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The construction of the free category on a quiver is standard in category theory.

Is there some $2$-dimensional analog of a quiver such that each $2$-quiver freely gives rise to a $2$-category? How about for $n>2$?

If a graph is a set of edges and a set of vertices, it seems reasonable that we could consider a set of edges, a set of vertices and and a set of 'faces', continuing on to 'volumes' and 'hypervolumes' etc., where we formalize it all using ordered pairs of lists. For example, the $2$-graph underlying the $2$-cell diagram

could be defined by setting the vertices $V$, edges $E$, and faces $F$ by

$$V=\{A,B,C,D,E\},$$

$$E=\{(A,B),(B,C),(C,D),(A,E),(E,D)\},$$

$$F=\Big\{\Big(\big((B,E),(A,B)\big),\big((A,E)\big)\Big),\Big(\big((B,C)\big),\big((E,C),(B,E)\big)\Big),\Big(\big((C,D),(E,C)\big),\big((E,D)\big)\Big)\Big\}.$$

The 'free $2$-category' on this $2$-graph would have $V$ for objects, lists of composable edges for $1$-cells, and lists of vertically composable faces for the $2$-cells in each category (where we add 'identity faces' and 'faces' corresponding to whiskerings with these identities).

I believe graphs are usually defined with so-called incidence functions instead of ordered pairs as above, but the incidence functions here are simply given by projection onto the first coordinate for sources and projection onto the second coordinate for targets.

Despite this, most attempts I'm familiar with to formalize the graph theory underlying pasting diagrams and such rely on geometric realizations and topological methods in dimension $2$, and Whitney stratifications in higher dimensions. It looks like CW complexes or abstract simplical complexes might be key words close to what I'm looking for. Has an approach similar to the above been developed anywhere in the literature? Any assistance is appreciated.

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  • $\begingroup$ You could be looking for globular sets. 1-globular sets are precisely quivers. See here ncatlab.org/nlab/show/globular+set $\endgroup$
    – Gjergji Zaimi
    Commented Mar 12, 2022 at 8:12
  • $\begingroup$ I believe Alec is after something like the more general concept of a (2-dimensional) computad (a.k.a. polygraph), introduced by Ross Street. $\endgroup$
    – Alexander Campbell
    Commented Mar 12, 2022 at 8:50
  • $\begingroup$ I’m looking for the ‘graph structure’ that ‘underlies a $2$-category’ in the same way that directed graphs underly ordinary categories. More precisely, there should be a forgetful functor from the $2$-category of $2$-categories to the $2$-category of ‘these structures’ whose left adjoint gives the ‘free $2$-category’ on one of ‘these structures’, categorifying one stage the fact that there is a forgetful functor from the $1$-category of categories to the $1$-category of directed graphs whose left adjoint gives the free category on a directed graph. $\endgroup$
    – Alec Rhea
    Commented Mar 12, 2022 at 18:17
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    $\begingroup$ Your description "the graph structure that underlies a 2-category" doesn't uniquely determine an answer. There is a free functor generating 2-categories from 2-globular sets, and also a free functor generated 2-categories from 2-computads, and other things in between. But I expect that computads (ncatlab.org/nlab/show/computad) are what you want. @AlexanderCampbell, I think you should post that as an answer. $\endgroup$
    – Mike Shulman
    Commented Mar 12, 2022 at 23:50
  • $\begingroup$ @AlexanderCampbell This is indeed what I was looking for, thank you for the reference and thank you Mike for the link. If you'd like to post this as an answer I'll accept it to close out the question. $\endgroup$
    – Alec Rhea
    Commented Mar 13, 2022 at 19:24

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