Simplicial sets, homotopies and trivial Kan fibrations

A simplicial set is a contravariant functor from the simplex category $\mathrm{\Delta }$ to the category of sets (Simplicial objects, simplicial commutative rings and homotopy groups, Covariant functor, identity functor, composite functor, and contravariant functor); thus a simplicial set $X$ assigns a set $X_k$ to each $[k]$ and a map $X_k\to X_l$ to each order-preserving $[l]\to [k]$, contravariantly.

For $n\ge 0$ put $\mathrm{\Delta }[n{]}_k={\mathrm{Hom}}_{\mathrm{\Delta }}([k],[n])$, the standard $n$-simplex. Its boundary $\mathrm{\partial }\mathrm{\Delta }[n]$ is the subfunctor consisting of the non-surjective maps $[k]\to [n]$; this is a simplicial set, and $\mathrm{\partial }\mathrm{\Delta }[0]$ is empty, since the only map $[0]\to [0]$ is surjective. A non-surjective order-preserving map factors through a proper face $[n-1]\to [n]$, and a surjective map contains the distinguished nondegenerate $n$-simplex and therefore lies outside the boundary. Thus, for $n\ge 1$, the boundary is exactly the union of the images of the proper face inclusions $\mathrm{\Delta }[n-1]\to \mathrm{\Delta }[n]$. Products and pullbacks of simplicial sets are computed degreewise, because the functor category $\mathrm{Fun}({\mathrm{\Delta }}^{\mathrm{o}\mathrm{p}},\mathrm{S}\mathrm{e}\mathrm{t})$ has limits and colimits formed objectwise.

A simplicial homotopy from $f$ to $g$, for maps $f,g\text{\hspace{0.17em}⁣}:X\to Y$ of simplicial sets, is a map $H\text{\hspace{0.17em}⁣}:X\times \mathrm{\Delta }[1]\to Y$ whose restrictions to $X\times \{0\}$ and $X\times \{1\}$ are $f$ and $g$; here $\mathrm{\Delta }[1]={\mathrm{Hom}}_{\mathrm{\Delta }}(-,[1])$ and $\{0\},\{1\}$ are the two vertices of $\mathrm{\Delta }[1]$. A simplicial set is contractible here when it is homotopy equivalent in this sense to the one-point constant simplicial set $\mathrm{\Delta }[0]$, i.e. when there are maps in both directions whose composites are simplicially homotopic to the identities.

A map $p\text{\hspace{0.17em}⁣}:X\to Y$ of simplicial sets is a trivial Kan fibration when every commutative square $\begin{array}{ccc}\mathrm{\partial }\mathrm{\Delta }[n]& ⟶& X\\ \downarrow & & \downarrow \\ \mathrm{\Delta }[n]& ⟶& Y\end{array}$ with $n\ge 0$ admits a diagonal lift $\mathrm{\Delta }[n]\to X$ making both triangles commute. In degree zero the left vertical map is the inclusion $\mathrm{\varnothing }\to \mathrm{\Delta }[0]$, so the lifting condition says exactly that $p_0\text{\hspace{0.17em}⁣}:X_0\to Y_0$ is surjective. The term thus specifies lifting of boundaries, not merely a quasi-isomorphism of the associated complexes, and no choice principle is needed to state it.