Finite Kan representatives and ordered iterated loops

The finite sources used below are explicit: nondegenerate simplices of ${\mathrm{\Delta }}^r\times {\mathrm{\Delta }}^1$ are strictly increasing chains in the finite grid $[r]\times [1]$; those of $({\mathrm{\Delta }}^1{)}^d$ are strictly increasing chains in $\{0,1{\}}^d$. These products have finitely many such chains, and their quotients give the sphere cylinders and $T_d$. A map from a finite simplicial set to a degreewise sequential colimit factors through a stage: choose representatives for its finitely many nondegenerate simplices, then advance until their finitely many face relations hold. Any equality of two such maps holds at a common later stage by the same finite argument. Apply this to a horn and its containing simplex to obtain a filler in $Z_{\mathrm{\infty }}$ from a Kan stage. By [F3], homotopy classes use the finite source ${\mathrm{\Delta }}^r\mathrm{/}\mathrm{\partial }{\mathrm{\Delta }}^r$, and equality uses its finite product with ${\mathrm{\Delta }}^1$, fixing the basepoint. Representatives therefore lift to stages and every equality occurs at a later stage. This proves bijectivity of the canonical colimit map; functoriality makes it a group isomorphism. For the ordered calculation, [F4] gives a pointed homotopy equivalence $Z\to \mathrm{Sing}\mathrm{\mid }Z\mathrm{\mid }$. Postcomposition, followed by the pointed adjunction, gives $[A,Z{]}_{\ast }\cong [\mathrm{\mid }A\mathrm{\mid },\mathrm{\mid }Z\mathrm{\mid }{]}_{\ast }$, naturally for any pointed simplicial $A$. Here brackets denote maps modulo homotopies fixing the basepoint. With $A=T_r$, [F2] identifies its realization with the ordered cube sphere $I^r\mathrm{/}\mathrm{\partial }{I}^r$, so these classes compute ${\pi }_r(Z)$. All constructions preserve the specified basepoints.

By [F3], the pointed mapping objects and hence iterated loops of a Kan complex are Kan. Direct currying gives $[A,\mathrm{\Omega }B{]}_{\ast }\cong [K\wedge A,B{]}_{\ast }$: a representative $f$ goes to $(t,a)\mapsto f(a)(t)$, and the same formula with an extra ${\mathrm{\Delta }}^1$ parameter gives the inverse correspondence on pointed homotopies. Apply this repeatedly with $A=T_r$, using step 1.1, to obtain the bijections $D_{r,m}$. They are group isomorphisms: realizing evaluation and using the ordered smash identification in [F2] gives a continuous map $S^1\wedge \mathrm{\mid }\mathrm{\Omega }B\mathrm{\mid }\to \mathrm{\mid }B\mathrm{\mid }$. Its iterates define the same evaluation operation on continuous cube representatives, agreeing with currying on every realized simplicial representative. This operation preserves concatenation in a retained sphere coordinate. Concatenation in distinct cube coordinates has a common constant unit and satisfies the four-quarter-cube interchange identity, so those operations agree, as in the group-law argument of [F2]. This agrees with the prescribed first-coordinate group law. For ${\mathrm{\Omega }}^{m}u$, applying $u$ after all old loop evaluations adds its loop as the new innermost one. Uncurrying puts its circle to the left of all old loop circles, giving exactly $v\circ (1_K\wedge \widehat{f})$, where $\widehat{f}:T_{r+m}\to Z$ is the fully uncurried representative. This proves the displayed identity with ${\mathrm{\Sigma }}_{\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{t}}$. Evaluation commutes with all pointed maps, so the isomorphisms and identity are natural.