Limits in categories of relations and limit-colimit commutation

Morse Homology on Noncompact Manifolds

“…Conditions under which the canonical homomorphism κ is an isomorphism were obtained by B. Eckmann and P. Hilton in [3] and by A. Frei and J. Macdonald in [5]. We first remark that in general κ is neither necessarily injective nor surjective.…”

Symplectic Tate homology

“…We will see below that this fails in the case of two different filtrations. This failure can be traced to the failure of the commutative square (1) to be exact in the sense of [26].…”

“…In general, the map κ is neither injective nor surjective; we will see examples of this in Section 5. Conditions under which the canonical homomorphism κ is an isomorphism were obtained by B. Eckmann and P. Hilton in [18] and by A. Frei and J. Macdonald in [26].…”

Rabinowitz Floer homology and mirror symmetry

“…REMARK 3) By Theorem 5.6 in [14], taking the limits in the opposite order creates an isomorphic complex. (Also see [10] for an application of this theorem to Morse Theory.…”

“…Define the completed symplectic cochains to be the limit over this bi‐directed system, and denote it by Note that, under our conventions, the limits take to negative infinity and to positive infinity. Remark By [, Theorem 5.6], taking the limits in the opposite order creates an isomorphic complex. (Also see for an application of this theorem to Morse Theory.…”