minor notation changes.

egal

]]>Add cross reference to commutative monoidal category.

]]>added pointer to:

- Saunders MacLane, Ch. XI of:
*Categories for the Working Mathematician*, Graduate Texts in Mathematics**5**Springer (second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

Added a subsection “Permutations” which defines the natural transformation associated to any permutation from the braiding.

]]>Added original reference.

]]>and full publication data for:

- Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik,
*Tensor Categories*, AMS Mathematical Surveys and Monographs**205**(2015) $[$ISBN:978-1-4704-3441-0, pdf$]$

added pointer to:

- Francis Borceux, Section 6.1 of:
*Handbook of Categorical Algebra*Vol. 2:*Categories and Structures*$[$doi:10.1017/CBO9780511525865$]$, Encyclopedia of Mathematics and its Applications**50**, Cambridge University Press (1994)

Added a subsection which discusses the relationship between symmetric monoidal categories and $\Gamma$-categories.

Amit Sharma

]]>The previous version said that $SymMonCat$ has no model category structure. In this version a comment is added which says that the subcategory of Permutative categories has a model category structure. Also noted that the coherence theorem states that each symmetric monoidal category is equivalent to a permutative category.

Amit Sharma

]]>Added the result discussed at the Cafe that the cartesian product of symmetric monoidal categories is their (weak) 2-biproduct.

]]>Well, that sort of abuse of language is ubiquitous in mathematics.

]]>the nerve is a simplicial set, but we can't group complete a simplicial set,

only an E_∞-space (or at least an E_1-space).

So it really should be the nerve equipped with the E_∞-space structure

induced from the symmetric monoidal structure. ]]>

I changed

The nerve of a symmetric monoidal category is always an infinite loop space

to

The group completion of the nerve of a symmetric monoidal category is always an infinite loop space

I think we need the group completion here since an infinite loops space is more like a “stable infinity-group” than a “stable infinity-monoid”. Also, Thomason uses this group completion (see around 1.6.1 in his paper Symmetric monoidal categories model all connective spectra.

I did not fix the diagram underneath this remark, if indeed it needs fixing as I claim.

]]>Hi Jade, it has not been fixed, it should be re-drawn in Tikz when somebody finds a spare moment!

]]>Ah yes, thanks!

It seems to me an interesting question (ref. #23 and #24) whether the (2-)functor itself exists, even if we don’t know whether it can be exhibited as a cofibrant replacement in some model structure.

]]>Finitary 2-monads on $Cat$ are themselves the algebras for a 2-monad on the category of finitary endo-2-functors of $Cat$, so they inherit a model structure by Lack’s transfer theorem in the same way. In this model structure, the algebras for the canonical cofibrant replacement $T'$ of a finitary 2-monad $T$ are the pseudo $T$-algebras. Unfortunately, pseudoalgebras for the 2-monad whose algebras are commutative monoidal categories are not equivalent to symmetric monoidal categories. What is true is that symmetric monoidal categories are pseudoalgebras for the 2-monad whose algebras are permutative categories.

]]>I mean, intuitively it should exist, n’est pas? Basically replace equality by an isomorphism, and freely throw in everything that is needed to still have a monoidal category?

]]>Something that I think is true is that there is a model structure on 2-monads on $Cat$ such that the symmetric monoidal category 2-monad is a cofibrant replacement of the 2-monad of what John called commutative monoidal categories.

I wouldn’t be surprised if one could cook up from this cofibrant replacement of 2-monads some 2-functor between the 2-categories of algebras which basically does what John is looking for.

]]>Yes, the canonical model structure on $Cat$ transfers along the free-forgetful adjunction to a model structure on the category of symmetric monoidal categories and strict symmetric monoidal functors. This is an instance of the model structure on algebras for a 2-monad in section 4 of Steve Lack’s Homotopy-theoretic aspects of 2-monads.

For the second question, if by “nontrivial braiding” you mean that the isomorphism $X\otimes X \cong X\otimes X$ is not the identity, then I think the answer should be no, because the map $C' \to C$ from a cofibrant replacement is a strict symmetric monoidal functor (hence preserves the braiding on the nose) and an equivalence of categories (hence faithful); thus if the braiding is trivial in $C$ it should also be trivial in $C'$.

]]>Is there a model category of symmetric monoidal categories? If I took the cofibrant replacement of a *commutative* monoidal category (a commutative monoid object in $Cat$), would I get something with a nontrivial braiding?

It is there in the source, but is being trimmed off. This trimming is what needs to be fixed. I see now that the issue seems also to be there in Firefox. Some tweaking of the SVG graphic is needed.

]]>There seems to be a left ( missing in that diagram. i.e. at (f/g).

]]>Thanks for all the work!

]]>