The n-Category Café

A major ingredient for everything that follows is the fact – already emphasized in yesterday’s lecture, but I hadn’t mentioned it – that

Theorem. There is a symmetric closed monoidal category structure on dendroidal sets such that

a) the tensor unit is $U = \Omega(|)$, the free operad on the tree with a single branch, which is the operad with a single color and only the identity operation on that;

b) the internal hom dendroidal set hom(A,B) is $$hom(A,B) : T \mapsto DendroidalSets(A \otimes T , B)$$

c) for $P$ and $Q$ two operads we have for the tensor product $N(P) \otimes N(Q)$ of their nerves that it is the nerve of an operad $P \otimes Q$ such that algebras for $P \otimes Q$ are precisely the $P$-algebras in the category of $Q$-algebras.

On representables this tensor product is given by a “shuffle product of trees”. This is best described maybe in terms of diagrams and I won’t attempt to do so at the moment.

Recall that we say

Definition. An $(\infty,1)$-operad (or $\infty$-operad if we are being colloquial) is a dendroidal set $A$ which satisfies the weak dendroidal Kan condition: all inner dendroidal horns of $A$ have fillers. An ordinary operad is precisely an $(\infty,1)$-operad for which all these lifts are unique.

In the following we will notationally identify operads $P$ with their dendroidal nerves $N(P)$.

Another concept from yesterday which I didn’t mention in my last installment is that of a normal dendroidal set: this is one which can be realized by iterative pushouts, so it’s the analog of a CW-complex. These have some nice properties:

Theorem. For $A$ a normal dendroidal set and $P$ an $(\infty,1)$-operad, the inernal hom dendroidal set $hom(A,P)$ is itself an $(\infty,1)$-operad.

For instance consider the symmetric monoidal category $E = Top$ with its standard interval object with $P = D(E)$ the homotopy coherent nerve $D(E)$ of the canonical operad on $E$ (recalled last time, here the choice of interval object enters), and for $Ass$ the operad of associtive algebras, we have

$$hom(N(Ass) \otimes N(Ass) , D(E))$$

is again an $(\infty,1)$-operad. It is at the moment a conjecture that it should be true that the vertices of this dendroidal set are the double loop spaces.

More generally, the Boardman-Vogt resolution of the $n$-fold tensor product of the associtive algebra operad with itself should be the little $n$-cube operad $D_n$:

$$W(N(Ass)^{\otimes n}) \simeq D_n \,.$$

As an example of this example: for $E = Cat$ with interval object the groupoid $\{a \stackrel{\simeq}{\to} b\}$ we have that

$$hom(N(Ass) \otimes N(Ass), D(E))$$

is the $(\infty,1)$-operad whose vertices are the braided monoidal categories.

Now some more homotopyy theory description of $DendroidalSets$.

Recall from the very conception of Quillen model categories that $SimplicialSets$ and $Top$ are Quillen model equivalent. Now

Theorem (Moerdijk, Cisinski) . On $DendroidalSets$ there is a monoidal model category structure. in which

- normal dendroidal sets (the CW-complex-like ones) are precisely the cofibrant objects

- $(\infty,1)$-categories are precisely the fibrant objects

- a morphism $A \to B$ between normal dendroidal sets is a weak equivalence precisely if for any $(\infty,1)$-operad $X$ the functor

$$j^* \tau hom(B,X) \to j^* \tau hom(A,X)$$

is an equivalence of categories. Here $\tau : DendroidaSets \to OrdinaryOperads$ is the left adjoint to the dendroidal nerve on ordinary operads and $j^*$ is pullback along the inclusion $j : Categories \to Operads$, so that $j^*$ applied to an operad picks out the category of unary operations inside the operad.

Observe that ordinary simplicial sets can be characterized as precisely those dendroidal sets which have a morphism to the tensor unit operad $U = \Omega(|)$ (the trivial operad on a single color), and that in fact the over-category $DendroidalSets\downarrow U$ is equivalent to $SimplicialSets$. We have

$$\array{ DendroidalSets\downarrow U &&\simeq&& SimplicialSets \\ & {}_{forgetful}\searrow && \swarrow_{Lan(i)} \\ && DendroidalSets } \,.$$

Under this identification the above model structure on $DendroidalSets$ induces a model structure on $SimplicialSets$ which is the one due to Joyal whose fibrant objects are the $(\infty,1)$-categories in their incarnation as quasi-categories.

With this observation in mind one can now try to generalize a couple of facts about the Joyal model structure on simplicial sets to the above model strucvture on $DendroidalSets$.

Theorem The homotopy coherent nerve functor $$SimplicialOperads \simeq TopologicalOperads \stackrel{D}{\to} DendroidalSets$$

induces a Quillen equivalence of model categories.