# The span-nerve of a category

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?

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Why are you using $D$ instead of $\Delta$? It seems in fact that this construction goes through for ANY presheaf category. Do you have a particular reason to consider this functor? – David Carchedi Jun 18 '10 at 18:09
Of course, this construction works for any indexing category D. But my motivation is "higher spans." Just like every category with fiber products has a bi-category of spans, every category with fiber products has a Kan complex of higher spans. – David Spivak Jun 18 '10 at 18:36
One terminological caveat: "$n$-spans" and "higher spans" are sometimes used for the globular analogue of this (which you may well already be familiar with, or I can give refs if not). I haven't heard of the simplicial version, but I don't know the simplicial work terribly well so that doesn't mean it's not out there... but it sounds like a very nice natural idea! – Peter LeFanu Lumsdaine Jun 20 '10 at 13:04
I think that $Gr(\Delta^n)= D\downarrow \underline{n}$ – Buschi Sergio May 5 '12 at 16:35