Extension (simplicial set)

In the context of simplicial sets, an extension typically refers to the process of adding new simplices to a given simplicial set while preserving the simplicial identities. This can be done to satisfy certain properties or to build larger, more complex simplicial sets from smaller ones. There are different types of extensions, each with specific characteristics and purposes.

One important type of extension involves filling horns. A horn in a simplicial set is a collection of all faces of a simplex except for one. An extension in this context is the process of adding the missing face to the horn, creating the full simplex. This is often referred to as filling the horn. Kan complexes, or fibrant simplicial sets, are characterized by the property that all horns can be filled. This property is crucial in homotopy theory, as it allows for the construction of liftings and extensions of maps, analogous to lifting and extension properties in topological spaces.

Another kind of extension might involve adding new simplices to realize a given homology class or to modify the homotopy groups of a simplicial set. The process of constructing a simplicial set with prescribed homotopy groups often involves iteratively attaching cells, which can be viewed as a type of extension.

Formally, given a simplicial set $X$ and a simplicial set $Y$, an extension could be considered a monomorphism (injective map) $X \hookrightarrow Y$. The extension adds "new simplices" to X, those simplices in Y which are not in the image of the monomorphism.

The precise definition and usage of "extension" can vary depending on the specific context in which it is employed within simplicial set theory and homotopy theory. However, the underlying idea remains the same: adding new simplices to a simplicial set while respecting the simplicial structure.