Standard simplex ( $Δ^{n}$ )

The standard simplicial

n

-simplex is the representable simplicial set

Δ_{m}^{n} = Hom_{Δ} ([m], [n]) .

Simplicial horn ( $Λ_{i}^{n}$ )

The

i

th horn

Λ_{i}^{n} \subseteq Δ^{n}

is the union of all codimension-one faces except the face opposite vertex

i

A horn

Λ_{i}^{n}

is inner when

0 < i < n

and outer when

i = 0

i = n

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