Spahn relative nerve

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

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

is required to commute.

If $J=[n]$ (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 $M^{op}(\phi)$.

$$\array{ N_f(J)&\to&M^{op}(\phi)\\ \searrow&&\swarrow\\ &\Delta^n }$$

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).

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, $(\infty,2)$-Categories and the Goodwillie Calculus, Theorem 0.0.3 (B5)