homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
Pasting schemes are one possibility to give a rigorous treatment of the (older) notational device of pasting diagrams.
More than one definition of pasting scheme has been used to give a justification to the practice of pasting diagrams.
For example, there are pasting schemes in the sense of Johnson 1987, and there are pasting schemes in the sense of Power 1990.
Power’s notion is based on planar embeddings of quivers that satisfy the following conditions:
For the free category generated by , every endomorphism is an identity ( has no directed cycles);
There are vertices of (the source and sink, respectively) such that for every vertex , the hom-sets and are inhabited (there exist directed paths in from to and from to ).
We will call such quivers progressive, in rough analogy with progressive string diagrams.
There is a restricted geometric realization functor
(into a convenient category of topological spaces ) which we again denote as .
(after Power) A pasting scheme consists of a progressive quiver together with an subspace embedding , such that is a embedding on each edge interior.
To each pasting scheme we may associate a 2-computad . Part of the computad structure is easily described:
The underlying quiver of is ;
The elements of (the 2-cells) are the bounded connected components of .
To complete the description of the computad, we need to define the source and target of a 2-cell . For each edge such that lies in the boundary of , and each interior point , we let be the tangent vector, and the inward-pointing normal (pointing into ); the embedding condition ensures that these make sense. Then say that is a negative edge of if
(the cell is “to the right” as is traversed along ) and a positive edge if
(the cell is to the left).
Power shows that under the progressivity conditions, for each there are unique distinct vertices such that the negative edges of form a directed path from to , and the positive edges of form a directed path . We then define and .
Power’s pasting theorem may be stated in the following form:
For every pasting scheme , the free 2-category on the 2-computad has exactly one 2-morphism whose -domain is the source of , whose -codomain is the sink of , whose -domain of is the directed path consisting of positive edges of the unique unbounded component of , and whose -codomain is the directed path consisting of negative edges of that component.
The notion of pasting in a 2-category was introduced in
Jean Bénabou, Introduction to bicategories, in Lecture Notes in Mathematics Vol. 47, pp. l-77, Springer-Verlag, New York/Berlin, (1967)
Michael Johnson: Pasting Diagrams in -Categories with Applications to Coherence Theorems and Categories of Paths, Doctoral Thesis, University of Sydney, 1987
Other notions of pasting presentations have been given by Street (parity complexes) and by Steiner,
Ross Street, Parity complexes, Cahiers Top. Géom Diff. Catégoriques 32 (1991), 315-343. (link) Corrigenda, Cahiers Top. Géom Diff. Catégoriques 35 (1994), 359-361. (link)
Richard Steiner, The algebra of directed complexes, Appl. Cat. Struct. 1 (1993), 247-284.
Last revised on August 6, 2017 at 18:54:49. See the history of this page for a list of all contributions to it.