nLab reflexive completion

Definition

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

Properties

• 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.

Related concepts

• Isbell duality

• Isbell envelope

• fixed point of an adjunction

• MacNeille completion

References

• 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