vInc
plain-language theorem explainer
Degree-n morphism from singular chains on V into the small (U,V)-chain group. Anyone assembling the concrete Mayer–Vietoris short complex cites it as the V-leg of the middle map. The construction is the coproduct universal property: each generator simplex of V is sent to the small generator named by the V-index map.
Claim. For each degree $n$, there is a $\mathbb{Z}$-module morphism $C_n(V)\to C_n^{U,V}$ sending the generator of a singular $n$-simplex $\sigma$ on $V$ to the generator of the corresponding small simplex (the image of $\sigma$ under the index map from $V$-simplices into the small-simplex index set).
background
In the singular Mayer–Vietoris setup one works with an open cover ${U,V}$ of a space $X$. Ordinary singular chains $C_n(U)$ and $C_n(V)$ are free on all singular $n$-simplices landing in $U$ or $V$. The small chain group $C_n^{U,V}$ is the free $\mathbb{Z}$-module on those simplices that are small for the cover (each lands entirely in $U$ or entirely in $V$), presented as a coproduct of copies of $\mathbb{Z}$.
The generator map attaches to each small-simplex index the corresponding summand inclusion into that coproduct. The V-index map takes an ordinary $V$-simplex index and packages it as a small index, using that the push of a $V$-simplex remains small for the right set of the cover.
Locally this module builds the degreewise short complex whose middle arrow is the biproduct descent of the two inclusions from $C_n(U)$ and $C_n(V)$ into small chains; exactness of that complex is the algebraic heart of Mayer–Vietoris.
proof idea
One-line definition by the coproduct universal property. The domain $C_n(V)$ is itself a coproduct of $\mathbb{Z}$ over $V$-simplex indices, so a morphism out of it is uniquely determined by where generators go. The body is Sigma.desc of the composite that first applies the V-index map and then the small-generator inclusion. No further lemmas are invoked at the definition site; the companion generator identity is proved separately by the standard Sigma.ι_desc computation.
why it matters
This is the V-component of the middle map in the concrete degree-$n$ Mayer–Vietoris short complex of $\mathbb{Z}$-modules. Downstream it appears in the biproduct descent that defines that short complex, in the proof that the two routes from intersection chains into small chains agree, in the middle-exactness lemma (a cancelling pair of $U$- and $V$-chains comes from the intersection), and in the epi and exactness statements for the short complex.
In the Recognition foundation stack, singular Mayer–Vietoris supplies the homological glue for covers used when comparing local recognition data on overlapping charts. The declaration itself is pure algebraic topology scaffolding; it does not invoke the forcing chain (T5–T8), the Recognition Composition Law, or the $\varphi$-ladder, but it is part of the singular-homology toolkit those geometric arguments rely on when they pass to homology of unions.
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