The n-Category Café

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.