Let $J$ be a category, let $f:J\to (\infty,1)Cat\hookrightarrow sSet$ be a functor. Then there is a cocartesian fibration $p:N_f(J)\to N(J)$ which is related to $N(f)$ by the Grothendieck construction. $N_f(J)$ is called the relative nerve of $J$ relative $f$.
Let $L$ be a linear order. A map $\Delta^L\to N_f(J)$ consists of the following data:
(1) A functor $s:L\to J$.
(2) For every nonempty subset $L^\prime\subseteq L$ with maximal element $j^\prime\in L^\prime$, a map $t(J^\prime):\to \Delta^{L^\prime}\to f(s(j^\prime))$.
(3) Coherence in the obvious way: For nonempty subsets $L^{\prime \prime}\subseteq L^\prime\subseteq L$ with maximal elements $j^{\prime\prime}$ resp. $j^\prime$, the diagram
is required to commute.
If $J=[n]$ (considered as a category), then any $f: J\to sSet$ corresponds to a composable sequence $\phi:A_0\leftarrow\dots \leftarrow A_n$, and there is a map over $\Delta^n$ to the mapping simplex $M^{op}(\phi)$.
If $f$ is constant on $\Delta^0$ there is a canonical isomorphism $N_f(J)\simeq N(J)$ to the nerve of $J$, since the nerve is just a (covariant) functor from the category of linear orders $\Delta$ to $J$ composed with the coYoneda embedding which is encoded in condition (1) above (and (2) and (3) are empty in this case).
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, $(\infty,2)$-Categories and the Goodwillie Calculus, Theorem 0.0.3 (B5)
Last revised on February 11, 2013 at 22:33:18. See the history of this page for a list of all contributions to it.