Topological rigidity as a monoidal equivalence

Laurent Poinsot (1, 2)

1 CReA - Centre de Recherche de l'École de l'air
2 LIPN - Laboratoire d'Informatique de Paris-Nord

A topological commutative ring is said to be rigid when for every set $X$, the topological dual of the $X$-fold topological product of the ring is isomorphic to the free module over $X$. Examples are fields with a ring topology, discrete rings, and normed algebras. Rigidity translates into a dual equivalence between categories of free modules and of ``topologically-free'' modules and, with a suitable topological tensor product for the latter, one proves that it lifts to an equivalence between monoids in this category (some suitably generalized topological algebras) and some coalgebras. In particular, we provide a description of its relationship with the standard duality between algebras and coalgebras, namely finite duality.

Mots clés: Topological dual space; topological basis; coalgebras; finite duality

Domaines: Algèbre commutative [math.AC]; Topologie générale [math.GN]; Catégories et ensembles [math.CT]

Laurent Poinsot. Topological rigidity as a monoidal equivalence. Communications in Algebra, 2019, 47 (9), pp.3457-3480. ⟨10.1080/00927872.2019.1566957⟩. ⟨hal-01833662v2⟩