category theory

Contents

Idea

The category of dendroidal sets is often denoted $dSet$ or similar, following the common denotation sSet for the category of simplicial sets.

Properties

Some structure carried by the category dSet of dendroidal sets:

$sSet$-enriched structure

Using the fact that dSet is a closed monoidal category with internal hom dendroidal sets $[C,D]$ for dendroidal sets $C$ and $D$, and using the functor $i^* : dSet \to SSet$ we obtain canonically the structure of an simplicially enriched category / sSet-enriched category on $dSet$ with the hom-simplicial set between $C$ and $D$ being $i^*[C,D]$.

Model category structure

The category $dSet$ of dendroidal sets carries a monoidal model category-structure – the model structure on dendroidal sets – which serves to present the (∞,1)-category of (∞,1)-operads:

Together with the fact that $i^*: dSet \to sSet$ is a right Quillen functor (with respect to the model structure for quasi-categories) this imples that dSet is an $sSet_{Joyal}$-enriched model category (but not, without further work, an $sSet_{Quillen}$-enriched model category!).

The entries of the following table display model categories and Quillen equivalences between these that present the (∞,1)-category of (∞,1)-categories (second table), of (∞,1)-operads (third table) and of $\mathcal{O}$-monoidal (∞,1)-categories (fourth table).

general pattern
strict enrichment(∞,1)-category/(∞,1)-operad
$\downarrow$$\downarrow$
enriched (∞,1)-category$\hookrightarrow$internal (∞,1)-category
(∞,1)Cat
SimplicialCategories$-$homotopy coherent nerve$\to$SimplicialSets/quasi-categoriesRelativeSimplicialSets
$\downarrow$simplicial nerve$\downarrow$
SegalCategories$\hookrightarrow$CompleteSegalSpaces
(∞,1)Operad
SimplicialOperads$-$homotopy coherent dendroidal nerve$\to$DendroidalSetsRelativeDendroidalSets
$\downarrow$dendroidal nerve$\downarrow$
SegalOperads$\hookrightarrow$DendroidalCompleteSegalSpaces
$\mathcal{O}$Mon(∞,1)Cat
DendroidalCartesianFibrations

References

See the list of references at dendroidal set.

For instance:

category: category