dSet

**homotopy theory, (∞,1)-category theory, homotopy type theory**

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…

models: topological, simplicial, localic, …

see also **algebraic topology**

**Introductions**

**Definitions**

**Paths and cylinders**

**Homotopy groups**

**Basic facts**

**Theorems**

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

Some structure carried by the category dSet of dendroidal sets:

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 $i^*[C,D]$.

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-categories | RelativeSimplicialSets | |

$\downarrow$simplicial nerve | $\downarrow$ | |||

SegalCategories | $\hookrightarrow$ | CompleteSegalSpaces | ||

(∞,1)Operad | ||||

SimplicialOperads | $-$homotopy coherent dendroidal nerve$\to$ | DendroidalSets | RelativeDendroidalSets | |

$\downarrow$dendroidal nerve | $\downarrow$ | |||

SegalOperads | $\hookrightarrow$ | DendroidalCompleteSegalSpaces | ||

$\mathcal{O}$Mon(∞,1)Cat | ||||

DendroidalCartesianFibrations |

See the list of references at *dendroidal set*.

For instance:

- Ieke Moerdijk and Ittay Weiss,
*Dendroidal Sets*, Algebraic and Geometric Topology, Volume 7, Number 3 (2007), 1441-1470. (doi:10.2140/agt.2007.7.1441, arXiv:math/0701293)

category: category