sdOpIter_comp_sdOp
plain-language theorem explainer
The k-fold iterate of the singular barycentric subdivision operator on C_n(X) commutes with one further subdivision step: S^k ∘ S = S ∘ S^k. Anyone building iterated subdivision chain homotopies cites this. The proof is induction on k, using the recursive definition of the iterate and category associativity.
Claim. For any topological space $X$ and degrees $n,k\in\mathbb{N}$, if $S$ is the singular barycentric subdivision operator on the degree-$n$ singular chain group of $X$ and $S^k$ is its $k$-fold iterate, then $S^k\circ S=S\circ S^k$ as morphisms $C_n(X)\to C_n(X)$.
background
In this module the singular barycentric subdivision operator $S=\mathrm{sdOp},X,n$ is the endomorphism of the free abelian group $C_n(X)$ of singular $n$-chains obtained by descending the generator-wise subdivision map. Its iterates are defined recursively: $S^0=\mathrm{id}$ and $S^{k+1}=S^k\circ S$.
The ambient setting is the singular chain complex of a space in $\mathbf{Top}$, with the usual boundary $\partial$ and the prism/cone constructions that produce chain homotopies. Commutation of $S$ past its own iterates is the elementary algebraic fact needed before one can telescope the prism operator into an iterated homotopy $T_k$ satisfying $\partial T_k+T_k\partial=\mathrm{id}-S^k$.
Upstream, the zero and successor clauses of the iterate (sdOpIter_zero, sdOpIter_succ) fix the recursion; category identities id_comp/comp_id and associativity rearrange composites.
proof idea
Induction on the iteration count $k$.
Base case $k=0$: both sides reduce to $S$ by the zero clause of the iterate together with left and right unit laws for composition.
Successor step: expand $S^{k+1}=S^k\circ S$ on the left, reassociate, apply the inductive hypothesis $S^k\circ S=S\circ S^k$, reassociate again, and reassemble the right-hand side as $S\circ S^{k+1}$ via the successor clause. Purely equational; no topology is used.
why it matters
This lemma is the commutation step inside the inductive proofs of the telescoped chain-homotopy identities tOpIter_chain_homotopy_succ and tOpIter_chain_homotopy_zero. Those state $\partial\circ T_k+T_k\circ\partial=\mathrm{id}-S^k$ (and the degree-zero truncation $\partial\circ T_k=\mathrm{id}-S^k$), which are the standard algebraic engine showing that iterated barycentric subdivision is chain-homotopic to the identity.
In the Recognition Science foundation stack this sits in the singular-homology layer that underwrites continuum and covering arguments feeding the forcing chain, rather than at a numbered T0–T8 landmark. It closes a purely formal gap: without $S^k S=SS^k$ the inductive step that builds $T_{k+1}$ from $T_k$ and one more prism cannot rearrange the composites.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.