Spahn
relative nerve (Rev #3, changes)

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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.

If J=[n]J=[n] (considered as a category), then any f:JsSetf: J\to sSet corresponds to a composable sequence ϕ:A 0A n\phi:A_0\leftarrow\dots \leftarrow A_n, and there is a map over Δ n\Delta^n to the mapping simplex M op(ϕ)M^{op}(\phi).

N f(J) M op(ϕ) Δ n\array{ N_f(J)&\to&M^{op}(\phi)\\ \searrow&&\swarrow\\ &\Delta^n }

If ff is constant on Δ 0\Delta^0 there is a canonical isomorphism N f(J)N(J)N_f(J)\simeq N(J) to the nerve of JJ, since the nerve is just a (covariant) functor from the category of linear orders Δ\Delta to JJ composed with the coYoneda embedding which is encoded in condition (1) above (and (2) and (3) are empty in this case).

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, Notation 6.2.0.1 (leading to the definition of a monad), 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 22:33:18 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.