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With topology in mind, one might imagine simply transporting the defini- tion of Euler characteristic from spaces to categories via the classifying space functor, as with
metric and approach spaces) are obtained in this way, and the category of closure spaces appears as a category of canonical ( P , V)-algebras, where P is the powerset
We saw in the prologue that the comonad Path on Cat has as its coalgebras free categories on graphs (the category of coalgebras is equivalent to the category of graphs), and that
there is a fibration (Definition 1.11) from G to its set of connected components (seen as a discrete category) given by the quotient functor generated by the collection of morphisms
As far as we are aware the duality between N -compact spaces and a class of Z -rings is new, as is the duality between a certain class of topological abelian groups and a class
d - Space of d-spaces (in the sense of [11]) is topological and its full subcategory generated by suitably ordered cubes is our proposed convenient category for directed homotopy..
Thus, we know that the presheaf functor Pos −→ GTop preserves exponentiable objects (since every poset P is exponentiable as is every presheaf topos PSh( P )) and
In [CKW] the theory was illustrated in detail with a section devoted to rel seen as a colax functor from the 2-category REG of regular categories, arbitrary functors, and
