## October 14, 2008

### Freely Generated ω-Categories

#### Posted by Urs Schreiber

I try to review some aspects of some of the literature on $\omega$-categories which are “freely generated”. Then I have some questions.

Ross Street has a definition of a freely generated $\omega$-category on p. 326 (45 of 54) of The algebra of oriented simplices.

I think this is equivalent to the concept of polygraphs that Francois Métayer describes in Resolutions by polygraphs (briefly mentioned in TWF 227) which in turn follows A. Burroni’s Higher dimensional word problems (but which is equivalent to Street’s concept of computads developed some 20 years earlier, as Michael Batanin kindly points out in a comment below):

Write $n Cat^+$ for the category of globular $(n+1)$-sets equipped with the structure of a strict $n$-category (not $(n+1)$!) on their truncation to a globular $n$-set. There is an obvious forgetful functor $W_{n+1} : (n+1)Cat \to n Cat^+$ and this has a left adjoint $L_{n+1} : n Cat^+ \to (n+1)Cat$. Heuristically, $L_{n+1}$ takes an $n$-category and a bunch of $(n+1)$-globes and creates an $(n+1)$-category whose $(n+1)$-morphisms are all possible pasting diagrams of these $(n+1)$-globes, i.e. all formal ways to stick these together. Composition of such pasting diagrams is by gluing.

Then a “freely generated” $\omega$-category $C$, or “polygraph, ” is one which in each degree $k$ is in the image of this $L_k$.

Métayer shows in Cofibrant complexes are free that these freely generated $\omega$-categories/polygraphs are precisely the cofibrant $\omega$-categories with respect to the folk model structure. Moreover, he constructs for each $\omega$-category $C$ a “free resolution”, hence a cofibrant replacement $C_{cof} \stackrel{\simeq}{\to}\gt C$, which is essentially the construction we had discussed here.

(In fact, it seems to me that the statement can be strengthened: This construction yields a functor $(-)_{cof} : \omega Cat \to \omega Cat$ together with a natural transformation $\rho : (-)_{cof} \to Id$ whose component at $C$ is the cofibrant replacemtent $\rho_C : C_{cof} \to C$.)

This notion of free generation is interesting, because it seems to allow to conceive as strict $\infty$-categories things that are otherwise thought of as being inherently “weak”. As Burroni remarks on p. 7-8:

“All of them (oo-graphs, simplicial sets, etc) can be interpreted as polygraphs. […] That has been achieved for simplicial sets in the work of R. Street [Algebra of oriented Simplices].”

This I want to see in more detail:

Street mentions on p. 333 that the $\omega$-nerve functor $N : \omega Cat \to SimpSet$ has a left adjoint $F : SimpSet \to \omega Cat$.

I’d like to understand this functor $F$ well. Street gives an “explicit” description in terms of coends. Unfortunately, despite some effort, I am still not really fluent with coends. Here is what i am guessing a more pedestrian definition of $F$ would be, using the above language:

Let $S$ be a simplicial set. Iteratively build an $\omega$-category $F(S)$ as follows:

in degree 0 $F(S)$ is just the set of 0-simplices of S. In degree 1 $F(S)$ has all the finite sequences of “composable” (attachable) 1-simplices in $F(S)$, composition being concatenation of such sequences.

Next, observe that using the second oriental, the “triangle”, $\array{ && b \\ & {}^{d_0}\nearrow & & \searrow^{d_2} \\ & & \Downarrow \\ a &&\to_{d_1}&& c }$ we get two maps $S_2 \stackrel{s,t}{\to} (F(S))_1$: $t$ sends every 2-simplex in $S$ to its 1st face, while $s$ sends every 2-simplex in $S$ to the “pasting composite” of its 0th face with its 2nd face.

So we can set $(F(S))_2 := L_2(S_2 \stackrel{s,t}{\to} (F(S)_1 ) \,.$

And so on. At level $k$ we find, using the $k$th-oriental, that there is a pasting composite of all the even-numbered faces of every $k$-simplex in $S$, and a pasting composite of all the odd-numbered faces. This gives two maps $S_k \stackrel{s,t}{\to} (F(S))_{k-1}$ and we set $(F(S))_k := L_k(S_k \stackrel{s,t}{\to} (F(S)_{k-1} ) \,.$ The directed limit of this operation is an $\omega$-category $F(S)$.

And I am thinking that the $F$ obtained this way is the left adjoint to the $\omega$-nerve. Is that right??

For every simplicial set $S$, $N(S(F))$ is the simplicial set whose $k$-simplices are pasting diagrams of $k$-simplices in $S$ and the unit of the adjunction $S \hookrightarrow N(F(S))$ regards any k-simplex as the trivial pasting diagram consisting just of itself. This should hence be a monomorphism.

Similarly, for every $\omega$-category $C$, $F(N(C))$ should be something which in degree $k$ has all pasting diagrams of $k$-morphisms in $C$ and the counit $F(N(C)) \to C$ should send each pasting diagram to the result of evaluating its composition in $C$.

Is that right? Is $F(N(C))$ the same as the free resolution $C_{cof}$ that Métayer describes?

It seems to me, at this somewhat rough level, that the functor $F : SimpSet \to \omega Cat$ should be faithful. Is that right? (Given that the unit is a monomorphism…)

Finally I should mention that I have seen on Jeffrey Morton’s blog in his entry on Octoberfest 08 that Harnik talked about something like $\omega$-categories obtained by generators. But I haven’t seen Harnik’s work yet.

Posted at October 14, 2008 11:32 AM UTC

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### Re: Freely Generated ω-Categories

Hi, Urs.

I just want to say that polygraphs are the same as computads of Ross Street. They were defined 20 years before Burroni. I find it quite annoying that people give credit to Burroni (he deserves many other credits, of course) for inventing this notion just because the terminology has been changed.

Coming back to mathematics. The Metayer’s resolution is just a counit of the adjunction Comp_n -> Cat_n (n= \omega for Metayer). It is, therefore, a natural transformation ((-)_cof –> Id) as you suggested) It is different from Street’s adjunction SSet -> Cat_n . For Street’s adjunction. I do not know if the result is cofibrant and the counit is a trivial fibration but I strongly suspect that the answers for both questions are negative.

As far as I know Makkai and Harnik’s work they are also talking about Street’s computads but thinking about them as \omega-caegories which contain a subset of indeterminate cells. Morphisms are \omega-functors preserving inditerminates. These are equivalent definitions.

There are also computads for weak \omega-categories which I defined in 1998. I belive they could be used to define folklore model structure on weak \omega-Cat. I would be very interested to have it.

Michael.

Posted by: Michael on October 15, 2008 2:11 AM | Permalink | Reply to this

### Re: Freely Generated ω-Categories

I just want to say that polygraphs are the same as computads of Ross Street. They were defined 20 years before Burroni. I find it quite annoying that people give credit to Burroni (he deserves many other credits, of course) for inventing this notion just because the terminology has been changed.

Thanks for pointing this out. While I am not sure if Burroni mentions this, certainly Métayer does in the article I linked to. I have added a respective remark to the above entry.

For Street’s adjunction. I do not know if the result is cofibrant and the counit is a trivial fibration but I strongly suspect that the answers for both questions are negative.

Okay, thanks. I should look at this in more detail then. It seemed to me that the construction $F: SSet \to Comp_\omega \to \omega Cat$ which I tried to describe above is beginning to look like a left adjoint to the $\omega$-nerve. But I didn’t try to check that in more detail than mentioned above.

As far as I know Makkai and Harnik’s work they are also talking about Street’s computads but thinking about them as $\omega$-categories which contain a subset of indeterminate cells. Morphisms are $\omega$-functors preserving inditerminates. These are equivalent definitions.

What would be a good reference describing their approach?

There are also computads for weak $\omega$-categories which I defined in 1998. I believe they could be used to define folklore model structure on weak $\omega$-Cat. I would be very interested to have it.

That would be interesting. Have you thought about $\omega$-categories where everything is strict except the units? (I am asking this in the context of Simpson’s conjecture which we talked about recently here.) I suppose these could be obtained just by suitably restricting your definition of general weak $\omega$-categories? Have you thought about whether every weak $\omega$-category might be equivalent to one which is strict except possibly for its units (as Simpson’s conjecture would suggest)?

Posted by: Urs Schreiber on October 15, 2008 9:59 AM | Permalink | Reply to this

### Re: Freely Generated ω-Categories

Hi, Urs.

You, probably, can find Makkai and Harnik’s peprints in Makkai’s homepage but I do not have a reference.

Concerning semistrict \omega-categories. I thought about Simpson’s conjecture, of course. I had similar conjecture for algebras over contractible n-operads, but I could not prove it. The combinatorics turned out to be too complicated. One conjecture which should lead to Simpson’s conjecture is formulated in my paper “Symmetrisation of $n$-operads and compactification of real configuration spaces.” (Conjecture 3.1).

Another conjecture about it: the definition of the operad B which acts on fundamental \omega-groupoid (this operad was used by Cisinski in his paper on homotopy conjecture) of a space can be modified in a way that everyhing is strict except for units. This should be a kind of Moore loop process. In B, for example, interchange is strict but associativity and units are weak. We should be able to kill associativity of vertical compositions. Yet, it is not clear, can we kill “higher associativity” for horizontal compositions. For example, we need to make two Yang-Baxter hexagons equal killing this 3-cell which I called in Minessota Breenator. This Breenator is responsible for an interesting element of order 3 in a homotopy group of spheres (I do not remember which homotopy group. There is a letter of Larry Breen to Mike Hopkins where he describes it). So we have to be very careful with the definition of “higher associativity”.

Finally, in another paper on computads, which is an archive I have another conjecture concerning semistrict \omega-categories. Given an \omega-operad we can construct a sequence of symmetric operads out of it, which I called slices. The conjecture is: There is a unique contractible operad such that all its slices are operads for monoids.

Finally, I hope we will understand better semistrictness by studying the multitensors generated by \omega-operads (see my joint paper with Mark Weber). We hope to be able to find some conditions when this multitensor is an actual tensor. Examples of such tensors are Gray-product, Crans-Product, higher Gray-product.

Michael.

Posted by: michael on October 16, 2008 8:05 AM | Permalink | Reply to this
Read the post Codescent and the van Kampen Theorem
Weblog: The n-Category Café
Excerpt: On codescent, infinity-co-stacks, fundamental infinity-groupoids, natural differential geometry and the van Kampen theorem
Tracked: October 21, 2008 9:32 PM

### Re: Freely Generated ω-Categories

I am starting to move some of my open questions to query boxes in $n$Lab-entries:

there is nowone in the entry on orientals.

All help is greatly appreciated.

Posted by: Urs Schreiber on December 3, 2008 9:35 PM | Permalink | Reply to this

### Re: Freely Generated ω-Categories

I tried a little editing here; I didn’t address your query but I did address some related issues.

This was the first time I’ve taken a look at the nLab. This is bound to be a tremendous resource. Some terms bandied about in the Café which usually made my eyes glaze over now have succinct descriptions; I’m sure I’ll turn to it time and again.

Posted by: Todd Trimble on December 4, 2008 1:59 AM | Permalink | Reply to this

### Re: Freely Generated ω-Categories

Thanks, Todd!

The reason why I am wondering if $F : SimpSet \to \omega Cat$ is faithful is that using that functor one sees one can carry over a lot of simplicial technology over to the world of $\omega$-categories. This is extremely useful, for instance for constructing universal characteristic classes in nonabelian cohomology, etc. where one needs to build suitable cofibrant replacements of $\omega$-categories.

I want to sort this out because people keep bugging me whenever they hear that $\omega$-categories are strict. It seems to me that it is not widely appreciated what a long way already strict $\infty$-categories already go for some applications, mainly precisely due to the fact that one can “free” the strict composition. It’s more like there is the option to specify strict composites, than a constraint.

So I want to nail this down more precisely: the obstruction to doing within $\omega$-categories what one can do with simplicial sets would seem to be measured by the failure of $F$ to be faithful. It looks pretty faithful to me, though, but I am not sure how to prove it precisely. So I am wondering

Posted by: Urs Schreiber on December 4, 2008 9:39 AM | Permalink | Reply to this

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