Let ${\bf D}$ denote the category of finite non-empty sets and monomorphisms between them; let $\underline{n}$= {$0,1,\ldots,n$}. I will call presheaves on ${\bf D}$ *simplicial sets* and denote it $sSets=Pre({\bf D})$ by abuse of terminology, but note that they do have a nice geometric realization; alternately a concerned reader can replace ${\bf D}$ by the category of finite ordinals and the discussion and question will still make sense. I will denote a Yoneda object in *sSets* by $\Delta^n:=y(\underline{n})$ and call it an $n$-simplex.

For any functor F whose codomain is ${\bf Sets}$, let Gr(F) denote its Grothendieck construction. So we can consider $Gr$ as a functor $Gr\colon sSets\rightarrow Cat$. Applied to an $n$-simplex, this functor returns the poset $Gr(\Delta^n)=P_+${$0,1,\ldots,n$} of non-empty subsets of $n$ (morphisms are opposite to inclusions). So $Gr(\Delta^0)$ is the terminal category, $Gr(\Delta^1)$ is the "span" category, and $Gr(\Delta^n)$ might be called an $n$-span.

The functor $Gr\colon sSets\rightarrow Cat$ has a right adjoint, which I'll call $Sp\colon Cat\to sSets$ defined in the obvious way: for a category $C$, define $Sp(C)_n:=Fun(Gr(\Delta^n),C)$ where $Fun(X,Y)$ is the set of functors.

From this we have constructed something that might be called a "span nerve" of a category. It receives a map from the usual nerve (given by a certain functor from $Gr(\Delta^n)\to[n]$). If $C$ has fiber products then $Sp(C)$ is a Kan complex. In general perhaps $Sp(C)$ (or its Kan- or Joyal- fibrant replacement) has interesting meaning as a space or quasi-category.

Question: This "Span nerve" has presumably been studied before, but I think it's faster to ask here than to try to find it myself. What should its name be, and where might I find information about it?