A Combinatorial-Topological Shape Category for Polygraphs

Abstract

We introduce globular posets, an oriented version of thin, recursively dividable posets, as a new combinatorial presentation of higher categories. We show that globular posets encompass many common classes of higher-categorical cell shapes, including standard globes, cubes, oriented simplices, and a large sub-class of opetopes, and that they are closed under lax Gray products and joins. Then, we prove that they form a wellbehaved sub-class of Steiner's directed complexes, and derive that each globular poset presents a polygraph with one generator for each of its elements. We define a regular polygraph to be a presheaf on a category of globular posets with a greatest element, and prove that the category of regular polygraphs is equivalent to a full subcategory of the category of polygraphs. We extend the lax Gray product and join constructions to regular polygraphs. Finally, we define a geometric realisation functor, and prove that the geometric realisation of a regular polygraph is a CW complex. In an appendix, we propose regular polygraphs satisfying a representability property as a new non-algebraic notion of weak higher category. ContentsA Lax Gray product of globes 62 B Representable regular polygraphs: an introduction 66Construction 4.7. Let X be a partial ω-category. The J-dual D J (X) of X is the partial ω-category with an n-cell D J (x) for each n-cell x of X, andwhere D J (x) # k D J (y) is defined whenever y # k x is defined and k + 1 ∈ J, or x # k y is defined and k + 1 / ∈ J.

The smash product of monoidal theories

“…In this section, we use results of [Had20a,Section 6]. These results are stated relative to the restricted class of constructible directed complexes, but the proofs do not involve any properties that are not satisfied by all regular directed complexes.…”

“…Suppose that m > 1 and that k = frdim(U ). Then we proceed as in the proof of [Had20a,Proposition 26] to produce i ∈ {1, . .…”

Diagrammatic sets and rewriting in weak higher categories

“…Instead, we base our revisitation on Richard Steiner's theory of directed complexes [Ste93], continuing the elaboration that we started in [Had20]. The basic data structure is a finite graded poset together with an orientation, that is, an edge-labelling of the poset's Hasse diagram in the set {+, −}.…”

“…Conversely, a closed embedding of posets that is compatible with the orientations is an inclusion of oriented graded posets. This was the definition of inclusion used in [Had20].…”

The Identification of Customary Rules in International Criminal Law

“…On the one hand, wilful killing in armed conflict may be prosecuted and punished by national courts as ordinary crimes of murder, war crimes, or crimes against humanity at the international level. 161 For example, in 1973, a United States Army Lieutenant was convicted of murder and assault for his involvement in the My Lai massacre during the Vietnam War. 162 On the other hand, intentional killing in peacetime is prohibited and criminalised as murder in national law.…”