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<i<n.
  • 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<i\leq n. 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<i<n, 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<i<n. 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<i<n. 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<i<n. There are then inclusions (\Lambda^n_i)^\flat\hookrightarrow(\Lambda^n_i)^\sharp and (\Delta^n)^\flat\hookrightarrow(\Delta^n)^\sharp for 0<i<n. 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<i\leq n.

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:

Capture

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:

Capture 2

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.

Advertisements

One thought on “Marked Simplicial Sets – I

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s