nLab
dSet

Contents

Context

Category theory

Homotopy theory

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

Contents

Idea

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

Properties

Some structure carried by the category dSet of dendroidal sets:

sSetsSet-enriched structure

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

Model category structure

The category dSetdSet 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 *:dSetsSeti^*: dSet \to sSet is a right Quillen functor (with respect to the model structure for quasi-categories) this imples that dSet is an sSet JoyalsSet_{Joyal}-enriched model category (but not, without further work, an sSet QuillensSet_{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 𝒪\mathcal{O}-monoidal (∞,1)-categories (fourth table).

general pattern
strict enrichment(∞,1)-category/(∞,1)-operad
\downarrow\downarrow
enriched (∞,1)-category\hookrightarrowinternal (∞,1)-category
(∞,1)Cat
SimplicialCategories-homotopy coherent nerve\toSimplicialSets/quasi-categoriesRelativeSimplicialSets
\downarrowsimplicial nerve\downarrow
SegalCategories\hookrightarrowCompleteSegalSpaces
(∞,1)Operad
SimplicialOperads-homotopy coherent dendroidal nerve\toDendroidalSetsRelativeDendroidalSets
\downarrowdendroidal nerve\downarrow
SegalOperads\hookrightarrowDendroidalCompleteSegalSpaces
𝒪\mathcal{O}Mon(∞,1)Cat
DendroidalCartesianFibrations

References

See the list of references at dendroidal set.

For instance:

category: category

Last revised on October 5, 2019 at 04:17:16. See the history of this page for a list of all contributions to it.