mapping simplex (Rev #2)

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 for all nn, X× SΔ nΔ nX\times_S \Delta^n\to \Delta^n is a cartesian fibration.

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:Δ LA f(j)\sigma:\Delta^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.

There is a natural map h:M(ϕ)Δ nh:M(\phi)\to \Delta^n (take J=mJ=m, then the Yoneda lemma gives a map Δ mΔ n\Delta^m\to \Delta^n).

An edge ee of M(ϕ)M(\phi) is defined by a pair of integers 0ijn0\le i\le j\le n and an edge e A je^\prime\in A^j. M(ϕ)M(\phi) becomes a marked simplicial set (M(ϕ),E)(M(\phi), E) by marking those edges for which e e^\prime is degenerated.


Let p:XΔ np:X\to \Delta^n be a cartesian fibration, let ϕ:A 0A n\phi:A^0\leftarrow\dots\leftarrow A^n be a composable sequence of (,1)(\infty,1)-categories and functors. Then A map q:M(ϕ)Xq:M(\phi)\to X is called a quasi-equivalence if it satisfies:

(1) The map hh commutes with pp and qq.

(2) qq sends marked edges of M(ϕ)M(\phi) to pp-cartesian ones.

(3) For every 0in0\le i\le n, the induced map A ip 1{i}A^i\to p^{-1}\{ i \} is a categorical equivalence?.


Let p:XΔ np:X\to \Delta^n be a cartesian fibration.

(1) There exists a composable sequence of (,1)(\infty,1)-categories and functors ϕ:A 0A n\phi:A^0\leftarrow\dots\leftarrow A^n and a quasi-equivalence q:M(ϕ)Xq:M(\phi)\to X.

(2) If ϕ:A 0A n\phi:A^0\leftarrow\dots\leftarrow A^n is a composable sequence and q:M(ϕ)Xq:M(\phi)\to X a quasi-equivalence. Then for any map TΔ nT\to \Delta^n, the induced map

M(ϕ)× Δ nTX× Δ nTM(\phi)\times_{\Delta^n}T\to X\times_{\Delta^n}T

is a categorical equivalence?.

  • relative nerve?

  • (infinity,1)-Grothendieck construction?


Revision on February 6, 2013 at 00:34:08 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.