nLab filler

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 $X$.

Related concepts

factorization system

homotopy lifting property

orthogonality