Paths and cylinders
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. 2 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 2 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).
A Kan fibration , 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, is minimal precisely if for every commuting diagram of the form
then the two composites
(e.g. Goerss-Jardine 96, chapter I, prop. 10.3, Joyal-Tierney 05, theorem 3.3.1, theorem 3.3.3).
(e.g. Goerss-Jardine 96, chapter I, lemma 10.4)
(e.g. Goerss-Jardine 96, chapter I, corollary 10.8)
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. 1 and lemma 1 they are already isomorphic. Write for this typical fiber.
Moreover, for all the morphisms are left homotopic to and so applying this lemma and prop. 1 once more yields that the fiber over each is isomorphic to .
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.
André Joyal, Myles Tierney, section 3.3 of Notes on simplicial homotopy theory, (pdf)