Category theory applied to Pontryagin duality

Abstract

A proof of the Pontryagin duality theorem for locally compact abelian (LCA) groups is given, using category-theoretical ideas and homological methods. The proof is guided by the structure within the category of LCA groups and does not use any deep results except for the Peter-Weyl theorem. The duality is first established for the subcategory of elementary LCA groups (those isomorphic with T* Θ Z J 0 R k φ F, where T is the circle group, Z the integers, R the real numbers, and F a finite abelian group), and through the study of exact sequences, direct limits and projective limits the duality is expanded to larger subcategories until the full duality theorem is reached.Introduction* In this note we present a fairly economical proof of the Pontryagin duality theorem for locally compact abelian (LCA) groups, using category-theoretic ideas and homological methods. This theorem was first proved in a series of papers by Pontryagin and van Kampen, culminating in van Kampen's paper [5], with methods due primarily to Pontryagin. In [10, pp. 102-109], Weil introduced the simplifying notion of compactly generated group and explored the functorial nature of the situation by examining adjoint homomorphisms and projective limits. Proofs along the lines of Pontryagin-van Kampen-Weil appear in the books by Pontryagin [7, pp. 235-279] and Hewitt and Ross [2,. A different proof based on abstract Fourier analysis was given by Cartan and Godemont [1]; similar methods are also used by Rudin [9, and Heyer [3,. Negrepontis [6, presented the theorem in light of category theory, but for the most part used different methods and more structure theory than is used here.The proof we present is based on ideas used previously by Hofmann [4, pp. 109-117] and the author [8]. It rests neither on the structure theorem for compactly generated LCA groups (see [2], [6], [7]) nor on the structure of the IΛalgebra of G (see [6] and [9]). The only deep result we require is the classical Peter-Weyl theorem, applied in our case to compact abelian groups, of course.

Arithmetic duality theorems for 1-motives over function fields

“…Let T be the toric part of M and let U À1 be associated to T as in the previous lemma. The first vertical map is an isomorphism by the proof of [15], Proposition II.2.9, p. 209, and the fact that [10], p. 112, lines [11][12][13][14][15][16][17]. There exists a natural exact commutative diagram The first vertical map in the above diagram is an isomorphism by the properness of A, the second one is an injection by the previous lemma and the rightmost one is an injection by [15], proof of Lemma II.…”

“…are the maps arising from sequences (15) and (16) in the proof of the theorem (with the roles of M and M * interchanged), then…”

“…where the first map is the surjection appearing in the exact sequence (16) for i = 0 and the second map is induced by the canonical map…”

Positive (Modal) Logic Beyond Distributivity

“…An approach towards duality for (non-distributive) meet-semilattices was developed by Hofmann, Mislove and Stralka (HMS) [27], along the same lines of the proof of the Van Kampen-Pontryagin duality for locally compact abelian groups given in [35]. This was later generalised to a duality for lattices by Jipsen and Moshier [34].…”

Commutants for Enriched Algebraic Theories and Monads

“…For example, if T is commutative and we take A = T T , then (T, A) = (T, T T ) is balanced. Since LCAb is an additive category [29], finite products in LCAb are biproducts, so for each n ∈ N, the power Z n in LCAb (which is again discrete) is also a copower in LCAb and is sent by the equivalence Hom(−, T) : LCAb op → LCAb to a power Hom(Z n , T) ∼ = Hom(Z, T) n ∼ = T n in LCAb. The Lawvere theory Mat(Z) is commutative (16.2.1), so the algebra (Mat(Z), T) in LCHaus is commutative (13.3), and hence there is a canonical morphism of Lawvere theories…”