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.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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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

1.1

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 $j$ .

L1given

2.1

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.

L1step 1.1∎

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)