Applied Categorical Structures – Official Blog

by Krishnan, Sanjeevi

As a practical foundation for a homotopy theory of abstract spacetime, we extend a category of certain compact partially ordered spaces to a convenient category of “locally preordered” spaces. We show that our new category is Cartesian closed that the forgetful functor to the category of compactly generated spaces creates all limits and colimits.

DOI: 10.1007/s10485-008-9140-9
Online Date: 6/28/2008
Print publication date: 10/1/2009
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by Dimov, Georgi D.

Generalizing Duality Theorem of H. de Vries, we define a category which is dually equivalent to the category of locally compact Hausdorff spaces and perfect maps.

DOI: 10.1007/s10485-008-9144-5
Online Date: 6/28/2008
Print publication date: 10/1/2009
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by Heymans, Hans

We consider the theory of categories enriched in an involutive quantaloid $\mathcal{Q}$ : the $\mathcal{Q}$-*-categories. After giving an introduction to involutive quantaloids and nuclei, we use matrices with entries in $\mathcal{Q}$ to define $\mathcal{Q}$-*-categories. Then we examine the relations between two kinds of morphisms between them, the functors and the *-maps, to provide a basis to study completeness properties. These results are used to provide a definition of pseudo-presheaves, presheaves and sheaves on involutive quantaloids in order to get a generalization of presheaves and sheaves on sites. Finally a characterization of these sheaves in terms of covers and compatible families is presented.

DOI: 10.1007/s10485-008-9149-0
Online Date: 6/26/2008
Print publication date: 2/1/2009
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