Rezk Completion in Category Theory

• Rezk Completion is a canonical construction in category theory that universally completes structures to achieve univalence, where equality coincides with isomorphism.
• It employs the Yoneda embedding and presheaf methods to form fully faithful and essentially surjective functors with robust universal properties.
• The construction extends to higher categories via localization techniques on simplicial and Θₙ-spaces, ensuring coherence in equivalence and descent properties.

A Rezk completion is a canonical construction in category theory and higher category theory that universally completes a given structure—category, enriched category, Segal-type object, or relative ∞-category—into a form satisfying univalence (or completeness), thereby enabling robust equivalence invariance and descent properties. The terminology arises from Charles Rezk’s foundational work on complete Segal spaces. In the univalent foundations and higher category framework, Rezk completion ensures that the notions of “isomorphism” and “equality” of objects coincide, and that every essentially surjective, fully faithful functor becomes a genuine equivalence. The construction has been systematically extended and generalized through presheaf/nerve formalisms, algebraic fibrant replacements, and left Bousfield localizations in model categories.

1. Classical Rezk Completion and Univalent Categories

Given a small category $C$, its Rezk completion $\widehat{C}$ is a univalent category together with a functor $y: C \to \widehat{C}$ (typically the Yoneda embedding) that is fully faithful and essentially surjective, with the following universal property: for any univalent category $D$, precomposition along $y$ yields an equivalence

$$y^* : \operatorname{Fun}(\widehat{C}, D) \longrightarrow \operatorname{Fun}(C, D).$$

The canonical construction forms the full subcategory of the presheaf category $[C^\mathrm{op}, \mathrm{Set}]$ consisting of presheaves (functors) merely isomorphic to representables. Morphisms in this completion are inherited from natural transformations among such functors. In univalent foundations, $\widehat{C}$ is characterized by the property that the canonical map

$(x = y) \longrightarrow (x \cong y)$

between the identity type and the type of isomorphisms is an equivalence for all $x, y$, meaning equality and isomorphism of objects coincide (Ahrens et al., 2013, Wullaert et al., 8 Jan 2026).

The Rezk completion is left adjoint (in the 2-categorical or bicategorical sense) to the inclusion of the full subcategory of univalent categories into all categories, and every functor from $C$ to any univalent category $D$ factors through $\widehat{C}$ up to essentially unique isomorphism (Wullaert et al., 8 Jan 2026). In the context of higher inductive types, the completion can be constructed without a universe shift, by freely adding equalities corresponding to all isomorphisms.

2. Rezk Completion for Enriched and Structured Categories

For categories enriched in a symmetric monoidal closed (univalent and complete) category $V$, the Rezk completion is constructed analogously. Given a $V$-enriched category $E$, the enriched Yoneda embedding

$y_E: E \longrightarrow [E,V]$

is fully faithful. The enriched Rezk completion $R(E)$ is realized as the image of this embedding, i.e., the full sub-$V$-category of $[E,V]$ consisting of presheaves isomorphic to representables. The inclusion $\tilde y: E \to R(E)$ is both essentially surjective and fully faithful, hence a weak equivalence in the bicategory of univalent enriched categories.

The universal property persists: for any univalent $V$-enriched category $D$, precomposition by $\tilde y$ yields an equivalence of functor categories: $[R(E), D] \longrightarrow [E, D].$ This construction, entirely algebraic, specializes to the classical case when $V = \mathrm{Set}$, and admits further analogs for monoidal categories, bicategories, and more elaborate algebraic theories, utilizing displayed bicategories to allow for unified lifting of structures such as finite limits, exponentials, subobject classifiers, or elementary topos properties (Weide, 2024, Wullaert et al., 2022, Wullaert et al., 8 Jan 2026).

3. Rezk Completion in Higher Category Theory: Segal and Θₙ-Spaces

In the setting of higher categories, the Rezk completion is realized via localization in categories of simplicial spaces or presheaves on Θₙ. For simplicial spaces $X: \Delta^{\mathrm{op}} \to \mathcal{S}$, the notions of Segal and completeness conditions translate categorical composition and identity/iso identification to homotopical analogs:

• The Segal condition, implemented via spine inclusions, enforces coherent associativity.
• Completeness ensures that homotopy equivalences become identities, corresponding to univalence.

The Rezk completion functor, at the model categorical level, is a left Bousfield localization forcing these conditions, yielding so-called complete Segal spaces (for $n=1$) or complete Segal Θₙ-spaces (for $n > 1$). For every Segal object, the Rezk completion is initial with respect to maps into complete Segal objects, and for Θₙ-spaces, one obtains model structures and Quillen adjunctions at each dimension, ultimately characterizing Dwyer–Kan equivalences in terms of the weak equivalences in the completed model (Tuominen, 3 Mar 2025, Stenzel, 2019).

Key technical construction for Segal and completeness localizations relies on “thickening” cell diagrams, such as the walking isomorphism $I$ and its nerves, and resolving these data through diagrams $Q^k$ and their mapping spaces.

4. The Relative Rezk Nerve and Localization in ∞-Categorical Models

The Rezk completion generalizes to relative ∞-categories $(C, W)$, where $W$ is a subcategory of “weak equivalences”: the Rezk nerve $N^R_\infty(C, W)$ (a simplicial space) encodes the data of functors from simplices with prescribed equivalence conditions along the morphisms. The resulting simplicial space is then localized (e.g., via Bousfield localization) to force completeness.

The universal property is precisely that the complete Segal space generated this way presents the ∞-categorical localization $C[W^{-1}]$: equivalently, functor categories from the Rezk nerve to any complete Segal space X correspond (via restriction) to functors from $(C,W)$ to the underlying relative category of X that preserve weak equivalences.

This relative perspective clarifies both local (for a single category) and global (across all relative categories) universal properties, and yields adjunctions between relative ∞-categories and complete Segal spaces (Mazel-Gee, 2015, Arakawa et al., 20 May 2025).

Structure Type	Objects Completed	Resultant Property
Small categories	all categories	univalence in objects
Enriched categories	all $V$-enriched categories	univalence in enrichment
Monoidal categories	all (lax) monoidal categories	univalence for underlying
Segal/Θₙ-spaces	all Segal-type objects	completeness, univalence
Relative ∞-categories	all pairs $(C, W)$	localization/inversion

5. Modular Lifting, Structured Rezk Completions, and Applications

In practical settings, Rezk completion must accommodate additional structures (limits, exponentials, topos structure, etc.). The modular formalism of displayed bicategories allows the systematic transfer of structures along the univalent completion: for each structure, one constructs its lift on the completed object and verifies preservation and reflection of universal properties and structure maps.

This enables, for instance, the extension of Rezk completion to elementary topoi, retaining finite limits, cartesian closure, subobject classifiers, and further structure as required (Wullaert et al., 8 Jan 2026).

Applications include:

• Cauchy (idempotent-split) completions as special cases (via discrete enrichment).
• Groupoids, fundamental groupoids, and stack completions as truncated or sheafified Rezk completions.
• Completion of (pre)toposes or categories with a logical or adjoint structure to their univalent analogs.
• Algebraic and type-theoretic settings: the strict Rezk completion of models of homotopy type theory provides models where identity types coincide with paths, essential for homotopy canonicity, and is constructed via algebraic fibrant replacement mechanisms (Bocquet, 2023).

6. Universal Property and Equivalence Invariance

The unifying feature of Rezk completion across contexts is a robust universal property: it provides the initial univalent (or complete) structure receiving a weak equivalence (fully faithful, essentially surjective functor, DK-equivalence, etc.) from the original object. Consequences include:

• Any fully faithful, essentially surjective functor between univalent objects is an equivalence, independently of the axiom of choice.
• All functors out of the original object into univalent/complete targets uniquely factor through the completion.
• The completion is functorial, commutes with further structure via modular lifting, and has well-defined model categorical and higher categorical adjointness (Ahrens et al., 2013, Weide, 2024, Mazel-Gee, 2015, Tuominen, 3 Mar 2025, Wullaert et al., 8 Jan 2026).

These properties provide foundational infrastructure for robust mathematical abstraction and the study of equivalence-invariant and descent phenomena in categorical, higher-categorical, and homotopical settings.