In mathematics, especially

homotopy theory In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipli ...

, a minimal fibration is used to approximate fibrations between

presheaves In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the da ...

. A minimal fibration has a defining property that an equivalence between them (in some sense) is an isomorphism. Thus, minimal fibrations can be used to study some coherence questions up to equivalences. Perhaps the most basic example is a minimal Kan fibration, which is a

Kan fibration In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category structure on simplicial sets and are therefore of fundamental importance. Kan complexes are ...

such that for each pair of ''n''-simplexes

\sigma, \tau

with the same boundary, if

\sigma, \tau

are fiberwise homotopic to each other relative to the boundary, then they are equal:

\sigma = \tau

. In particular, a fiber homotopy equivalence between minimal Kan fibrations is an isomorphism. A minimal Kan fibration is a fiber bundle (in the simplicial sense). Quillen's original approach to establishing the standard

model category A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , . Models can be divided in ...

structure on the category of simplicial sets (as well as more recent accounts) uses minimal Kan fibrations.

References

* * * * Barratt, Michael G., and J. C. Moore. "On semisimplicial fibre-bundles." American Journal of Mathematics 81.3 (1959): 639–657.

References

Further reading