tOpIter
plain-language theorem explainer
Defines the telescoped chain homotopy T_k : C_n(X) → C_{n+1}(X) by T_0 = 0 and T_{k+1} = T + S ∘ T_k, where T is the barycentric subdivision homotopy and S is the subdivision operator. Algebraic topologists cite it when converting the identity-minus-iterated-subdivision into an explicit boundary. The definition is a plain recursive equation on morphisms of singular chain groups.
Claim. For a topological space $X$ and degree $n$, the telescoped homotopy $T_k : C_n(X;\mathbb{Z}) \to C_{n+1}(X;\mathbb{Z})$ is defined by $T_0 = 0$ and $T_{k+1} = T + S \circ T_k$, where $S$ is barycentric subdivision on $C_n$ and $T$ is the standard subdivision chain homotopy $C_n \to C_{n+1}$.
background
In singular homology one works with the free $\mathbb{Z}$-modules $C_n(X)$ on continuous maps $\Delta^n \to X$. The barycentric subdivision operator $S : C_n(X) \to C_n(X)$ replaces each simplex by the sum of its barycentric pieces; it is chain-homotopic to the identity via a prism operator $T : C_n(X) \to C_{n+1}(X)$.
Iterating $S$ produces chains whose simplices become arbitrarily small in diameter. The algebraic identity that makes this useful is the telescoping relation $\mathrm{id} - S^k = \partial T_k + T_k \partial$ (up to degree conventions), so that on cycles one has $z - S^k z = \partial(T_k z)$. The present definition packages the operators $T_k$ that appear on the right-hand side.
The ambient module is the Recognition Science singular-chain formalization: $C_n(X)$ is realized as a coproduct of copies of $\mathbb{Z}$ indexed by singular simplices, and both $S$ and $T$ are induced by generators on those simplices.
proof idea
Pure recursive definition on the iteration index. Base case $k=0$ is the zero morphism $C_n(X)\to C_{n+1}(X)$. The successor clause adds the basic prism operator $tOp$ to the composite of the subdivision operator $sdOp$ with the already-constructed $T_k$. No lemmas are invoked; the equation is the content.
why it matters
This operator is the algebraic engine behind the small-simplex argument for Mayer–Vietoris in the Recognition Science singular-homology stack. Downstream lemmas apply it elementwise: on cycles one obtains $z - S^k z = \partial(T_k z)$, and on boundaries the same identity holds after a degree split. Those identities feed injectivity and surjectivity of the small-chain inclusion into ordinary chains, which in turn show that the small-span map induces an isomorphism on homology.
In the broader foundation layer this is classical algebraic topology scaffolding rather than a Recognition-Science forcing step (T0–T8). It exists so that open-cover arguments and excision can be stated inside the same Lean development that later hosts the eight-tick and dimensional constraints. Closing the small-simplex isomorphism is a prerequisite for any homology-level comparison theorems that the framework may need.
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