Let $\Gamma^{op}$ be the category of finite pointed sets. A special $\Gamma$-category is a functor $Y:\Gamma^{op}\to Cat$ such that the canonical maps $Y[n]\to Y[1]^n$, called Segal maps, are equivalences of categories, for all $n\geq 0$.

There is a canonical way of associating to an unbiased symmetric monoidal category a special $\Gamma$-category (called a “homotopy monoidal category” by Tom Leinster in Higher Operads, Higher Categories).

I think that Moritz Groth, in his Example 3.4 of his course in infinity categories, is saying that this association is an equivalence.

However, Tom Leinster, pages 120-121 of the aforementioned book, says that “there is reasonable hope” that this is true.

I would like to know whether this result is true or not, and where do the delicate points lie. If it is true, I what are some references where the result is proven in detail?

EDIT: I would like to add some context. A commutative monoid in a cartesian monoidal category $\mathcal C$ is, equivalently, a functor $\Gamma^{op}\to \mathcal C$ that satisfies that the Segal maps are isomorphisms.

My point is that I would like to apply this to the concept of symmetric monoidal category itself. However, a (small) symmetric monoidal category is not monoid, but a symmetric pseudomonoid in the cartesian 2-category $Cat$.

So I guess I would like to know whether it is true that symmetric pseudomonoids in 2-categories $D$ are equivalently pseudofunctors $\Gamma^{op} \to D$ that satisfy that the canonical maps $Y[n]\to Y[1]^n$ are 1-equivalences. Note that this is not exactly what I wrote in the first line (here I'm letting $Y$ be a pseudofunctor).

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    $\begingroup$ What's an unbiased symmetric monoidal category if it's not a functor on $\Gamma^{op}$? $\endgroup$ – Marc Hoyois Nov 12 '17 at 17:44
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    $\begingroup$ @MarcHoyois In Leinster's book an unbiased symmetric monoidal category is a family of functors $\bigotimes_n:C^n\to C$ plus a ton of coherence data. I agree that functors from $\Gamma^{op}$ are much more intutive. $\endgroup$ – Denis Nardin Nov 12 '17 at 17:55
  • $\begingroup$ Ah I see. The arXiv version of the book only seems to discuss the non-symmetric case though. $\endgroup$ – Marc Hoyois Nov 12 '17 at 20:55
  • $\begingroup$ @DenisNardin Thanks, yes, that's what I had in mind. I think it's pretty intuitive: it's a weak algebra for a very explicit 2-monad in $Cat$. $\endgroup$ – Bruno Stonek Nov 13 '17 at 12:12
  • $\begingroup$ Sorry for the edits: I've been confused by this. I'm pretty convinced that what I wrote at the end is true. I also now realize that this is the statement underlying what Groth wrote. However, I still wonder if what Leinster was thinking is true, i.e. that special $\Gamma$-categories are equivalent to symmetric monoidal categories. $\endgroup$ – Bruno Stonek Nov 13 '17 at 17:05

Tom Leinster's book is very old. For higher category theory 2003 is like a previous epoch. In those times there were many competing definitions of higher category theory and higher algebra, for most of them it was unclear if they are equivalent or whether they even model all homotopy types. For some of them this was eventually proven false, e.g. the once-favoured derivators are unsufficient to properly work with higher algebra and the strict $\omega$-categories are obviously unsufficient for the homotopy hypothesis (homotopy types = $\infty$-groupoids). The main delicate points are in giving the proper definitions and handling all coherence conditions, so that the theory is flexible enough for all purposes but at the same time is still tractable. For this reason the aforementioned statement is stated not as a theorem (probably it is so in Leinster's foundations, but I don't have the print copy at hand) but rather as a guiding principle which should be true in any proper higher category theory. This is certainly true in the now-accepted foundations via quasicategories and higher Segal spaces. In fact, it is usually taken as the definition of a monoidal higher category, so the only question is whether it covers all classical example (it does). J. Lurie's "Higher Algebra" is nowadays the go-to reference for all questions about the foundations of higher algebra, and his "Higher Topos Theory" lays the used foundations of $(\infty, 1)$-category theory.

If you are working within classical 1-category theory, then the statement is very easy to prove explicitly. I am unaware of any specific reference for it. For $A_\infty$-monoids in homotopy theory I believe that the approach via $\Gamma$-spaces originates from the work of G. Segal. You can check the references in the relevant page on nLab.

Looking at the arXiv version of Leinster's paper, it also doesn't really discuss $\infty$-categories, nor does it even define an equivalence of $\omega$-categories (see par. 9.2), so the equivalence between monoidal categories and $\Gamma$-spaces was likely never really proved in that foundations.

  • $\begingroup$ I think your third sentence is rather misleading. It's been known since the mid-20th-century that strict $\omega$-categories are insufficient, and I've never read anyone proposing that derivators are sufficient for all homotopy coherent algebra, only that they're good enough for a lot and are easier to work with. The only proposed definition of higher category that I know of to have been seriously suggested and then proven "wrong" was the original version of Penon's, but even that one can be slightly modified to fix the problem (though I don't think it has been proven "correct" yet either). $\endgroup$ – Mike Shulman Nov 13 '17 at 16:55

I think this can be seen by an indirect route using much more general results. Probably there's an easier and more elementary way than the following. Also I'm not quite sure of the details here, and moreover I don't really have any insight into what the main difficulties are in doing this.

  1. Lurie's definition of a symmetric monoidal $\infty$-category is a cocartesian fibration of $\infty$-categories $C^\otimes \to \Gamma^{op}$. Starting with a special $\Gamma$-category, we take the Grothendieck construction and apply the ordinary nerve to obtain one of these things.

  2. Lurie's model for $\infty$-operads has been shown to be equivalent to dendroidal sets. Symmetric monoidal categories are just operads with a certain representability property, so in particular this gives an equivalence between symmetric monoidal $\infty$-categories in Lurie's sense and a certain subcategory of dendroidal sets.

  3. Dendroidal sets are Quillen equivalent to simplicial operads via a dendroidal homotopy coherent nerve. I'm assuming this includes the colored case.

So we can turn a special $\Gamma$-category into a colored simplicial operad. Because we were transporting across equivalences, this colored simplicial operad should still be representable (i.e. it's a symmetric monoidal simplicially-enriched category), and it should still have the property that its hom-spaces are essentially discrete (I'm banking on these properties being model-independent notions). So by applying $\pi_0$ to the hom-spaces, we should obtain a symmetric monoidal category. Presumably one can check that this process is inverse to one you described.

  • $\begingroup$ Presumably there is a Quillen equivalence between ∞-(pre)operads and simplicial operads using the operadic nerve (HA., although I don't know if anyone has written that down. $\endgroup$ – Denis Nardin Nov 12 '17 at 16:53
  • $\begingroup$ It's easy to check directly (without invoking anything higher-categorical) that the category of coloured operads (in Set) is equivalent to a subcategory of categories over $\Gamma^{op}$ (with objects those functors satisfying 1-categorical analogues of Lurie's definition of an infinity-operad, and morphisms functors that preserve cocartesian morphisms over inert maps in $\Gamma^{op}$), via the "category of operators" of an operad. $\endgroup$ – Rune Haugseng Nov 12 '17 at 23:14
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    $\begingroup$ Under this equivalence, the coloured operads that come from a symmetric monoidal category will correspond to the subcategory where the maps to $\Gamma^{op}$ are Grothendieck opfibrations, corresponding to functors to Cat that are "special $\Gamma$-objects". So we're reduced to checking that (unbiased) symmetric monoidal categories (and lax symmetric monoidal functors) give a full subcategory of coloured operads. Hopefully Leinster's definition is set up so that this is straightforward... $\endgroup$ – Rune Haugseng Nov 12 '17 at 23:18

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