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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 contravariant simplicial mapping space functor takes the given pushout to the stated strict pullback. Since $X$ is a Kan complex, all four mapping spaces are Kan complexes. Moreover, the monomorphism $A\hookrightarrow B$ induces a Kan fibrationIndeed, a lifting problem against a horn is adjoint to a lifting problem for $X$ against the pushout-product of $A\hookrightarrow B$ with that horn inclusion; this pushout-product is an anodyne monomorphism, and $X$ fills it.

A strict pullback of fibrant simplicial sets along a fibration computes the homotopy pullback. The displayed pullback square is therefore also a homotopy pullback square.

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.

Products of Eilenberg–MacLane spaces satisfywhere the last isomorphism uses the Chinese remainder theorem. This space is $2$-connected, so the Hurewicz theorem identifiesThe assumed surjection on $H_3$ is an isomorphism because the group is finite. Hence $f$ is an isomorphism on ${\pi }_3$; all other homotopy groups of the source and target vanish. Thus $f$ is a weak homotopy equivalence, and the Whitehead theorem for Kan complexes makes it a homotopy equivalence.

Solved by gpt-5.6-sol high.

Regard the given short exact sequence as a degreewise short exact sequence of chain complexes concentrated in degree $n$. The inverse functor in the Dold–Kan correspondence is exact, so it produces a degreewise short exact sequence of simplicial abelian groupsThe last map is degreewise surjective and hence a Kan fibration. Its strict fiber is $K(A_1,n)$, and a strict fiber of a fibration computes the homotopy fiber. This proves the asserted homotopy fiber sequence of pointed Kan complexes.

Solved by gpt-5.6-sol high.

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

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 weak homotopy equivalence between connected CW complexes is a homotopy equivalence. The analogous statement holds for Kan complexes.