nLab minimal Kan fibration

Idea

The concept of minimal Kan fibration is the specialization of the concept of minimal fibrations to the Kan fibrations in the classical model structure on simplicial sets. Hence a minimal Kan fibration is a Kan fibration whose fibers are, in some sense, as small as possible in its homotopy class. Moreover, every Kan fibration has a strong deformation retract to a minimal Kan fibration (prop. below). Hence minimal Kan complexes (i.e. minimal fibrations over the point) are the analogue in simplicial sets of minimal Sullivan models in rational homotopy theory.

Minimal Kan fibrations over connected bases happen to be fiber bundles, locally trivial over each simplex (lemma below). This implies that their geometric realization into any convenient category of topological spaces is also a fiber bundle and hence in particular a Serre fibration. This is what makes minimal fibrations play a key role in all available proofs of the Quillen equivalence between the model structure on topological spaces and the standard model structure on simplicial sets (see at homotopy hypothsis – for Kan complexes).

Definition

Definition

A Kan fibration $\phi \colon S \longrightarrow T$, is called a minimal Kan fibration if for any two cells in the same fiber with the same boundary if they are homotopic relative their boundary, then they are already equal.

More formally, $\phi$ is minimal precisely if for every commuting diagram of the form

$$\array{ (\partial \Delta[n]) \times \Delta[1] &\stackrel{p_1}{\longrightarrow}& \partial \Delta[n] \\ \downarrow && \downarrow \\ \Delta[n] \times \Delta[1] &\stackrel{h}{\longrightarrow}& S \\ \downarrow^{\mathrlap{p_1}} && \downarrow^{\mathrlap{\phi}} \\ \Delta[n] &\longrightarrow& T }$$

then the two composites

$$\Delta[n] \stackrel{\overset{d_0}{\longrightarrow}}{\underset{d_1}{\longrightarrow}} \Delta[n] \times \Delta[1] \stackrel{h}{\longrightarrow} S$$

are equal.

Properties

Proposition

The pullback (in sSet) of a minimal Kan fibration, def. , along any morphism is again a mimimal Kan fibration.

Proposition

For every Kan fibration, there exists a fiberwise strong deformation retract to a minimal Kan fibration, def. .

(e.g. Goerss-Jardine 96, chapter I, prop. 10.3, Joyal-Tierney 05, theorem 3.3.1, theorem 3.3.3).

Proof idea

Choose representatives by induction, use that in the induction step one needs lifts of anodyne extensions against a Kan fibration, which exist.

Lemma

A morphism between minimal Kan fibrations, def. , which is fiberwise a homotopy equivalence, is already an isomorphism.

(e.g. Goerss-Jardine 96, chapter I, lemma 10.4)

Proof idea

Show the statement degreewise. In the induction one needs to lift anodyne extensions agains a Kan fibration.

Lemma

Every minimal Kan fibration, def. , over a connected base is a simplicial fiber bundle, locally trivial over every simplex of the base.

(e.g. Goerss-Jardine 96, chapter I, corollary 10.8)

Proof

By assumption of the base being connected, the classifying maps for the fibers over any two vertices are connected by a zig-zag of homotopies, hence by this lemma the fibers are connected by homotopy equivalences and then by prop. and lemma they are already isomorphic. Write $F$ for this typical fiber.

Moreover, for all $n$ the morphisms $\Delta[n] \to \Delta[0] \to \Delta[n]$ are left homotopic to $\Delta[n] \stackrel{id}{\to} \Delta[n]$ and so applying this lemma and prop. once more yields that the fiber over each $\Delta[n]$ is isomorphic to $\Delta[n]\times F$.

Related concepts

minimal fibration

• minimal Joyal fibration

• minimal dendroidal fibration

References

• Pierre Gabriel, Michel Zisman, chapter VI.5 of Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer (1967) (pdf)

• E. Curtis, Simplicial Homotopy Theory, Advances in Math., 6, (1971), 107 – 209.

• Paul Goerss, Rick Jardine, chapter I, section 10 of Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1996)

• André Joyal, Myles Tierney, section 3.3 of Notes on simplicial homotopy theory, (pdf)