Applied Categorical Structures – Official Blog

by Gumm, H. Peter

We define an out-degree for F-coalgebras and show that the coalgebras of outdegree at most κ form a covariety. As a subcategory of all F-coalgebras, this class has a terminal object, which for many problems can stand in for the terminal F-coalgebra, which need not exist in general. As examples, we derive structure theoretic results about minimal coalgebras, showing that, for instance minimization of coalgebras is functorial, that products of finitely many minimal coalgebras exist and are given by their largest common subcoalgebra, that minimal subcoalgebras have no inner endomorphisms and show how minimal subcoalgebras can be constructed from Moore-automata. Since the elements of minimal subcoalgebras must correspond uniquely to the formulae of any logic characterizing observational equivalence, we give in the last section a straightforward and self-contained account of the coalgebraic logic of D. Pattinson and L. Schröder, which we believe is simpler and more direct than the original exposition.

DOI: 10.1007/s10485-007-9116-1
Online Date: 12/14/2007
Print publication date: 6/1/2008
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by Banaschewski, B.

DOI: 10.1007/s10485-007-9119-y
Online Date: 12/8/2007
Print publication date: 12/1/2007
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by Colebunders, E.; Hušek, M.; Künzi, H.-P.; Tholen, W.

DOI: 10.1007/s10485-007-9117-0
Online Date: 12/8/2007
Print publication date: 12/1/2007
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by Bourn, D.; Rosický, J.

We characterize cosieves in locally presentable categories which are generated by a set of objects or are even principal. We apply our results to the category of algebraic theories where they are related to Mal’cev conditions dealt with in universal algebra.

DOI: 10.1007/s10485-007-9114-3
Online Date: 12/8/2007
Print publication date: 12/1/2008
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by Ball, Richard N.; Pultr, Aleš; Sichler, Jiří

We show that the Priestley sum of finite trees contains no cyclic finite poset.

DOI: 10.1007/s10485-007-9118-z
Online Date: 12/7/2007
Print publication date: 12/1/2008
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