nLab relative Rezk nerve

Idea

One of the central theorems in (∞,1)-category theory is the fully faithfulness of the Rezk nerve

$$\begin{aligned} \mathsf{Cat}_{(\infty,1)} & \longrightarrow \mathsf{sAn} \\ C&\mapsto \Big([n]\mapsto \operatorname{Map}\big([n],C\big)\Big) \mathrlap{\,,} \end{aligned}$$

where $\mathsf{sAn}$ denotes the (∞,1)-category of simplicial animae.

The mapping anima $Map\big([n],C\big)$ is simply the full subcategory of the functor $(\infty,1)$-functor $(\infty,1)$-category $\operatorname{Fun}([n],C)$ consisting of equivalences.

The relative Rezk nerve replaces equivalences by maps in a prescribed wide subcategory $W\subset C$ and gives an alternative presentation of the localization of an (∞,1)-category of $C$. (See below)

Definition

Let $C$ be an (∞,1)-category and $W\subset C$ be a wide sub (∞,1)-category. The relative Rezk nerve of $(C,W)$ is the simplicial anima $N^{\mathrm{rel}}(C,W)$ given by

$$[n] \mapsto \Big\vert\operatorname{Fun}\big([n],C\big)\times_{C^{n+1}} W^{n+1}\Big\vert,$$

where $|-|:\mathsf{Cat}_{(\infty,1)}\to \mathsf{An}$ is the left adjoint to the inclusion.

The relative Rezk nerve was introduced by Rezk 2001, Section 3 under the name classification diagram.

Properties

One of the fundamental properties of the relative Rezk nerve is that its associated (∞,1)-category computes localization of an (∞,1)-category:

Theorem

For every (∞,1)-category $C$ and every wide subcategory $W\subset C$, we have

$$\mathrm{ac}\circ N^{\mathrm{rel}}(C,W) \simeq C[W^{-1}],$$

where $\mathrm{ac}\colon \mathsf{sAn}\to \mathsf{Cat}_{(\infty,1)}$ denotes the left adjoint to the Rezk nerve (computing associated ($\infty$,1)-category).

This was first established by Mazel-Gee in MG19. Shorter proofs are in Ara23 and AC25. In special cases, Rezk proved a version of this theorem when $(C,W)$ is part of a simplicial model category in Rez01, Theorem 8.3; Bergner proved this for model categories in Ber09, Theorem0.2.

Another fundamental property of the relative Rezk nerve is that its Segalification? admits an explicit presentation:

Theorem

The Segalification? of $N^{\mathrm{rel}}(C,W)$ can be described in terms of zig-zags in $C$ in which one is allowed to go backward only via maps in $W$.

The mapping animae of the Segalification agrees (or generalizes) with the one appearing in hammock localization.

(AC26, Theorems A and B)

Related concepts

• localization of an (∞,1)-category

References

Classical references:

• Charles Rezk, A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353 (2001), no. 3, 973–1007 (arXiv:math/9811037, doi:10.1090/S0002-9947-00-02653-2).

• Julia E. Bergner, Complete Segal spaces arising from simplicial categories, Trans. Amer. Math. Soc. 361 (2009), no. 1, 525–546 (arXiv:0704.1624), doi:10.1090/S0002-9947-08-04616-3

Relation to localization of an (∞,1)-category:

• Aaron Mazel-Gee, The universality of the Rezk nerve, Algebr. Geom. Topol. 19 (2019), no. 7, 3217–3260 (arXiv:1510.03150, doi:10.2140/agt.2019.19.3217).

• Kensuke Arakawa, Bastiaan Cnossen, A short proof of the universality of the relative Rezk nerve, Proc. Amer. Math. Soc. 154 (2026), no. 5, 1849–1853 (arXiv:2505.14123, doi:10.1090/proc/17483).

• Kensuke Arakawa, Classification diagrams of marked simplicial sets (2023), arXiv:2311.01101 [math.AT].

AC25 offers a rather “model-agnostic” viewpoint, while Ara23 offers a generalization using quasicategories.

Relation to hammock localization:

• Kensuke Arakawa, Bastiaan Cnossen, Hammock localization via Segal animae (2026), arXiv:2608.19870 [math.CT].

A generalization to dendroidal complete Segal spaces:

• Kensuke Arakawa, Victor Carmona, Francesca Pratali, Relative dendroidal Rezk nerve and applications (2026), arXiv:2606.11895 [math.AT].