There are many ways to present the theory of -categories. One is via the Joyal model structure on :

**Theorem: **There exists a model structure on where:

- The cofibrations are the monomorphisms.
- The weak equivalences are those maps such that the induced map is an equivalence.

There is also a presentation via complete Segal spaces, which are simplicial objects of . 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 be a simplicial set. A marked simplicial set is a pair , where is a collection of edges of such that the collection of degenerate edges of is a subcollection of . The edges in 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 be a simplicial set. Then we can form a marked simplicial set where only the degenerate edges are marked. We may form another marked simplicial set where all of the edges of are marked.

Suppose and are marked simplicial sets. A map from to is a map such that . If it is obvious, we will not specify the collection of marked edges. In other words, we will usually write instead of if there is no risk of confusion. The collection of marked simplicial sets and maps between them forms a category, . Suppose is a simplicial set. We will write to denote .

In ordinary homotopy theory, anodyne maps play a very important role. Let 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 . Recall that the horn inclusions for are generating cofibrations for the Kan model structure. This means that a map is a fibration if it has the right lifting property with respect to the horn inclusions for .

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 for .
- The inclusion .
- The inclusion .
- The map for every Kan complex .

Let us recall the concept of a -Cartesian morphism for an inner fibration . We will elaborate on this more in a future post.

**Definition: **Suppose is an inner fibration and an edge in . is said to be -Cartesian if the map is a trivial Kan fibration.

**Proposition: **A map is a marked anodyne map if and only if it has the left lifting property with respect to every map in such that:

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

Therefore if is an -category, and both have the right lifting property with respect to every marked anodyne map. We can attempt to prove this statement in the case when is a Kan complex directly as well.

**Lemma: **There is only one unique map .

*Proof. *There is a canonical map given by the inclusion. It suffices to show that the only map is the identity. In the category there is only one map given by the identity (the morphisms in are nonincreasing maps); since is simply we observe that there is therefore only one map from to itself given by the identity.

**Theorem: **Let be a map of simplicial sets with the right lifting property with respect to for . Then the maps and have the right lifting property with respect to the class of all maps generating the class of marked anodyne maps. In addition, if has a unique lift with respect to for , then so do and .

*Proof. *Let us first prove that the maps and have the right lifting property with respect to the inclusions for . Assume is a map that satisfies the conditions in the proposition. It is then obvious that has the right lifting property with respect to the inclusions for . Consider the map . The map then has the right lifting property with respect to the inclusions for . There are then inclusions and for . We can then construct the lifting as the composition . Proving that and have the right lifting property with respect to the inclusion is obvious since has the right lifting property with respect to the inclusions for .

Let us now prove that and have the right lifting property with respect to the inclusion . There is an inclusion . Since we observe that since has the right lifting property with respect to . Indeed, this is obvious from the following diagram:

We will now approach the last class of maps, namely the maps for a Kan complex. Consider the diagram:

Asking that the lift exists amounts to constructing a map . Consider the map , and denote by the map on the underlying simplicial sets. Then the map can be chosen to be the map , and the diagram commutes.

For the last part we observe that if the lift is unique, then so are the maps and . The above Lemma completes the proof.

Suppose . Then is a Kan complex, and and both have the right lifting property with respect to every marked anodyne map.

Let 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.

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