Simplicial homotopy theory

Simplicial homotopy theory models spaces and higher categories using simplicial sets.

The simplex category has objects the finite nonempty ordered sets $[n]=\{0<\cdots <n\}$ and morphisms the order-preserving maps.

A simplicial set is a functor from the opposite of the simplex category to sets. Its face and degeneracy maps satisfy the simplicial identities.

The geometric realization of a simplicial set glues one topological simplex for each simplicial simplex, using the face and degeneracy maps as the gluing relations.

The singular simplicial set of a topological space $X$ has the continuous maps ${\Delta }^n\to X$ as its $n$-simplices. Evaluation gives a natural weak homotopy equivalence $|\mathrm{Sing}X|\to X$.

The standard simplicial $n$-simplex is the representable simplicial set

The $i$th horn ${\Lambda }_i^n\subseteq {\Delta }^n$ is the union of all codimension-one faces except the face opposite vertex $i$.

A horn ${\Lambda }_i^n$ is inner when $0<i<n$ and outer when $i=0$ or $i=n$.

A quasicategory is a simplicial set in which every inner horn has a filler.

A Kan complex is a simplicial set in which every simplicial horn, including every outer horn, has a filler.

The nerve of a category $\mathcal{C}$ hasso an $n$-simplex is a string of $n$ composable morphisms. Every inner horn in a nerve has a unique filler.

The simplicial mapping space is defined byIf $A\hookrightarrow B$ is a monomorphism and $X$ is a Kan complex, restriction $\underline{\mathrm{Hom}}(B,X)\to \underline{\mathrm{Hom}}(A,X)$ is a Kan fibration.

A simplicial abelian group is a simplicial object in abelian groups. Its underlying simplicial set is always a Kan complex.

For a simplicial abelian group $A$, one convention for normalized chains isEquivalently, it is the quotient of the unnormalized chain complex by its degenerate subcomplex.

The normalized-chain functor gives an equivalence between simplicial abelian groups and nonnegatively graded chain complexes of abelian groups. Its inverse is the Eilenberg–MacLane denormalization functor $K$.