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Pasting In Multiple Categories
 THEORY APPL. CATEG
, 1998
"... In the literature there are several kinds of concrete and abstract cell complexes representing composition in ncategories, ωcategories or 1categories, and the slightly more general partial ωcategories. Some examples are parity complexes, pasting schemes and directed complexes. In this paper we ..."
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In the literature there are several kinds of concrete and abstract cell complexes representing composition in ncategories, ωcategories or 1categories, and the slightly more general partial ωcategories. Some examples are parity complexes, pasting schemes and directed complexes. In this paper we give an axiomatic treatment: that is to say, we study the class of `ωcomplexes' which consists of all complexes representing partial ωcategories. We show that !complexes can be given geometric structures and that in most important examples they become wellbehaved CW complexes; we characterise ωcomplexes by conditions on their cells; we show that a product of ωcomplexes is again an ωcomplex; and we describe some products in detail.
Presentations of OmegaCategories By Directed Complexes
, 1997
"... The theory of directed complexes is extended from free !categories to arbitrary ! categories by defining presentations in which the generators are atoms and the relations are equations between molecules. Our main result relates these presentations to the more standard algebraic presentations; we ..."
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The theory of directed complexes is extended from free !categories to arbitrary ! categories by defining presentations in which the generators are atoms and the relations are equations between molecules. Our main result relates these presentations to the more standard algebraic presentations; we also show that every !category has a presentation by directed complexes. The approach is similar to that used by Crans for pasting presentations. 1991 Mathematics Subject Classification: 18D05. 1 Introduction There are at present three combinatorial structures for constructing !categories: pasting schemes, defined in 1988 by Johnson [8], parity complexes, introduced in 1991 by Street [16, 17] and directed complexes, given by Steiner in 1993 [15]. These structures each consist of cells which have collections of lower dimensional cells as domain and codomain; see for example Definition 2.2 below. They also have `local' conditions on the cells, ensuring that a cell together with its boundin...
Computads and 2 dimensional pasting diagrams (April 23, 2007)
"... §1 Types, shapes and occurrences p. 3 §2 Factorization and geometry p. 13 §3 Cuts in partial orders, and planar arrangements p. 32 ..."
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§1 Types, shapes and occurrences p. 3 §2 Factorization and geometry p. 13 §3 Cuts in partial orders, and planar arrangements p. 32
The word problem for computads
, 2005
"... 1. Concrete presheaf categories p. 20 2. ωgraphs p. 26 3. ωcategories p. 27 ..."
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1. Concrete presheaf categories p. 20 2. ωgraphs p. 26 3. ωcategories p. 27
A Geometry for Diagrammatic Computads
"... This talk is a preliminary report on a project suggested in [3]. The goal is to give a (combinatorially) geometric description of a rather large class of computads ( = free ∞categories with distinguished generators; all higherdimensional categories mentioned here are strict), much as [3] does for ..."
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This talk is a preliminary report on a project suggested in [3]. The goal is to give a (combinatorially) geometric description of a rather large class of computads ( = free ∞categories with distinguished generators; all higherdimensional categories mentioned here are strict), much as [3] does for the class of manytoone computads. Sources of inspiration were the articles [1], [6] and [5], in which computads are constructed from certain “complexes”. In each of them a condition of global acyclicity is imposed (allowing composition to be represented by plain settheoretic union), which the present work will avoid. The central notion here is that of a hypotopic set. It differs from the notion of an oriented polytopic set in that facets of a cell are allowed (and by further constraints in fact demanded) to occur in arbitrary lower dimensions (hence the prefix ‘hypo’). Simplicial and cubical sets as well as dendrotopic sets all can be interpreted as instances of hypotopic sets. Legitimate cell configurations in a hypotopic set, the tissues, form a category, which the expected pasting operations render ωcategorial. (So the ωcategory laws,
Pasting Presentations for OmegaCategories
, 1995
"... The pasting theorem showed that pasting schemes are useful in studying free !categories. It was thought that their inflexibility with respect to composition and identities prohibited wider use. This is not the case: there is a way of dealing with identities which makes it possible to describe !ca ..."
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The pasting theorem showed that pasting schemes are useful in studying free !categories. It was thought that their inflexibility with respect to composition and identities prohibited wider use. This is not the case: there is a way of dealing with identities which makes it possible to describe !categories in terms of generating pasting schemes and relations between generated pastings, i.e., with pasting presentations. In this chapter I develop the necessary machinery for this. The main result, that the !category generated by a pasting presentation is universal with respect to respectable families of realizations, is a generalization of the pasting theorem. Contents 1 Introduction 3 2 Pasting schemes according to Johnson 4 2.1 Graded sets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 !categories : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.3 Pasting schemes : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.4 The pasting theorem : : ...