Whitney-sum coalgebra on stable unoriented Thom cohomology

Assume AC, inherited from the cited bundle, cohomology, or operation suppliers. Let $\mathcal{P}={\mathbb{F}}_2[w_1,w_2,\dots ]$ with $\mathrm{\mid }{w}_k\mathrm{\mid }=k$, and use the graded identification $M=\mathcal{P}U$ with $\mathrm{\mid }U\mathrm{\mid }=0$ and $w_0=1$. Extend ${\mathrm{\Delta }}_P(w_k)={\sum }_{i+j=k}{w}_i\otimes w_j$ uniquely to a unital algebra homomorphism ${\mathrm{\Delta }}_P:\mathcal{P}\to \mathcal{P}\otimes \mathcal{P}$. Define the candidate maps ${\mathrm{\Delta }}_M(fU)={\mathrm{\Delta }}_P(f)(U\otimes U)$, ${ϵ}_M(fU)=f(0)$, and $\eta (1)=U$. These formulas specify the proposed Whitney-sum operations; their well-definedness, inverse-limit compatibility, and coalgebra axioms are proved in Whitney sum defines the connected coalgebra on stable Thom cohomology ↗.

Each polynomial has finite support and each generator coproduct is a finite sum, so these are degree-preserving linear maps on $M$ and the ordinary graded tensor product. The rankwise geometric interpretation is the Thom pullback of an external Whitney sum, with

$w_{k}U⟼{\sum }_{i+j=k}(w_{i}U)\otimes (w_{j}U),U⟼U\otimes U\mathrm{.}$

The cited well-definedness lemma proves agreement with that geometric pullback, stabilization compatibility, coassociativity, and both counit identities; these are obligations of the construction rather than additional premises.