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Bimorphisms in pro-homotopy and proper homotopy

Jerzy Dydak, Francisco Ruiz del Portal (1999)

Fundamenta Mathematicae

A morphism of a category which is simultaneously an epimorphism and a monomorphism is called a bimorphism. The category is balanced if every bimorphism is an isomorphism. In the paper properties of bimorphisms of several categories are discussed (pro-homotopy, shape, proper homotopy) and the question of those categories being balanced is raised. Our most interesting result is that a bimorphism f:X → Y of t o w ( H 0 ) is an isomorphism if Y is movable. Recall that ( H 0 ) is the full subcategory of p r o - H 0 consisting of...

Birkhoff's Covariety Theorem without limitations

Jiří Adámek (2005)

Commentationes Mathematicae Universitatis Carolinae

J. Rutten proved, for accessible endofunctors F of Set, the dual Birkhoff’s Variety Theorem: a collection of F -coalgebras is presentable by coequations ( = subobjects of cofree coalgebras) iff it is closed under quotients, subcoalgebras, and coproducts. This result is now proved to hold for all endofunctors F of Set provided that coequations are generalized to mean subchains of the cofree-coalgebra chain. For the concept of coequation introduced by H. Porst and the author, which is a subobject of...

Booleanization of uniform frames

Bernhard Banaschewski, Aleš Pultr (1996)

Commentationes Mathematicae Universitatis Carolinae

Booleanization of frames or uniform frames, which is not functorial under the basic choice of morphisms, becomes functorial in the categories with weakly open homomorphisms or weakly open uniform homomorphisms. Then, the construction becomes a reflection. In the uniform case, moreover, it also has a left adjoint. In connection with this, certain dual equivalences concerning uniform spaces and uniform frames arise.

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