interesting question as to how to relate the symmetric and non-symmetric worlds, and also how to relate them to the finite-product/Lawvere world.
The blog post that I linked to has some thoughts about how to move between worlds.
Did you read my blog post? When you have a morphism of doctrines D1→D2, the left adjoint from D1-theories to D2-theories takes the D1-theory of an X to the D2-theory of an X. For instance, the morphism of doctrines from semicartesian symmetric monoidal categories to cartesian monoidal categories takes the semicartesian theory of a commutative monoid, namely Γ, to the cartesian (Lawvere) theory of a commutative monoid. Similarly, we can start from the symmetric monoidal theory of a commutative monoid and extend it up to the semicartesian situation to obtain Γ.
By the properties of adjunctions, a (strict) model of a semicartesian theory in a cartesian category is the same as a (strict) model of the cartesian theory that it freely generates. This ought to also be true at the homotopical level as long as the adjunctions are homotopically meaningful. I don’t know to what extent that is known to be true. One might hope for them to be Quillen adjunctions, in which case it would be true as long as the theory you are looking at is sufficiently cofibrant. Probably Γ is sufficiently cofibrant.
As for why we use a semicartesian theory and not a Lawvere theory, that’s a good question! The fact that people usually do it that way confused me for a long time, until I figured out what a semicartesian theory was and that Γ is one. I believe the answer is partly tradition, but there are also reasons for that tradition. Γ-spaces worked well for Segal’s infinite loop space machine, and the more general semicartesian theories associated to operads (= monoidal theories) worked well for May and Thomason’s comparison of infinite loop space machines. Operads, in turn — or more precisely, semicocartesian operads — worked well for May’s infinite loop space machine. I don’t know to what extent any of that machinery would work with Lawvere theories; my guess would be some, but not all, at least not as easily. I do know, for instance, that if you use ordinary operads rather than semicocartesian ones, then May’s infinite loop space machine fails. So even if at a fully (∞,1)-categorical level there should be no difference, when working with concrete presentations one choice can be easier to work with than another.
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