nLab reflexive completion




The reflexive completion (also called the Isbell completion) of a category AA is the fixed point of the Isbell duality adjunction for AA.


  • The reflexive completion is idempotent: ((A))(A)\mathcal{R}(\mathcal{R}(A)) \simeq \mathcal{R}(A).

  • The reflexive completion is Cauchy complete. The Cauchy completion A¯\bar A of a small category AA is a full subcategory of the reflexive completion (A)\mathcal{R}(A). We have that (A¯)(A)\mathcal{R}(\bar A) \simeq \mathcal{R}(A). Hence Morita equivalent categories have equivalent reflexive completions. See §10 of Avery and Leinster.

  • A(A)A \to \mathcal{R}(A) creates limits and colimits. (A)A^\mathcal{R}(A) \to \hat{A} preserves limits and reflects limits and colimits. See §11 of Avery and Leinster.

  • The reflexive completion of a small category has an initial object and a terminal object.

  • The reflexive completion is a full subcategory of the Isbell envelope, and can be concretely described as a 2-pullback:

See envelope of an adjunction.


Last revised on August 11, 2023 at 14:59:44. See the history of this page for a list of all contributions to it.