nLab
marked simplicial set

Idea

Marked simplicial sets are simplicial sets with a little bit of extra structure that allows to model Cartesian fibrations of simplicial sets precisely as the fibrations in the model structure on marked simplicial over-sets: the marked edges in a (fibrant) marked simplicial set are the Cartesian morphisms in the coprresponding (∞,1)-category.

Definition

A marked simplicial set is

• a pair $(S,E)$ consisting of

  • a simplicial set $S$

  • and a subset $E\subset S_1$ of edges of $S$, called the marked edges,

• such that

  • all degenerate edges are marked edges.

A morphism $(S,E)\to (S\prime ,E\prime )$ of marked simplicial sets is a morphism $f:S\to S\prime$ of simplicial sets that carries marked edges to marked edges in that $f(E)\subset E\prime$.

Notation

• The category of marked simplicial sets is denoted ${\mathrm{sSet}}^{+}$.

• for $S$ a simplicial set let

  • $S^{♭}$ or $S^{\min }$ be the minimally marked simplicial set: only the degenerate edges are marked;

  • $S^{\#}$ or $S^{\max }$ be the maximally marked simplicial set: every edges is marked.

• for $p:X\to S$ a Cartesian fibration of simplicial sets let

  • $X^{♯}$ or $X^{\mathrm{cart}}$ be the cartesian marked simplicial set: precisely the $p$-cartesian morphisms are marked

• For $X$ and $Y$ marked simplicial sets let

  • ${\mathrm{Map}}^{♭}(X,Y)$ be the simplicial set underlying the internal hom $Y^X$

  • ${\mathrm{Map}}^{\#}(X,Y)$ the simplicial set consisting of all simplices $\sigma \in {\mathrm{Map}}^{♭}(X,Y)$ such that every edge of $\Sigma$ is a marked edge of $Y^X$.

Enrichment

The category of marked simplicial sets is cartesian closed.

• The $n$-simplices of the internal hom $Y^X$ are simplicial maps $X\times {\Delta }^n\to Y$ such that when you restrict $X_1\times {\Delta }_1^n\to Y_1$ to $E\times {\Delta }_0^n$ (where $E$ is the set of marked edges of $X$), this morphism factors through the marked edges of $Y$.

• The marked edges of $Y^X$ are those simplicial maps $X\times {\Delta }^1\to Y$ such that the restriction of $X_1\times {\Delta }_1^1\to Y_1$ to $E\times {\Delta }_1^1$ factors though the marked edges of $Y$. In the presence of the previous condition, this says that when you apply the homotopy $X\times {\Delta }^1\to Y$ to a marked edge of $X$ paired with the identity at $[1]$, the result should be marked.

There are functors

$$\begin{array}{ccc}& \stackrel{(-{)}^{♭}}{\to }& \\ & \stackrel{(-{)}^{♭}}{\leftarrow }& \\ \mathrm{sSet}& \stackrel{(-{)}^{♯}}{\to }& {\mathrm{sSet}}^{+}\\ & \stackrel{(-{)}^{♯}}{\leftarrow }& \end{array}$$\array{ & \stackrel{(-)^{\flat}}{\to} & \\ & \stackrel{(-)^{\flat}}{\leftarrow} & \\ sSet & \stackrel{(-)^{\sharp}}{\to} & sSet^+ \\ & \stackrel{(-)^{\sharp}}{\leftarrow} & }

with $(-{)}^{♭}⊣(-{)}^{♭}⊣(-{)}^{♯}⊣(-{)}^{♯}$.

• The hom-objects ${\mathrm{Map}}^{\#}(X,Y)=(Y^X{)}^{\#}$ make ${\mathrm{sSet}}^{+}$ a simplicial category.

Applications

The main point of marked simplicial sets is to carry the model structure on marked simplicial over-sets. This is a model for the (∞,1)-category of cartesian fibrations of (∞,1)-categories.

References

Section 3.1 of

• Jacob Lurie, Higher Topos Theory