mapping simplex (Rev #1)

In the theory of cartesian fibrations of simplicial sets cartesian fibrations XΔ nX\to \Delta^n over a simplex play an important role since an arbitrary morphism XSX\to S is a cartesian fibration iff X× SXX\times_S X is a cartesian fibration.

XΔ nX\to \Delta^n is by the (,1)(\infty,1)-Grothendieck construction equivalently a functor Δ n(,1)Cat\Delta^n\to (\infty,1)Cat; i.e. a composable sequence of (,1)(\infty,1)-categories and functors ϕ:A 0A n\phi:A^0\leftarrow\dots\leftarrow A^n.

The mapping simplex M(ϕ)M(\phi) of ϕ\phi is defined by:

  • For a nonempty finite finite linear order LL with greatest element jj, a map Δ LM(ϕ)\Delta^L\to M(\phi) consists of a order preserving map f:L[n]f:L\to [n] and a morphism σ LA f(j)\sigma^L\to A^{f(j)}.

  • Given two such linear orders LL and L L^\prime with greatest elements jj resp. j j^\prime there is a natural map M(ϕ)(Δ L )M(ϕ)(Δ L)M(\phi)(\Delta^{L^\prime})\to M(\phi)(\Delta^{L}) sending (f,σ)(f,\sigma) to (fp,eσ)(f\circ p, e\circ \sigma), where e:A f(j )A f(p(j))e:A^{f(j^\prime)}\to A^{f(p(j))} is obtained by ϕ\phi.

Revision on February 5, 2013 at 22:52:00 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.