Spahn
relative nerve (Rev #1)

Let JJ be a category, let f:J(,1)CatsSetf:J\to (\infty,1)Cat\hookrightarrow sSet be a functor. Then there is a cocartesian fibration p:N f(J)N(J)p:N_f(J)\to N(J) which is related to N(f)N(f) by the Grothendieck construction. N f(J)N_f(J) is called the relative nerve of JJ relative ff.

Let LL be a linear order. A map Δ LN f(J)\Delta^L\to N_f(J) consists of the following data:

(1) A functor s:LJs:L\to J.

(2) For every nonempty subset L LL^\prime\subseteq L with maximal element j L j^\prime\in L^\prime, a map t(J ):Δ L f(s(j ))t(J^\prime):\to \Delta^{L^\prime}\to f(s(j^\prime)).

(3) Coherence in the obvious way: For nonempty subsets L L LL^{\prime \prime}\subseteq L^\prime\subseteq L with maximal elements j j^{\prime\prime} resp. j j^\prime, the diagram

Δ L t(L ) f(s(j )) Δ L t(L ) f(s(j )\array{ \Delta^{L^{\prime\prime}}&\stackrel{t(L^{\prime\prime})}{\to}&f(s(j^{\prime\prime}))\\ \downarrow&&\downarrow\\ \Delta^{L^{\prime}}&\stackrel{t(L^{\prime})}{\to}&f(s(j^{\prime}) }

is required to commute.

References

  • Jacob Lurie, Higher Topos Theory, §3.2.5

  • Jacob Lurie, Derived Algebraic Geometry II, Noncommutative Algebra, §3.1, p.94-97

  • The relative nerve appears en passant also in

    • Jacob Lurie, Higher Algebra, Construction 2.2.5.12

    • Jacob Lurie, (,2)(\infty,2)-Categories and the Goodwillie Calculus, Theorem 0.0.3 (B5)

Revision on February 11, 2013 at 07:44:50 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.