LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The affine face maps satisfy the cosimplicial identities

Statement

For every $n \geq 2$ and every pair of integers $0 \leq i < j \leq n$ , the affine face maps of The standard topological simplex and its affine face maps satisfy $δ_{j} \circ δ_{i} = δ_{i} \circ δ_{j - 1} : Δ^{n - 2} \to Δ^{n} .$

Facts & Assumptions

Given: An integer $n \geq 2$ and indices $0 \leq i < j \leq n$ .

[L1]

For $n \geq 1$ , the face map $δ_{k} : Δ^{n - 1} \to Δ^{n}$ inserts a zero in the $k$ th coordinate (The standard topological simplex and its affine face maps).

Proof

technique · direct

1.1L1given

Let $t = (t_{0}, \dots, t_{n - 2}) \in Δ^{n - 2}$ . By [L1], the composite $δ_{j} δ_{i} (t)$ is obtained by first inserting $0$ in slot $i$ and then inserting $0$ in slot $j$ . Since $i < j$ , the second insertion happens strictly to the right of the first one, so the resulting $(n + 1)$ -tuple has zeros exactly in positions $i$ and $j$ .

2.1L1step 1.1∎

Again by [L1], the composite $δ_{i} δ_{j - 1} (t)$ first inserts $0$ in slot $j - 1$ and then inserts $0$ in slot $i$ . After the second insertion, the earlier zero has shifted one place to the right, so the final tuple also has zeros exactly in positions $i$ and $j$ , with every other coordinate of $t$ appearing in the same relative order as in step 1.1. Therefore the two composites agree coordinatewise.

Depends on

The standard topological simplex and its affine face maps

Used by

The singular boundary squares to zero Theorem

Dependency tree · one level

1 result within one dependency step of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Allen Hatcher, Algebraic Topology (standard reference, not scraped)