The semi-model structure and the homotopy hypothesis

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

The main theorem of the previous lesson — elementary expansions of cellular models are weak equivalences — has two consequences. First, the models of the coherator \(\mathcal C\) carry a canonical left semi-model structure, with the weak equivalences of the lesson on the homotopy calculus, cofibrations generated by the boundary inclusions, and every object fibrant (Section 2). This is proved here in full. Second, this semi-model category is Quillen equivalent to topological spaces: Grothendieck's homotopy hypothesis holds for the \(\mathcal C\)-\(\infty\)-groupoids (Section 3). This step uses Henry's comparison theorem for cylinder categories, which is stated with its exact hypotheses and used without proof.

Prerequisites. Partial cylinders and the main theorem (Theorem 0.1), The homotopy calculus (Lemmas 3.2 and 4.2), Globular theories and their models (Theorem 4.3, Proposition 5.1).

1. Lifting and factorization

For a set \(K\) of morphisms of \(\mathcal A=\operatorname{Mod}(\mathcal C)\), \(K\)-\(\mathrm{inj}\) is the class of morphisms with the right lifting property against every member of \(K\), and \(K\)-\(\mathrm{cof}\) the class of morphisms with the left lifting property against every member of \(K\)-\(\mathrm{inj}\). Relative \(K\)-cellular extensions belong to \(K\)-\(\mathrm{cof}\), and \(K\)-\(\mathrm{cof}\) is closed under retracts, pushouts and transfinite composites.

Lemma 1.1 (retract argument). If \(i=pj\) and \(i\) has the left lifting property against \(p\), then \(i\) is a retract of \(j\).

Proof. Write \(i:X\to Y\), \(j:X\to Z\), \(p:Z\to Y\). The square with \(j\) on top, \(i\) on the left, \(p\) on the right and \(\mathrm{id}_Y\) at the bottom has a diagonal \(s:Y\to Z\) with \(si=j\) and \(ps=\mathrm{id}_Y\). Then \((\mathrm{id}_X,s):i\to j\) and \((\mathrm{id}_X,p):j\to i\) are maps of arrows composing to the identity of \(i\). \(\square\)

By the small object argument (Proposition 5.1 of the first lesson), every morphism factors as a relative \(I\)-cellular extension followed by a morphism in \(I\)-\(\mathrm{inj}\). With Lemma 1.1, \(I\)-\(\mathrm{cof}\) consists of the retracts of relative \(I\)-cellular extensions. A model \(X\) is cofibrant if \(\varnothing\to X\) lies in \(I\)-\(\mathrm{cof}\); factoring \(\varnothing\to X\) shows that the cofibrant models are the retracts of cellular models.

2. The canonical semi-model structure

We use the following convention. A left semi-model structure on a complete and cocomplete category consists of classes \(\mathcal W\) (weak equivalences), \(\mathrm{Cof}\) (cofibrations) and \(\mathrm{Fib}\) (fibrations) such that \(\mathcal W\) has the two-out-of-three property, all three classes are closed under retracts, and:

  1. every morphism factors as a cofibration followed by a morphism in \(\mathcal W\cap\mathrm{Fib}\), and cofibrations have the left lifting property against \(\mathcal W\cap\mathrm{Fib}\);
  2. every morphism with cofibrant domain factors as a morphism in \(\mathrm{Cof}\cap\mathcal W\) followed by a fibration, and morphisms in \(\mathrm{Cof}\cap\mathcal W\) with cofibrant domain have the left lifting property against fibrations.

Theorem 2.1 (the canonical semi-model structure). For every Grothendieck coherator \(\mathcal C\), the category \(\mathcal A\) of \(\mathcal C\)-\(\infty\)-groupoids carries a cofibrantly generated left semi-model structure with \[ \begin{aligned} \text{weak equivalences}&=\mathcal W,\\ \text{cofibrations}&=I\text{-}\mathrm{cof},\\ \text{fibrations}&=J\text{-}\mathrm{inj},\\ \text{trivial fibrations}&=\mathcal W\cap J\text{-}\mathrm{inj}=I\text{-}\mathrm{inj}. \end{aligned} \] The cofibrant objects are the retracts of cellular objects, and every object is fibrant.

Proof. \(\mathcal A\) is complete and cocomplete (Theorem 4.3 of the first lesson), and \(\mathcal W\) has two-out-of-three and is closed under retracts (Lemma 3.2 of the lesson on the homotopy calculus); lifting classes are closed under retracts. The equality \(\mathcal W\cap J\)-\(\mathrm{inj}=I\)-\(\mathrm{inj}\) is Lemma 4.2 of the same lesson. Axiom (1) follows from the factorization of Section 1 and the definition of \(I\)-\(\mathrm{cof}\).

Cofibrant domains. Theorem 0.1 of the previous lesson extends from cellular to cofibrant models. Let \(X\xrightarrow{u}Z\xrightarrow{v}X\) with \(vu=\mathrm{id}_X\) and \(Z\) cellular, and \(a:D_j\to X\). The maps \(u,v\), with the identity on the new \(D_{j+1}\), induce maps between \(X^+=X\amalg_{D_j}D_{j+1}\) along \(a\) and \(Z^+=Z\amalg_{D_j}D_{j+1}\) along \(ua\) that exhibit \(X\to X^+\) as a retract of \(Z\to Z^+\); so \(X\to X^+\) is a weak equivalence. Each \(J_j\) is a composite of two boundary attachments, so it lies in \(I\)-\(\mathrm{cof}\) and pushouts of it preserve cofibrancy. Hence every relative \(J\)-cellular extension with cofibrant domain is a weak equivalence: attach the cells of a coproduct one at a time, and use Lemma 3.2 of the lesson on the homotopy calculus at limit stages.

Axiom (2). The small object argument for \(J\) factors a morphism \(f:X\to Y\) with \(X\) cofibrant as \(f=pj\) with \(j\) relative \(J\)-cellular, hence in \(I\)-\(\mathrm{cof}\cap\mathcal W\) by the preceding paragraph, and \(p\in J\)-\(\mathrm{inj}\). Let \(i:X\to Y\) be in \(I\)-\(\mathrm{cof}\cap\mathcal W\) with \(X\) cofibrant, and factor \(i=pj\) in this way. By two-out-of-three \(p\in\mathcal W\cap J\)-\(\mathrm{inj}=I\)-\(\mathrm{inj}\), so \(i\) has the left lifting property against \(p\), and by Lemma 1.1 \(i\) is a retract of \(j\). As a relative \(J\)-cellular extension, \(j\) has the left lifting property against every fibration, and so does its retract \(i\).

Fibrant objects. Every \(X\to1\) lifts against each \(J_j\): extend a \(j\)-cell to its unit. \(\square\)

3. The comparison with spaces

Henry's approach constructs semi-model categories from small cylinder categories: categories with cofibrations and weak equivalences in which pushouts along cofibrations exist, weak equivalences satisfy two-out-of-six, every object has a cylinder, and trivial cofibrations have retractions. The completion of a small cylinder category \(\mathcal B\) is the category of presheaves on \(\mathcal B\) that send the initial object to a point and pushouts along cofibrations to pullbacks; it carries a semi-model structure. A cylinder coherator is a cylinder category with a marked object such that the inclusion of the free pre-cylinder category on one object is a trivial cofibration in Henry's sense.

Let \(\mathcal A_{\mathrm{fin}}\) be a small skeleton of the finitely cellular \(\mathcal C\)-\(\infty\)-groupoids, with finite boundary-cell extensions as cofibrations and the restriction of \(\mathcal W\) as weak equivalences.

Named result H (Henry's comparison theorem). Suppose that for the coherator \(\mathcal C\) every elementary expansion of a finitely cellular model is a weak equivalence. Then:

  1. \(\mathcal A_{\mathrm{fin}}\) is a cylinder category (Henry 2016, Proposition 5.3.6);
  2. its completion is \(\mathcal A\), with the semi-model structure of Theorem 2.1 (Proposition 5.3.7);
  3. with the marked object \(D_0\), \(\mathcal A_{\mathrm{fin}}\) is a cylinder coherator (Lemmas 5.3.9 and 5.3.11, Corollary 5.3.12);
  4. for every cylinder coherator, there is a fundamental \(\infty\)-groupoid functor from topological spaces to its completion which is a right Quillen equivalence, with geometric realization as left adjoint (Theorem 5.1.2(5)).

This result is used without proof in the programme. The hypothesis is Theorem 0.1 of the previous lesson. The cylinder used in (1) for the generating cofibration \(\partial D_j\to D_j\) is \(D_j\amalg_{\partial D_j}D_j\to D_{j+1}\to D_j\): the first map is \(I_{j+1}\), the end at the first face is \(J_j\), a weak equivalence by Theorem 0.1, and the last map is induced by a unit. The marked object in (3) is homotopically terminal because \(D_0\) is contractible (Proposition 2.2 of the lesson on cutoffs), so \(D_0\to1\) lies in \(I\)-\(\mathrm{inj}\). In the proof of Henry's Lemma 5.3.11, the projection from the slice over \(D_0\) is an acyclic fibration, as his Proposition 4.4.4 states; the words "trivial cofibration" there are a misprint.

Corollary 3.1 (the homotopy hypothesis). For every Grothendieck coherator \(\mathcal C\), assuming the named result H, there is a Quillen equivalence \[ |-|:\operatorname{Mod}(\mathcal C)\rightleftarrows\mathbf{Top}:\Pi_\infty \] between the semi-model category of Theorem 2.1 and topological spaces with weak homotopy equivalences. In particular \(\operatorname{Mod}(\mathcal C)[\mathcal W^{-1}]\simeq\operatorname{Ho}(\mathbf{Top})\), and the homotopy theory of \(\mathcal C\)-\(\infty\)-groupoids does not depend on the coherator.

Proof. Theorem 0.1 of the previous lesson is the hypothesis of H. By H(2) the completion of \(\mathcal A_{\mathrm{fin}}\) is \(\operatorname{Mod}(\mathcal C)\) with the structure of Theorem 2.1, and by H(3) and H(4) the fundamental \(\infty\)-groupoid functor into it is a right Quillen equivalence. Any two coherators give Quillen equivalent semi-model categories, both being Quillen equivalent to \(\mathbf{Top}\). \(\square\)

4. Exercises

4.1 (easy). Show that every model is fibrant and that \(D_j\) is cofibrant.

4.2 (medium). Show that a morphism between cofibrant models that is a weak equivalence and lies in \(I\)-\(\mathrm{cof}\) has a retraction.

4.3 (easy). Show that a pushout of a morphism in \(I\)-\(\mathrm{cof}\) is in \(I\)-\(\mathrm{cof}\).

4.4 (medium). Show that the composite \(D_j\amalg_{\partial D_j}D_j\to D_{j+1}\to D_j\) is the codiagonal, and that \(D_{j+1}\to D_j\) is a weak equivalence.

5. Solutions

4.1. Fibrancy is the last part of Theorem 2.1. \(D_j=F(d_j)\) is cellular (Section 5 of the first lesson), hence cofibrant.

4.2. Let \(i:X\to Y\) be such a morphism. Since \(X\) is fibrant, \(X\to1\) is a fibration, and the square with \(\mathrm{id}_X\) on top and \(Y\to1\) at the bottom has a diagonal \(Y\to X\) by axiom (2), as \(X\) is cofibrant. It is a retraction of \(i\).

4.3. If \(i\) has the left lifting property against \(p\), so does every pushout of \(i\): a square from the pushout to \(p\) restricts to a square from \(i\), whose diagonal, together with the given map, defines a diagonal on the pushout by its universal property.

4.4. The composite sends both copies of \(D_j\) to \(D_j\) by the identity, because the unit has source and target the given cell; so it is the codiagonal. The map \(D_{j+1}\to D_j\) is a retraction of \(J_j\), which is a weak equivalence by Theorem 0.1 of the previous lesson (\(D_j\) is cellular); two-out-of-three gives the claim.

Sources