## Marked Simplicial Sets – I

There are many ways to present the theory of $\infty$-categories. One is via the Joyal model structure on $\mathscr{S}\mathrm{et}_\mathbf{\Delta}$:

Theorem: There exists a model structure on $\mathscr{S}\mathrm{et}_\mathbf{\Delta}$ where:

• The cofibrations are the monomorphisms.
• The weak equivalences are those maps $f:X\to Y$ such that the induced map $\mathfrak{C}[f]:\mathfrak{C}[X]\to\mathfrak{C}[Y]$ is an equivalence.

There is also a presentation via complete Segal spaces, which are simplicial objects of $\mathscr{S}\mathrm{et}_\mathbf{\Delta}$. It is natural to ask for a simpler definition, something which requires us to specify a minimal amount of information. One such model is the theory of \textit{marked simplicial sets}, introduced by Lurie. A marked simplicial set is a simplicial set with a collection of distinguished edges. More precisely:

Definition: Let $X$ be a simplicial set. A marked simplicial set is a pair $(X,\Gamma)$, where $\Gamma$ is a collection of edges of $X$ such that the collection $\mathrm{deg}_1(X)$ of degenerate edges of $X$ is a subcollection of $\Gamma$. The edges in $\Gamma$ are called marked edges.

It should be obvious that there are two canonical ways of presenting any simplicial set as a marked simplicial set. Let $X$ be a simplicial set. Then we can form a marked simplicial set $X^\flat$ where only the degenerate edges are marked. We may form another marked simplicial set $X^\sharp$ where all of the edges of $X$ are marked.

Suppose $(X,\Gamma)$ and $(Y,\Omega)$ are marked simplicial sets. A map from $(X,\Gamma)$ to $(Y,\Gamma)$ is a map $f:X\to Y$ such that $f(\Gamma)\subseteq \Omega$. If it is obvious, we will not specify the collection of marked edges. In other words, we will usually write $X$ instead of $(X,\Gamma)$ if there is no risk of confusion. The collection of marked simplicial sets and maps between them forms a category, $\mathscr{S}\mathrm{et}_\mathbf{\Delta}^+$. Suppose $X$ is a simplicial set. We will write $(\mathscr{S}\mathrm{et}_\mathbf{\Delta}^+)_{/X}$ to denote $(\mathscr{S}\mathrm{et}_\mathbf{\Delta}^+)_{/X^\sharp}$.

In ordinary homotopy theory, anodyne maps play a very important role. Let $\mathscr{S}\mathrm{et}_\mathbf{\Delta}$ denote the model category of simplicial sets with the Kan model structure. A map is called \textit{anodyne} if it has the right lifting property with respect to every fibration. In other words, the collection of anodyne maps is simply the collection of trivial cofibrations in $\mathscr{S}\mathrm{et}_\mathbf{\Delta}$. Recall that the horn inclusions $\Lambda^n_i\hookrightarrow\Delta^n$ for $0\leq i\leq n$ are generating cofibrations for the Kan model structure. This means that a map $X\to {\Delta^0}$ is a fibration if it has the right lifting property with respect to the horn inclusions $\Lambda^n_i\hookrightarrow\Delta^n$ for $0\leq i \leq n$.

A very similar construction can be done in the setting of marked simplicial sets:

Definition: The class of marked anodyne maps is the smallest weakly saturated class of maps generated by:

• The inclusions $(\Lambda^n_i)^\flat\hookrightarrow(\Delta^n)^\flat$ for $0.
• The inclusion $(\Lambda^n_n,(\mathrm{deg}_1(\Delta^n)\cup\Delta^{\{n-1,n\}})\cap(\Lambda^n_n)_1)\hookrightarrow(\Delta^n,\mathrm{deg}_1(\Delta^n)\cup\Delta^{\{n-1,n\}})$.
• The inclusion $(\Lambda^2_1)^\sharp\coprod_{(\Lambda^2_1)^\flat}(\Delta^2)^\flat\to(\Delta^2)^\sharp$.
• The map $K^\flat\to K^\sharp$ for every Kan complex $K$.

Let us recall the concept of a $p$-Cartesian morphism for an inner fibration $p:X\to S$. We will elaborate on this more in a future post.

Definition: Suppose $p:X\to S$ is an inner fibration and $f:x\to y$ an edge in $X$. $f$ is said to be $p$-Cartesian if the map $X_{/f}\to X_{/y}\times_{S_{/p(y)}}S_{/p(f)}$ is a trivial Kan fibration.

Proposition: A map $p:X\to S$ is a marked anodyne map if and only if it has the left lifting property with respect to every map $i: Y\to Z$ in $\mathscr{S}\mathrm{et}_\mathbf{\Delta}^+$ such that:

• $i$ is an inner fibration on the underlying simplicial sets.
• An edge of $Y$ is marked if and only if its image under $i$ is and it is $i$-Cartesian.
• If $f$ is a map in $Z$ whose target is the image of a $0$-simplice of $Y$, then $f$ can be pulled back via $i$ to a map in $Y$.

Therefore if $\mathscr{C}$ is an $\infty$-category, $\mathscr{C}^\flat\to{\Delta^0}^\flat$ and $\mathscr{C}^\sharp\to{\Delta^0}^\sharp$ both have the right lifting property with respect to every marked anodyne map. We can attempt to prove this statement in the case when $\mathscr{C}$ is a Kan complex directly as well.

Lemma: There is only one unique map $(\Delta^n)^\flat\to(\Delta^n)^\sharp$.

Proof. There is a canonical map $(\Delta^n)^\flat\to(\Delta^n)^\sharp$ given by the inclusion. It suffices to show that the only map $\Delta^n\to\Delta^n$ is the identity. In the category $\mathbf{\mathbf{\Delta}}$ there is only one map $[n]\to[n]$ given by the identity (the morphisms in $\mathbf{\Delta}$ are nonincreasing maps); since $\Delta^n$ is simply $\mathrm{N}([n])$ we observe that there is therefore only one map from $\Delta^n$ to itself given by the identity.

Theorem: Let $p:X\to S$ be a map of simplicial sets with the right lifting property with respect to $\Lambda^n_i\hookrightarrow\Delta^n$ for $0. Then the maps $p^\sharp$ and $p^\flat$ have the right lifting property with respect to the class of all maps generating the class of marked anodyne maps. In addition, if $p$ has a unique lift with respect to $\Lambda^n_i\hookrightarrow\Delta^n$ for $0, then so do $p^\flat$ and $p^\sharp$.

Proof. Let us first prove that the maps $p^\sharp$ and $p^\flat$ have the right lifting property with respect to the inclusions $(\Lambda^n_i)^\flat\hookrightarrow(\Delta^n)^\flat$ for $0. Assume $p$ is a map that satisfies the conditions in the proposition. It is then obvious that $p^\flat$ has the right lifting property with respect to the inclusions $(\Lambda^n_i)^\flat\hookrightarrow(\Delta^n)^\flat$ for $0. Consider the map $p:X\to S$. The map $p^\sharp$ then has the right lifting property with respect to the inclusions $(\Lambda^n_i)^\sharp\hookrightarrow(\Delta^n)^\sharp$ for $0. There are then inclusions $(\Lambda^n_i)^\flat\hookrightarrow(\Lambda^n_i)^\sharp$ and $(\Delta^n)^\flat\hookrightarrow(\Delta^n)^\sharp$ for $0. We can then construct the lifting $(\Delta^n)^\flat\to X$ as the composition $(\Delta^n)^\flat\hookrightarrow(\Delta^n)^\sharp\to X^\sharp$. Proving that $p^\flat$ and $p^\sharp$ have the right lifting property with respect to the inclusion $(\Lambda^n_n,(\mathrm{deg}_1(\Delta^n)\cup\Delta^{\{n-1,n\}})\cap(\Lambda^n_n)_1)\hookrightarrow(\Delta^n,\mathrm{deg}_1(\Delta^n)\cup\Delta^{\{n-1,n\}})$ is obvious since $p$ has the right lifting property with respect to the inclusions $\Lambda^n_i\hookrightarrow\Delta^n$ for $0.

Let us now prove that $p^\flat$ and $p^\sharp$ have the right lifting property with respect to the inclusion $(\Lambda^2_1)^\sharp\coprod_{(\Lambda^2_1)^\flat}(\Delta^2)^\flat\to(\Delta^2)^\sharp$. There is an inclusion $(\Lambda^2_1)^\sharp\hookrightarrow(\Lambda^2_1)^\sharp\coprod_{(\Lambda^2_1)^\flat}(\Delta^2)^\flat$. Since $(\Lambda^2_1)^\sharp\coprod_{(\Lambda^2_1)^\flat}(\Delta^2)^\flat\hookrightarrow(\Delta^2)^\sharp$ we observe that since $p^\sharp$ has the right lifting property with respect to $(\Lambda^2_1)^\sharp\coprod_{(\Lambda^2_1)^\flat}(\Delta^2)^\flat$. Indeed, this is obvious from the following diagram:

We will now approach the last class of maps, namely the maps $K^\flat\to K^\sharp$ for $K$ a Kan complex. Consider the diagram:

Asking that the lift exists amounts to constructing a map $K^\sharp\to X^\sharp$. Consider the map $K^\flat\to X^\sharp$, and denote by $f$ the map on the underlying simplicial sets. Then the map $K^\sharp\to X^\sharp$ can be chosen to be the map $f^\sharp$, and the diagram commutes.

For the last part we observe that if the lift $\Delta^n\to X$ is unique, then so are the maps $(\Delta^n)^\sharp\to X^\sharp$ and $(\Delta^n)^\flat\to X^\flat$. The above Lemma completes the proof.

Suppose $S={\Delta^0}$. Then $X$ is a Kan complex, and $X^\flat\to{\Delta^0}$ and $X^\sharp\to{\Delta^0}$ both have the right lifting property with respect to every marked anodyne map.

Let $X\to Y$ be a map of marked simplicial sets. It is a cofibration if the underlying map of simplicial sets is a monomorphism. We will conclude this section with an important fact to remember about marked anodyne maps.

Proposition: Marked anodyne maps are closed under smash products with arbitrary cofibrations.

In a future post, we will talk about Cartesian fibrations.