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Pasting Schemes for the Monoidal Biclosed Structure on
, 1995
"... Using the theory of pasting presentations, developed in chapter 2, I give a detailed description of the tensor product on !categories, which extends Gray's tensor product on 2categories and which is closely related to BrownHiggins's tensor product on !groupoids. Immediate consequences are a gen ..."
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Using the theory of pasting presentations, developed in chapter 2, I give a detailed description of the tensor product on !categories, which extends Gray's tensor product on 2categories and which is closely related to BrownHiggins's tensor product on !groupoids. Immediate consequences are a general and uniform definition of higher dimensional lax natural transformations, and a nice and transparent description of the corresponding internal homs. Further consequences will be in the development of a theory for weak ncategories, since both tensor products and lax structures are crucial in this. Contents 1 Introduction 3 2 Cubes and cubical sets 5 2.1 Cubes combinatorially : : : : : : : : : : : : : : : : : : : : : : : : 5 2.2 A model category for cubes : : : : : : : : : : : : : : : : : : : : : 6 2.3 Generating the model category for cubes : : : : : : : : : : : : : : 7 2.4 Cubical sets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.5 Duality : : : : : : : : : : : : : ...
Quillen Closed Model Structures for Sheaves
, 1995
"... In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2groupoids, the category of bisim ..."
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Cited by 14 (0 self)
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In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2groupoids, the category of bisimplicial sheaves and the category of simplicial sheaves of groupoids. Subsequently, the homotopy theories of these categories are related to the homotopy theory of simplicial sheaves. 1 Introduction There are two ways of trying to generalize the well known closed model structure on the category of simplicial sets to the category of simplicial objects in a Grothendieck topos. One way is to concentrate on the local aspect, and to use the Kanfibrations as a starting point. In [14] Heller showed that for simplicial presheaves there is a local (there called right) closed model structure. In [2] K. Brown showed that for a topological space X the category of "locally fibrant" sheaves of spectra on ...
Sketches: Outline with References
 Dept. of Computer Science, Katholieke Universiteit Leuven
, 1994
"... This document is an outline of the theory of sketches with pointers to the literature. An extensive bibliography is given. Some coverage is given to related areas such as algebraic theories, categorial model theory and categorial logic as well. An appendix beginning on page 11 provides definitions o ..."
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Cited by 2 (0 self)
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This document is an outline of the theory of sketches with pointers to the literature. An extensive bibliography is given. Some coverage is given to related areas such as algebraic theories, categorial model theory and categorial logic as well. An appendix beginning on page 11 provides definitions of some of the less standard terms used in the paper, but the reader is expected to be familiar with the basic ideas of category theory. A rough machine generated index begins on page 21. I would have liked to explain the main ideas of all the papers referred to herein, but I am not familiar enough with some of them to do that. It seemed more useful to be inclusive, even if many papers were mentioned without comment. One consequence of this is that the discussions in this document often go into more detail about the papers published in North America than about those published elsewhere. The DVI file for this article is available by anonymous FTP from ftp.cwru.edu in the directory
DISTRIBUTIVE LAWS IN PROGRAMMING STRUCTURES
, 2009
"... Generalised Distributive laws in Computer Science are rules governing the transformation of one programming structure into another. In programming, they are programs satisfying certain formal conditions. Their importance has been to date documented in several isolated cases by diverse formal approac ..."
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Generalised Distributive laws in Computer Science are rules governing the transformation of one programming structure into another. In programming, they are programs satisfying certain formal conditions. Their importance has been to date documented in several isolated cases by diverse formal approaches. These applications have always meant leaps in understanding the nature of the subject. However, distributive laws have not yet been given the attention they deserve. One of the reasons for this omission is certainly the lack of a formal notion of distributive laws in their full generality. This hinders the discovery and formal description of occurrences of distributive laws, which is the precursor of any formal manipulation. In this thesis, an approach to formalisation of distributive laws is presented based on the functorial approach to formal Category Theory pioneered by Lawvere and others, notably Gray. The proposed formalism discloses a rather simple nature of distributive laws of the kind found in programming structures based on lax 2naturality and Gray’s tensor product of 2categories. It generalises the existing more specific notions of distributive