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Γ-spaces were introduced by Segal in 1969 as models for what can be now described as commutative ∞-monoids and ∞-groups in cartesian symmetric monoidal ∞-categories, e.g., E_∞-spaces and connective spectra. The word “cartesian” means that the tensor product coincides with the categorical product, i.e., A⊗B := A×B.

In Lurie's Higher Algebra one finds Definition 2.1.3.1, which, combined with Definition 2.0.0.7 and Example 2.1.2.18, gives a model for commutative ∞-monoids in any symmetric monoidal quasicategory. In Theorem 4.1.4.4 Lurie proves that in the case of left proper tractable symmetric monoidal model categories that satisfy the monoid axiom his notion of a commutative monoid is equivalent to the traditional one.

Using Example 4.1.3.6 or Proposition 4.1.3.10, which construct a symmetric monoidal quasicategory from a symmetric monoidal (simplicial) model category, one can try to unfold Lurie's definition and see what it means for the case of a symmetric monoidal quasicategory coming from a symmetric monoidal model category C.

Very roughly speaking, we obtain the data of a functor X from the category of finite (unpointed) sets to C and for any finite family p of finite sets we have a weak equivalence (also known as a Segal map) X_p: X_{∐p} → ⨂_i X_{p_i}, satisfying the obvious associativity conditions.

The above description is simplified quite a bit compared to what actually results from expanding Lurie's definition, but should suffice as the first approximation.

Needless to say, this description closely matches that of a Γ-object in the case of a cartesian monoidal model category. Recall that a Γ-object is, roughly speaking, a functor Y from finite pointed sets to some model category C such that for any finite family p of finite pointed sets the canonical Segal map Y_p: Y_{∐p} → ∏_i Y_{p_i} induced by the canonical maps ∐p → p_i (the ith component is the identity map and all other components are trivial) is a weak equivalence.

It is easy to see how one can construct an object X satisfying the conditions above from a Γ-object Y. The functor X is constructed from the functor Y by adding disjoint basepoints to all finite (unpointed) sets. Likewise, the Segal maps are obtained from the corresponding Segal maps of Y by adding basepoints to the elements of p.

The main difference between these two descriptions is that in the cartesian case all Segal maps can be canonically recovered from the maps ∐p → p_i of pointed finite sets, whereas in the general case they must be supplied as an additional data. (This also explains the choice of unpointed (as opposed to pointed) finite sets in the above description.)

Two earlier papers by Leinster (http://arXiv.org/abs/math/9912084v2, http://arXiv.org/abs/math/0002180v1) also discuss this definition, in the context of categories with weak equivalences that are closed under monoidal products, though unlike Lurie's book they do not construct an equivalence to the usual definition, and the condition on weak equivalences excludes virtually any known example of a monoidal model category (apart from those in which all objects are cofibrant).

Although the above definition of a “monoidal Γ-object” seems to be rather natural, I was unable to locate any references (apart from Lurie's book and Leinster's papers) that mention it or any similar construction in the framework of monoidal model categories.

I am specifically interested in statements that show an equivalence between such a model and the traditional definition. (Lurie's Theorem 4.1.4.4 proves such a statement for a significantly weaker model, with many additional homotopy coherences.)

Another question is whether one can construct a model structure on monoidal Γ-objects so that its bifibrant objects have cofibrant components and all Segal maps are weak equivalences.

Are there any references that discuss models for commutative ∞-monoids similar to Segal's Γ-spaces in the framework of (noncartesian) monoidal model categories?

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The following paper by Tom Leinster

http://arxiv.org/abs/math/0002180

defines for any symmetric monoidal model category $M$ and any (symmetric) operad $P$ (in $Set$) the notion of an $\infty$-algebra (which he calls homotopy algebra) over $P$ in $M$, in a way similar to Segal's $\Gamma$-spaces description of $\infty$-commutative monoids in cartesian monoidal model categories.

The main point is the following:

To any operad $P$ one can associate its prop $Prop(P)$. This is a symmetric monoidal category with objects $\{0,1,...\}$, such that the monoidal product is given on objects by $n\otimes m = n+m$. $Prop(P)$ has the property that for any symmetric monoidal category $M$, the category of algebras over $P$ in $M$ is equivalent to the category of symmetric monoidal functors from $Prop(P)$ to $M$. If $M$ is a symmetric monoidal model category, Leinster defines a homotopy algebra over $P$ in $M$ to be a colax symmetric monoidal functor $A:Prop(P)\to M$ such that the structure maps $$A(I)\to I,\:\:A(n+m)\to A(n)\otimes A(m)$$ are always weak equivalences (see Definition 2.2.1). There is also an obvious generalization to the enriched case.

The case of the commutative operad $Com$ is treated in Section 3.2. The prop $Prop(Com)$ is equivalent the category of (unpointed) finite sets and functions, with the monoidal product given by disjoint union. In Proposition 3.1.1 it is shown that if $M$ is a cartesian monoidal category, then there is an isomorphism of categories between the category of colax symmetric monoidal functors from $Prop(Com)$ to $M$ and the category of all functors from $\Gamma^{op}$ to $M$. In Proposition 3.1.2 it is shown that if $M$ is a cartesian monoidal model category then a colax symmetric monoidal functor $Prop(Com)\to M$ has weak equivalences as structure maps (as above) iff the corresponding functor $\Gamma^{op}\to M$ satisfies Segal's condition. Thus, a homotopy commutative monoid in $M$ in the sense of Leinster is precisely a $\Gamma$ object in $M$ satisfying Segal's condition. However, Leinster's definition generalizes and makes sense also if our monoidal model category $M$ is not cartesian.

The only problem is that Leinster doesn't prove the equivalence of his definition of a homotopy algebra over an operad $P$ in symmetric monoidal model category $M$, with the more known definition using a cofibrant replacement of $P$.

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  • $\begingroup$ Yes, I am aware of the existence of this paper and the companion expository paper arXiv.org/abs/math/9912084v2 (which treats only the case of commutative monoids). However, in Definition 2.1.1 Leinster requires that weak equivalences are closed under tensor products, which excludes almost all examples of monoidal model categories except for those in which all objects are cofibrant. This also applies to cartesian model categories. $\endgroup$ – Dmitri Pavlov Jul 17 '15 at 8:02
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    $\begingroup$ Does anyone know a reference which relates this notion of homotopy algebra to the more classical notion using a cofibrant replacement of the operad? Perhaps just in the cartesian case? $\endgroup$ – Chris Schommer-Pries Jul 17 '15 at 8:05
  • $\begingroup$ @ChrisSchommer-Pries: This notion is actually quite similar to what Lurie's definition amounts to, and Lurie does prove a rectification statement for commutative or associative monoids. A more general rectification statement for arbitrary operads can be found in Theorem 9.3.11 in dmitripavlov.org/operads.pdf. The definition in Leinster's paper (or in the original post) is slightly more strict, however. $\endgroup$ – Dmitri Pavlov Jul 17 '15 at 8:41
  • $\begingroup$ @DmitriPavlov If $M$ is a symmetric monoidal model category then the full subcategory of $M$ on the cofibrant objects $M^{cof}$ is a symmetric monoidal category with equivalences (in the sense of Definition 2.1.1). Thus, you can define a homotopy algebra over $P$ in $M$ to be a homotopy algebra over $P$ in $M^{cof}$. My guess is that there should be a model structure on the category of all colax symmetric monoidal functors $A:Prop(P)\to M$, such that the fibrant cofibrant objects are the homotopy algebras (perhaps satisfying some extra conditions). $\endgroup$ – Ilan Barnea Jul 17 '15 at 14:58
  • $\begingroup$ @IlanBarnea: This is certainly true, but unfortunately doesn't bring us closer to constructing a model structure on monoidal Γ-objects or proving a comparison result (using either model structures or any other formalism). $\endgroup$ – Dmitri Pavlov Jul 17 '15 at 16:41

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