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1. • CommentRowNumber1.
   • CommentAuthorJon Beardsley
   • CommentTimeDec 16th 2020
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   Author: Jon Beardsley
   Format: MarkdownItexApologies is this is a well known fact, but it seems to me that a symmetric monoidal pseudofunctor from FinSet to Cat is precisely the data of a symmetric monoidal category. I wanted to ask if this was proven in detail anywhere (or maybe not true). I'd also be interested to know how this might relate to various ideas of something like 2-PROPS or different types of weak algebras for some kind of operad/monad/PROP, assuming any of that makes sense.

   Apologies is this is a well known fact, but it seems to me that a symmetric monoidal pseudofunctor from FinSet to Cat is precisely the data of a symmetric monoidal category. I wanted to ask if this was proven in detail anywhere (or maybe not true). I’d also be interested to know how this might relate to various ideas of something like 2-PROPS or different types of weak algebras for some kind of operad/monad/PROP, assuming any of that makes sense.

2. • CommentRowNumber2.
   • CommentAuthorJohn Baez
   • CommentTimeDec 16th 2020
   • (edited Dec 16th 2020)
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   Author: John Baez
   Format: MarkdownItexI've never heard of this fact; Joe Moeller was just explaining it to me. It seems pretty cool and I hope you folks can prove this gold-plated version: **Conjecture.** The 2-category $\mathbf{SymMonCat}$ is equivalent to the 2-category of symmetric monoidal pseudofunctors $(\mathsf{FinSet}, +) \to (\mathbf{Cat}, \times)$.

   I’ve never heard of this fact; Joe Moeller was just explaining it to me. It seems pretty cool and I hope you folks can prove this gold-plated version:

   Conjecture. The 2-category $\mathbf{SymMonCat}$ is equivalent to the 2-category of symmetric monoidal pseudofunctors $(\mathsf{FinSet}, +) \to (\mathbf{Cat}, \times)$.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeDec 16th 2020
   • (edited Dec 16th 2020)
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   Author: Urs
   Format: MarkdownItexSeen under the Grothendieck construction (from xyz-functors out of $FinSet_\ast$ to xyz-fibrations over $FinSet_\ast$) this is the motivating example of the definition of $\infty$-operads in Chapter 2 of Jacob Lurie's _[[Higher Algebra]]_. See from construction 2.0.0.1 on [p. 165](https://www.math.ias.edu/~lurie/papers/HA.pdf#page=165) to Def. 2.0.0.7 on [p. 169](https://www.math.ias.edu/~lurie/papers/HA.pdf#page=169), where it says: > One of our main goals in this book is to show that Definition 2.0.0.7 is reasonable: that is, it provides a robust generalization of the classical theory of symmetric monoidal categories.

   Seen under the Grothendieck construction (from xyz-functors out of $FinSet_\ast$ to xyz-fibrations over $FinSet_\ast$) this is the motivating example of the definition of $\infty$-operads in Chapter 2 of Jacob Lurie’s Higher Algebra.

   See from construction 2.0.0.1 on p. 165 to Def. 2.0.0.7 on p. 169, where it says:

   One of our main goals in this book is to show that Definition 2.0.0.7 is reasonable: that is, it provides a robust generalization of the classical theory of symmetric monoidal categories.

4. • CommentRowNumber4.
   • CommentAuthorRichard Williamson
   • CommentTimeDec 16th 2020
   • (edited Dec 16th 2020)
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   Author: Richard Williamson
   Format: MarkdownItexRe #1: This seems to me to be fairly obvious, one is basically just parametrising bracketing. One can strictify everything as well by replacing $FinSet$ by a skeleton of itself and using strict 2-functors. Note that $FinSet$ (or its skeleton) is the free symmetric monoidal category on a commutative monoid, and a commutative monoid in $\mathsf{Cat}$, up to coherence issues, is a symmetric monoidal category (expressing this precisely proves the conjecture in #2). Thus we can certainly generalise this in various ways. E.g. the simplex category $\Delta$ is the free monoidal category on a monoid, so a monoidal category will be the same as a monoidal pseudofunctor from $\Delta$ to $\mathsf{Cat}$. There is a general pattern here that the _core_ of $FinSet$ is the free symmetric monoidal category on an object; retaining the non-automorphisms upgrades the universal property to being free on a commutative monoid. The analogous statement holds for $\Delta$ too. There is probably some very general 2-categorical theorem along these lines: take some 2-monad on $\mathsf{Cat}$ (e.g. that of symmetric monoidal categories), take the 1-monad corresponding to the one-object version of the algebras for this 2-monad (the monad of commutative monoids in this case), and then the 2-category of structure-preserving functors from the free 2-algebra (for the 2-monad) on a 1-algebra (for this 1-monad) to $\mathsf{Cat}$ should be equivalent to the 2-category of algebras for the 2-monad. I'd guess that's in the literature, but I'd have no idea where to find it.

   Re #1: This seems to me to be fairly obvious, one is basically just parametrising bracketing. One can strictify everything as well by replacing $FinSet$ by a skeleton of itself and using strict 2-functors. Note that $FinSet$ (or its skeleton) is the free symmetric monoidal category on a commutative monoid, and a commutative monoid in $\mathsf{Cat}$, up to coherence issues, is a symmetric monoidal category (expressing this precisely proves the conjecture in #2). Thus we can certainly generalise this in various ways. E.g. the simplex category $\Delta$ is the free monoidal category on a monoid, so a monoidal category will be the same as a monoidal pseudofunctor from $\Delta$ to $\mathsf{Cat}$.

   There is a general pattern here that the core of $FinSet$ is the free symmetric monoidal category on an object; retaining the non-automorphisms upgrades the universal property to being free on a commutative monoid. The analogous statement holds for $\Delta$ too. There is probably some very general 2-categorical theorem along these lines: take some 2-monad on $\mathsf{Cat}$ (e.g. that of symmetric monoidal categories), take the 1-monad corresponding to the one-object version of the algebras for this 2-monad (the monad of commutative monoids in this case), and then the 2-category of structure-preserving functors from the free 2-algebra (for the 2-monad) on a 1-algebra (for this 1-monad) to $\mathsf{Cat}$ should be equivalent to the 2-category of algebras for the 2-monad. I’d guess that’s in the literature, but I’d have no idea where to find it.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeDec 16th 2020
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   Author: Urs
   Format: MarkdownItexYeah, the idea is well-known in algebraic topology. For instance it's used in the definition of [[Gamma-spaces]] (symmetric monoidal $\infty$-categories that happen to be $\infty$-groupoids), dating back to 1974. Specifically for the case of symmetric monoidal categories, there ought to be earlier accounts than _[[Higher Algebra]]_, but I would have to dig around to find them.

   Yeah, the idea is well-known in algebraic topology. For instance it’s used in the definition of Gamma-spaces (symmetric monoidal $\infty$-categories that happen to be $\infty$-groupoids), dating back to 1974.

   Specifically for the case of symmetric monoidal categories, there ought to be earlier accounts than Higher Algebra, but I would have to dig around to find them.

6. • CommentRowNumber6.
   • CommentAuthorDylan Wilson
   • CommentTimeDec 16th 2020
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   Author: Dylan Wilson
   Format: MarkdownItexAs mentioned in the homotopy theory chatroom, this is immediate from the fact that Fin is the symmetric monoidal envelope of Fin_* (so actually a symmetric monoidal functor from Fin to any symmetric monoidal C is the same as a commutative algebra object in C). (HA.2.2.4.9). I guess then the question becomes: "why are 'pseudo-gamma categories' the same as symmetric monoidal categories (classically defined)?" but that is classical... I don't know the earliest reference. Presumably at this low-categorical level one could check by hand. Another proof is to use the Barr-Beck theorem: there is an evident construction that takes a symmetric monoidal category and spits out a 'pseudo-gamma category', and this commutes with the forgetful functor to the (2,1)-category of categories. So by nonsense we are reduced to checking that the 'free symmetric monoidal category' is the same as the 'free pseudo-gamma category', which, if need be, can be checked using the formula in HA.3.1.3.13. [This, by the way, builds an equivalence of (2,1)-categories of 'symmetric monoidal cats' and 'pseudo-gamma cats'.]

   As mentioned in the homotopy theory chatroom, this is immediate from the fact that Fin is the symmetric monoidal envelope of Fin_* (so actually a symmetric monoidal functor from Fin to any symmetric monoidal C is the same as a commutative algebra object in C). (HA.2.2.4.9). I guess then the question becomes: “why are ’pseudo-gamma categories’ the same as symmetric monoidal categories (classically defined)?” but that is classical… I don’t know the earliest reference. Presumably at this low-categorical level one could check by hand. Another proof is to use the Barr-Beck theorem: there is an evident construction that takes a symmetric monoidal category and spits out a ’pseudo-gamma category’, and this commutes with the forgetful functor to the (2,1)-category of categories. So by nonsense we are reduced to checking that the ’free symmetric monoidal category’ is the same as the ’free pseudo-gamma category’, which, if need be, can be checked using the formula in HA.3.1.3.13. [This, by the way, builds an equivalence of (2,1)-categories of ’symmetric monoidal cats’ and ’pseudo-gamma cats’.]

7. • CommentRowNumber7.
   • CommentAuthorJon Beardsley
   • CommentTimeDec 16th 2020
   • (edited Dec 16th 2020)
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   Author: Jon Beardsley
   Format: MarkdownItexI don't know how to reply to comments in the correct way, so I'll just try to label things sensibly. @Urs: One does need to be a bit careful since the framework Lurie uses is actually based on taking the free semi-Cartesian symmetric monoidal category on an operad (also known as the May-Thomason category of operators, which is I think the earliest reference for this precise construction) rather than the free symmetric monoidal category. Presumably there are sort of parallel theories here, i.e. Segal-type functors out of finite pointed sets and symmetric monoidal functors out of finite sets. And also, in your reference to Γ-spaces I think some care must still be taken. These are functors out of finite pointed sets satisfying the Segal condition, not symmetric monoidal functors out of finite sets. I suspect that some of the difference is obscured by the fact that the usual codomains in these cases, Cat and Top, are Cartesian monoidal. But anyway, I think when the codomain is Cartesian monoidal it doesn't matter. @Richard: Indeed, I also think it's somewhat "obvious" which is why I'm a bit surprised that it's not in some well known place in the literature. Maybe it's just too "obvious" for anyone to bother writing down, which I think is unfortunate (and also why I suspect there's some more general theorem which has this statement as a special case, as you suggest but are sadly not able to provided a reference for). @Dylan yes, I definitely agree that this statement can be proven using the symmetric monoidal envelope machinery described in higher algebra. Ultimately I was hoping there was some low-brow proof/description of it which predated that kind of stuff (and moreover I really did just want to talk about pseudofunctors of 2-categories, not ∞-categories). And, yes, one can check by hand, and Joe Moeller and I have basically done that.

   I don’t know how to reply to comments in the correct way, so I’ll just try to label things sensibly.

   @Urs: One does need to be a bit careful since the framework Lurie uses is actually based on taking the free semi-Cartesian symmetric monoidal category on an operad (also known as the May-Thomason category of operators, which is I think the earliest reference for this precise construction) rather than the free symmetric monoidal category. Presumably there are sort of parallel theories here, i.e. Segal-type functors out of finite pointed sets and symmetric monoidal functors out of finite sets. And also, in your reference to Γ-spaces I think some care must still be taken. These are functors out of finite pointed sets satisfying the Segal condition, not symmetric monoidal functors out of finite sets. I suspect that some of the difference is obscured by the fact that the usual codomains in these cases, Cat and Top, are Cartesian monoidal. But anyway, I think when the codomain is Cartesian monoidal it doesn’t matter.

   @Richard: Indeed, I also think it’s somewhat “obvious” which is why I’m a bit surprised that it’s not in some well known place in the literature. Maybe it’s just too “obvious” for anyone to bother writing down, which I think is unfortunate (and also why I suspect there’s some more general theorem which has this statement as a special case, as you suggest but are sadly not able to provided a reference for).

   @Dylan yes, I definitely agree that this statement can be proven using the symmetric monoidal envelope machinery described in higher algebra. Ultimately I was hoping there was some low-brow proof/description of it which predated that kind of stuff (and moreover I really did just want to talk about pseudofunctors of 2-categories, not ∞-categories). And, yes, one can check by hand, and Joe Moeller and I have basically done that.

8. • CommentRowNumber8.
   • CommentAuthorRichard Williamson
   • CommentTimeDec 17th 2020
   • (edited Dec 17th 2020)
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   Author: Richard Williamson
   Format: MarkdownItexJust to emphasise that the fact that $\mathsf{FinSet}$ is the free symmetric monoidal category on a commutative monoid is certainly in the literature (googling turned up a recent paper of Hyland and Power, but of course it is much older). If one restricts to fully strict symmetric monoidal categories, then the conjecture in #2 is just the statement of this universal property plus the observation that a commutative monoid in $\mathsf{Cat}$ is exactly a (fully strict) symmetric monoidal category. Extracting a weak version of this is just a matter of interpreting 'commutative monoid' weakly 2-categorically rather than 1-categorically, and again is really just (one version of) the statement of the universal property. I should say that I did not really try to find a reference for the 2-monadic formulation I mentioned! I don't know the literature too well on 2-monads, and tend to just follow my nose. If Mike sees this, he would know better whether there is a reference. One could begin by trying to make sense of the first step in my formulation, namely: given a 2-monad on $\mathsf{Cat}$, how do we rigorously construct the 1-monad whose algebras are the 'one-object' versions of the algebras for the 2-monad? I suspect there is some nice abstract way to do this using 2-categorical techniques. That must be extractable from the literature!

   Just to emphasise that the fact that $\mathsf{FinSet}$ is the free symmetric monoidal category on a commutative monoid is certainly in the literature (googling turned up a recent paper of Hyland and Power, but of course it is much older). If one restricts to fully strict symmetric monoidal categories, then the conjecture in #2 is just the statement of this universal property plus the observation that a commutative monoid in $\mathsf{Cat}$ is exactly a (fully strict) symmetric monoidal category. Extracting a weak version of this is just a matter of interpreting ’commutative monoid’ weakly 2-categorically rather than 1-categorically, and again is really just (one version of) the statement of the universal property.

   I should say that I did not really try to find a reference for the 2-monadic formulation I mentioned! I don’t know the literature too well on 2-monads, and tend to just follow my nose. If Mike sees this, he would know better whether there is a reference. One could begin by trying to make sense of the first step in my formulation, namely: given a 2-monad on $\mathsf{Cat}$, how do we rigorously construct the 1-monad whose algebras are the ’one-object’ versions of the algebras for the 2-monad? I suspect there is some nice abstract way to do this using 2-categorical techniques. That must be extractable from the literature!