The reflexive completion (also called the Isbell completion) of a category $A$ is the fixed point of the Isbell duality adjunction for $A$.
The reflexive completion is idempotent: $\mathcal{R}(\mathcal{R}(A)) \simeq \mathcal{R}(A)$.
The reflexive completion is Cauchy complete. The Cauchy completion $\bar A$ of a small category $A$ is a full subcategory of the reflexive completion $\mathcal{R}(A)$. We have that $\mathcal{R}(\bar A) \simeq \mathcal{R}(A)$. Hence Morita equivalent categories have equivalent reflexive completions. See §10 of Avery and Leinster.
$A \to \mathcal{R}(A)$ creates limits and colimits. $\mathcal{R}(A) \to \hat{A}$ 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.
Tom Avery, Tom Leinster. Isbell conjugacy and the reflexive completion. Theory and Applications of Categories, 36 12 (2021) 306-347 [tac:36-12, pdf]
Richard Garner, Topological functors as total categories, TAC
Last revised on August 11, 2023 at 14:59:44. See the history of this page for a list of all contributions to it.