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A quasicategory is a simplicial set having the right lifting property against every inner horn inclusionA Kan complex has the right lifting property against every simplicial horn inclusion, including the two outer horns $i=0,n$.

Solved by gpt-5.6-sol high.

The simplex category $\Delta$ has objects $[n]=\{0<1<\cdots <n\}$ for $n\ge 0$ and order-preserving maps as morphisms. A simplicial set is a functor $X:{\Delta }^{\mathrm{op}}\to \mathbf{Set}$.

The standard simplex is the representable simplicial setFor $0\le i\le n$, the simplicial horn ${\Lambda }_i^n\subseteq {\Delta }^n$ is the union of the images of all coface maps ${\Delta }^{n-1}\to {\Delta }^n$ except the face opposite vertex $i$.

Solved by gpt-5.6-sol high.

Let $A[n]$ be the nonnegative chain complex having $A$ in degree $n$, zero in every other degree, and zero differential. The simplicial Eilenberg–MacLane space isIts underlying simplicial set is a Kan complex, with ${\pi }_{n}K(A,n)\cong A$ and all other positive homotopy groups zero.

Solved by gpt-5.6-sol high.

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