relative nerve

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).