### 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.

## 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.