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Birkhoff's Covariety Theorem without limitations

Jiří Adámek — 2005

Commentationes Mathematicae Universitatis Carolinae

J. Rutten proved, for accessible endofunctors F of , 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 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 a member...

On pure quotients and pure subobjects

Jiří AdámekJiří Rosický — 2004

Czechoslovak Mathematical Journal

In the theory of accessible categories, pure subobjects, i.e. filtered colimits of split monomorphisms, play an important role. Here we investigate pure quotients, i.e., filtered colimits of split epimorphisms. For example, in abelian, finitely accessible categories, these are precisely the cokernels of pure subobjects, and pure subobjects are precisely the kernels of pure quotients.

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