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added to symmetric monoidal category a new Properties-section As models for connective spectra with remarks on the theorems by Thomason and Mandell.
The pentagon at monoidal category has some ommitted object $z$ letters, and some half-pronounced in my rendering at firefox.
I have moved the references to the end/bottom (they were, strangely, in the middle of the long web page).
Here’s a question on graphical calculus in symmetric monoidal categories (SMCs).
The graphical calculus for SMCs is usually presented in a way which makes the diagrams (sort of) planar. For example, in Joyal and Street’s The geometry of tensor calculus, planarity gets reflected in the assumption that a diagram in a SMC is given by a polarised progressive graph:
A polarised graph is an oriented graph together with a choice of linear order on each in($x$) and out($x$).
My question is: does there exist a graphical calculus for SMCs in which one does not need to choose such orderings? And on a related note: has anyone worked out an unbiased definition of symmetric monoidal category in which one could make sense of a monoidal product $\bigotimes_{i\in I} A_i$ without having a linear order on the index set $I$?
Background: I am working in a situation in which I have an (abstract) directed acyclic graph, and I would like to be able to interpret this graph as a diagram in a SMC.
An unbiased symmetric monoidal category $\mathcal{C}$ should be equipped with functors $T_n : \mathcal{C}^n // S_n \to \mathcal{C}$, where $\mathcal{C}^n // S_n$ is the pseudo-quotient of $\mathcal{C}^n$ by the evident $S_n$-action. (Concretely, it is a certain Grothendieck construction.) There should also be isomorphisms relating e.g. $T_n$, $T_{m_1}, \ldots, T_{m_n}$, and $T_{m_1 + \cdots + m_n}$, and identities relating compositions of these.
Or one could cheat and say that an unbiased symmetric monoidal category is a pseudoalgebra for the symmetric strict monoidal category 2-monad. Unpacking this should yield a description of the form above.
3-5: instead of choosing order or quotienting by order, one can simply have thing for all orders simultaneously behaving covariantly with respect to the order. In combinatorics this is achieved via Joyal’s species. Maybe one could incorporate them here ?
Re #6: yes, the word “anafunctor” or “clique” comes to mind.
You may be interested in Appendix A of Leinster’s book.
Thanks, everyone, for the answers! So it seems that there are several options available for how to solve this, and they are all canonically isomorphic and hence form a clique ;)
Optimally, I was hoping to simply be able to put a reference to some paper proving that any directed acyclic graph, appropriately labelled with objects and morphisms in a SMC, can be interpreted as a diagram in that category; but you seem to be saying that such a paper doesn’t exist (yet). I will report back here as soon as I figured out which solution fits my needs best.
any directed acyclic graph, appropriately labelled with objects and morphisms in a SMC, can be interpreted as a diagram in that category
I don’t quite understand how this labeling is supposed to work. In a polarized graph, if I have a vertex whose incoming strings are labeled by objects $a,b$ in order and whose outgoing strings are labeled by $c,d,e$ in order, then I label it by a morphism $a\otimes b \to c\otimes d\otimes e$. But without orderings on the incoming and outgoing strings, how do I know what type of morphism to label the vertex by? It seems that an unpolarized graph could only be labeled by objects and morphisms in an unbiased SMC. Which is okay, but most SMCs that people define in practice are biased, and making a biased one into an unbiased one involves making a bunch of choices essentially equivalent to polarizing your graph — or else redefining the SMC structure from the get-go to be unbiased.
Mike wrote:
It seems that an unpolarized graph could only be labeled by objects and morphisms in an unbiased SMC.
Yes, I’m aware of that – and that’s exactly what I was hoping would already have been done by someone! Or at least something equivalent, like using covariant labellings in terms of cliques, as previously suggested in this thread. Directed acyclic graphs are quite common in applied mathematics for describing various kinds of information flow, and it would be very useful to have a formal theory for interpreting them as diagrams in categories of processes. As a case in point, my particular application concerns (a variant of) Bayesian networks and is closely related to a recent unbiased reformulation of quantum mechanics based on exactly that idea.
What I will probably end up doing is to introduce an arbitrary ordering on the in- and outgoing edges at each node, explain how to regard an unbiased tensor product $\bigotimes_{i\in I} A_i$ as a biased one with respect to the chosen ordering, apply the results of Joyal and Street on polarized graphs, and finally show that the resulting composite, which is a scalar $I\to I$ in my case, is independent of the chosen orderings. Not too pretty, but for a mathematical physics paper it should do, and it will allow me to stick with the usual definition of SMC.
I think that’s the right thing to do.
Just for the record: this recent paper by Rupel and Spivak also gets close to what I was looking for, although they don’t seem to prove anything on how to interpret their diagrams in symmetric monoidal categories.
after the words “which says that this diagram commutes” the included picture is not rendered properly.
The relevant code is
+--{: style="text-align:center"}
[[!include monoidal category > pentagon]]
=--
Possibly some of the recent changes to the behaviour of !include
is causing the problem?
This was a tricky one. I have now fixed this in this case. There are other examples of broken SVG includes I believe, but hopefully what I have done in this case can be applied generally.
There were three issues.
1) The SVG page (monoidal category > pentagon) had double dollars and a \begin{svg}
environment around the actual SVG graphic. The behaviour of this is obscure; it was probably some hack used in the past to get things to work, but things should work perfectly well without it. I suggest to remove such occurrences wherever this is seen. I did so in this case.
2) After 2), there were no longer any inclusion issues, but the SVG did not display correctly. In debugging this, I discovered that Instiki was not set up to handle the SVG file mime type. This meant that if one tried to use <img src="some.svg"/>
nothing would be displayed. I have fixed this now. This means that if a particular SVG graphic is problematic, one solution if it does not contain any MathML is to just upload and use it as an image.
3) In this case, 2) did not help, because the diagram contains MathML. After some debugging, I discovered that either Instiki or Maruku is, bizarrely, removing a crucial part of the header from the SVG graphic (the xlmns attribute). I have fixed this by putting this back at the end of rendering if it is missing.
This particular graphic seems to display and include correctly now. We’ll have to treat other problematic ones on a case by case basis. Just raise it here if you find one and cannot fix it by 1) or both 1) and 2).
It doesn’t display properly in Chrome on an Android phone. But I don’t think that’s an issue which has been introduced with these fixes.
Have checked that the fix also works at comma object and at knot (the picture of the trefoil). In these cases, one only needed to reload the picture pages (comma object > 2cell and trefoil knot - SVG, although in the latter case the inclusion had been removed earlier due to the rendering failing.
The picture at comma object > 2cell needs adjusting a bit to make it display well on non-Firefox browers. I can try and do that at some point, but great if someone else fancies trying it before then.
Thanks for all the work!
There seems to be a left ( missing in that diagram. i.e. at (f/g).
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.
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?
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 $C'$.
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.
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?
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.
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.
Hi Jade, it has not been fixed, it should be re-drawn in Tikz when somebody finds a spare moment!
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.
Well, that sort of abuse of language is ubiquitous in mathematics.
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
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