In the theory of cartesian fibrations of simplicial sets cartesian fibrations over a simplex play an important role since an arbitrary morphism is a cartesian fibration iff for all , is a cartesian fibration.
A cartesian fibration is by the -Grothendieck construction equivalently a functor ; i.e. a composable sequence of -categories and functors .
The mapping simplex of is defined by:
For a nonempty finite finite linear order with greatest element , a map consists of a order preserving map and a morphism .
Given two such linear orders and with greatest elements resp. there is a natural map sending to , where is obtained by .
There is a natural map (take , then the Yoneda lemma gives a map ).
An edge of is defined by a pair of integers and an edge . becomes a marked simplicial set by marking those edges for which is degenerated.
Let be a cartesian fibration, let be a composable sequence of -categories and functors. Then A map is called a quasi-equivalence if it satisfies:
(1) The map commutes with and .
(2) sends marked edges of to -cartesian ones.
(3) For every , the induced map is a categorical equivalence?.
Let be a cartesian fibration.
(1) There exists a composable sequence of -categories and functors and a quasi-equivalence .
(2) If is a composable sequence and a quasi-equivalence. Then for any map , the induced map
is a categorical equivalence?.
relative nerve?
Created on February 6, 2013 at 00:27:33. See the history of this page for a list of all contributions to it.