An acyclic categories can be seen as a generalization of poset, so there must be some relationship between abstract polytope and acyclic category.
I'm working on this subject, and I don't know if there are any similar works. Unfortunately, I'm not a mathematician, I know nothing about algebraic geometry, so I asked chatGPT and got this:
An acyclic category is a type of mathematical structure that is used to model certain aspects of geometry. In particular, it is often used to model the relationships between different geometric objects, such as lines, points, and planes in a three-dimensional space.
In an acyclic category, the objects are the geometric objects and the morphisms are the relationships between them. For example, a morphism from a line to a point might represent the fact that the line passes through the point. A morphism from a plane to a line might represent the fact that the plane contains the line.
The acyclic property of the category ensures that there are no loops or cycles in the relationships between the objects, which is a key property of many geometric systems. This allows the category to accurately model the geometric relationships between the objects, and enables mathematicians to study and analyze these relationships using the tools of category theory.
And it provides some references:
"Acyclic categories and the foundations of geometry" by R. Brown and J.M. Clement, published in the journal Mathematical Proceedings of the Cambridge Philosophical Society.
"Acyclic categories and geometric models" by S. Mac Lane, published in the journal Bulletin of the American Mathematical Society.
"Acyclic categories and the algebraic geometry of projective spaces" by J.M. Clement, published in the journal Advances in Mathematics.
"Acyclic categories and the algebraic geometry of Grassmannians" by R. Brown, J.M. Clement, and S. Mac Lane, published in the journal Advances in Mathematics.
But I can't find any of them.
Do these papers really exist? Is there some accessible reference?
EDIT
I feel that the direction of algebraic geometry is different from what I am facing, so I will explain what I want to do.
The relation between a normal regular CW complex and a CW poset has been given by Anders Björner (1984). A regular CW complex is a CW complex whose gluing maps are homeomorphisms onto images. It is natural to generalize this correspondence to a normal semi-regular CW complex, whose gluing maps are local homeomorphisms onto images. In this case, the incidence relation is no longer just a inclusion relation.
For example, consider a digon:
The top left is the digon, which is a regular CW complex. The top right is its face lattice, which is a CW poset. The barycentric subdivision of a digon is built by dividing facets according to the barycenters (top left to bottom left), which results in a simplicial complex. Taking geometric realization of the face lattice (the order complex of a poset) also results in the same complex (top right to bottom left). The face lattice of this complex (bottom left to bottom right) is just the poset of chains of the face lattice of a digon (top right to bottom right).
Similar relations arise for normal semi-regular CW complexes. For example, consider a crescent:
The top left is the crescent, which is a normal semi-regular CW complex. The incidence structure of a crescent is an acyclic category (top right). The barycentric subdivision is defined in a similar way (top left to bottom left), which results in a geometrical simplicial set. Taking geometric realization of the incidence structure (the geometric realization of its nerve) also results in the same complex (top right to bottom left). The incidence structure of this complex (bottom left to bottom right) is just the nerve of the bounded acyclic category (top right to bottom right).
There seems to be more interesting properties on this correspondences, and there seems to be a more general form of incidence relation. A superficial discussion has been published on arXiv.
The current direction in algebraic geometry/topology seems to be to endow categories a geometrical view, which helps us study categories. For my purpose, the generalized incidence relations form acyclic categories, and some properties of acyclic categories can be interpreted as geometrical properties. Unfortunately, this is not compatible to the current study, or maybe there is some literature I'm missing. If there is literature on this generalization of incidence relation please answer me.
Associated to any category $C$, acyclic or not, is its nerve $N(C)$, which is a type of object called a simplicial set that can be seen as a generalization of an abstract polytope. When $C$ is a poset this reproduces a version of the order complex of a poset, so when $C$ is an acyclic category we get some kind of generalized order complex.
I'm not aware of any references discussing this exact case. Once you go past posets you might as well go all the way to considering categories in full generality, and the results are quite interesting. For example the nerve of the category $BG$ with one object with automorphism group $G$ models the classifying space of $G$, and taking the free abelian group on it produces a singular abelian group whose homology is the group homology of $G$. So there is some quite interesting stuff going on here and you need non-acyclic categories to get to it.