= Second barycentric subdivision of a delta-complex
{title2=$\operatorname{sd}^2X\text{ is simplicial}$}
The first <barycentric subdivision> of a <delta-complex> has distinct vertices within each simplex. Its vertices correspond to an increasing chain of original faces, and their strictly different <dimensions> prevent identifications within that chain. Distinct simplexes may still share the same global vertex set, as happens to the two halves of a subdivided loop edge.
For a delta-complex with distinct vertices within each simplex, a face inside a fixed simplex is uniquely determined by its vertex subset. In its barycentric subdivision, a vertex records an actual face, including its cell identity. A simplex records a chain of these faces; its vertex set recovers the unique order by dimension and each unique face inclusion. Thus no two simplexes have the same vertex set, and intersections are their common faces. Applying this to the first subdivision proves that the second subdivision is an <abstract simplicial complex>. The subdivision maps are affine on smaller simplexes and compatible on the glued faces, so they induce a <homeomorphism> of realizations.<Algebraic Topology by Allen Hatcher>[§2.1, exercise 23, p. 133: the two successive subdivision steps]{ref}
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