Partial model categories and their simplicial nerves

In this note we consider partial model categories, by which we mean relative categories that satisfy a weakened version of the model category axioms involving only the weak equivalences. More precisely, a partial model category will be a relative category that has the two out of six property and admits a 3-arrow calculus. We then show that Charles Rezk’s result that the simplicial space obtained from a simplicial model category by taking a Reedy fibrant replacement of its simplicial nerve is a complete Segal space also holds for these partial model categories. We also note that conversely every complete Segal space is Reedy equivalent to the simplicial nerve of a partial model category and in fact of a homotopically full subcategory of a category of diagrams of simplicial sets.

💡 Research Summary

This paper introduces the notion of a partial model category, a weakening of the usual model category axioms that focuses solely on the class of weak equivalences. A relative category (C, W) is called a partial model category if it satisfies two conditions: (1) the two‑out‑of‑six property, which implies the usual two‑out‑of‑three and guarantees that all isomorphisms lie in W; (2) a 3‑arrow calculus, i.e. the existence of sub‑categories U and V inside W such that every map w∈W admits a functorial factorisation w = v ∘ u with u∈U, v∈V, U is closed under pushouts, and V is closed under pullbacks. These axioms capture the essential parts of the model‑category structure that involve only weak equivalences, discarding the full cofibration/fibration machinery.

The authors first give several basic examples: any model category, any homotopically full sub‑category of a partial model category, and relative functor categories all inherit a partial model structure. They then prove a generalisation of Rezk’s theorem: for any partial model category (C, W), any Reedy fibrant replacement of its simplicial nerve N(C, W) is a complete Segal space. The proof follows Rezk’s original strategy but simplifies the Segal part dramatically. For each k≥2 they define categories A_k (chains of k composable arrows) and B_k (zig‑zags with a middle weak equivalence) and show that the inclusion A′_k→B_k is a homotopy equivalence. This uses the 3‑arrow calculus to construct explicit zig‑zag homotopies and the Quilen B³ theorem to identify ordinary pullbacks with homotopy pullbacks. The completeness condition follows from the fact that partial model categories are saturated (a map is in W iff it becomes an isomorphism in the homotopy category), exactly as in Rezk’s original argument.

The second main result is a converse: every complete Segal space X is Reedy (hence Rezk) equivalent to the simplicial nerve of some partial model category. The construction proceeds via a relative Yoneda embedding y: (C, W) → S^{C^{op}, W^{op}} which sends an object A to the simplicial set of maps L_H(C, W)(–, A) (L_H denotes the hammock localisation). The essential image E_y of y is shown to be a homotopically full sub‑category of a diagram category of simplicial sets, and it inherits a partial model structure. Moreover, y is a DK‑equivalence, i.e. its simplicial localisation is a weak equivalence of simplicial categories. Using the known adjunction K ⊣ N between relative categories and simplicial spaces, together with the fact that the unit of this adjunction is a Reedy equivalence, the authors assemble a zig‑zag of Reedy (and hence Rezk) equivalences linking X to N(E_y). Thus any complete Segal space arises from a partial model category.

The paper concludes by emphasizing that the two simple axioms—two‑out‑of‑six and 3‑arrow calculus—are sufficient to develop a robust homotopy theory of “homotopy theories” without invoking the full machinery of cofibrations and fibrations. This opens the way for applying model‑categorical ideas in contexts where only a class of weak equivalences is naturally available, such as certain relative categories, ∞‑categorical settings, or localisation problems. The results also clarify the minimal structural requirements needed for Rezk’s complete Segal space model of homotopy theories, providing a cleaner and more flexible foundation for future work in higher‑categorical homotopy theory.