# Contents

## Fillers of commutative squares

If

$\array { O_{0,1} & \overset{f_1}{\longrightarrow} & O_{1,1} \\ \downarrow h_0 & & \downarrow h_1 \\ O_{0,0} & \overset{f_0}{\longrightarrow} & O_{1,0} \\ }$

is a commutative diagram in a category $\mathcal{C}$, then a filler (synonyms: diagonal fill-in, lift) is:

• morphism $j\colon O_{0,0}\rightarrow O_{1,1}$ in $\mathcal{C}$ making both triangles created commute. (That is, $f_0 = h_1 j$ and $f_1 = j h_0$.)

In certain contexts, the problem of whether there exists a filler in this sense is called a lifting problem.

If $j$ is uniquely determined (in an appropriate sense), then $h_0$ is said to be orthogonal to $h_1$: see orthogonality.

The concept of filler plays an important role in homotopy theory; see for example model category.

## Horn fillers for simplicial sets

The term is used in the context of horns in simplicial sets and related structures. The term makes it possible to summarize the definition of a Kan complex in one sentence: a Kan complex is a simplicial set in which every horn has a filler.

Of course this is a special case of the preceding notion of filler: a horn filler for a horn $f: \Lambda_n^j \to X$ in a simplicial set $X$ is a diagonal fill-in of a commutative square

$\array{ \Lambda_n^j & \hookrightarrow & \Delta_n \\ \mathllap{f} \downarrow & & \downarrow \\ X & \to & 1 }$

going from $\Delta_n$ to $X$.

factorization system

homotopy lifting property

orthogonality

Last revised on July 19, 2017 at 14:20:44. See the history of this page for a list of all contributions to it.