Consider the category of semi-smplicial sets, ie simplicial sets without degeneracies. It is a preseaf category over $\Delta_{inj}$, the category of ordinals together with injective maps.
The usual definition of simplicial sets is to add morphisms to the base category $\Delta_{inj}$, to get the category $\Delta$, which is the category of ordinals together with order-preserving maps, and define simplicial sets to be the presheaves over $\Delta$.
It is my intuition that one could equivalently add objects to the category $\Delta_{inj}$, and give an equivalent definition of simplicial sets. The idea is to define $\Delta'$, such that simplicial sets are presheaves over $\Delta'$, but that instead of encoding the extra degeneracy data in the maps of $\Delta'$, I can encode them in the objects.
I can give an explicit definition for $\Delta'$ in smaller dimensions :
- There is only one object of dimension $0$, which is to be thought of as a point
- There are two objects of dimension $1$, one is the usual $1$-simplex and the other one is the degenerate $1$-simplex. The usual $1$-simplex has two maps from the point, whereas the degenerate $1$-simplex has only one map to the point.
- There are eleven objects of dimension $2$, eight of them are the usual $2$ simplex, in which any number of faces among its three are degenerate ($\left(^3_0\right) + \left(^3_1\right) + \left(^3_2\right) + \left(^3_3\right) = 8$). Two are the once degenerate $1$-simplex, and the last one is the twice degenerate point. The morphsims coming to these objects are what one can imagine, a $2$-simplex with one degenerate face has two maps coming from the non-degenerate $1$-simplex, and one map coming from the degenerate $1$-simplex, and so on...
The category I am looink for has objects in all dimensions, and the number of objects increases with the dimension, but this gives a good idea in lower dimensions of what I expect it to look like.
This category I am describing looks to me very similar to marked semi-simplicial sets, in which all the points are marked. If I denote $Y : \Delta_{inj} \to Set^{\Delta_{inj}^{op}}$ the Yoneda embedding of $\Delta_{inj}$ into the category of semi-simplicial sets, then the category of marked semi-simplicial sets would be the category whose points are couples $([n],X)$, where $[n]\in\Delta_{inj}$ is a ordinal and $X$ is a sub-semi-simplicial set of $Y([n])$ containing all the points. The morphsims are then given by $Hom(([n],X)([m],Z)) = \{f \in Hom_{\Delta_{inj}}([n],[m]), Y(f)(X) \subset Z\}$
It is not quite the same, since already in dimension $2$, there are only $9$ marked semi-simplicial sets, whereas the category I am after has $11$ objects,but I am wondering if an iteration of this construction could work.
So this is moslty a reference request, has this construction ever been studied? If so I would be very interested in knowing where I could find such techniques, if there are indeed links with marked semi-simplicial complexes, or if there is any combinatorial description of the category $\Delta'$ I am after.
Edit :
I am adding a picture of what I am thinking of, up to dimension 2. On this picture, the blue arrows and surfaces are the degenerate ones. The red arrows are the arrows of the category, and they are subject to the usual semisimplicial relations, but there are also tuples of purple arrows, that are also in the category, but are now supposed to be equal.
