Jiřı́ Adámek

TCS Journal 2004 Journal Article

On coalgebra based on classes

• Jiřı́ Adámek
• Stefan Milius
• Jiřı́ Velebil

The category Class of classes and functions is proved to have a number of properties suitable for algebra and coalgebra: every endofunctor is set-based, it has an initial algebra and a terminal coalgebra, the categories of algebras and coalgebras are complete and cocomplete, and every endofunctor generates a free completely iterative monad. A description of a terminal coalgebra for the power-set functor is provided.

TCS Journal 2003 Journal Article

Infinite trees and completely iterative theories: a coalgebraic view

• Peter Aczel
• Jiřı́ Adámek
• Stefan Milius
• Jiřı́ Velebil

Infinite trees form a free completely iterative theory over any given signature—this fact, proved by Elgot, Bloom and Tindell, turns out to be a special case of a much more general categorical result exhibited in the present paper. We prove that whenever an endofunctor H of a category has final coalgebras for all functors H(_)+X, then those coalgebras, TX, form a monad. This monad is completely iterative, i. e. , every guarded system of recursive equations has a unique solution. And it is a free completely iterative monad on H. The special case of polynomial endofunctors of the category Set is the above mentioned theory, or monad, of infinite trees. This procedure can be generalized to monoidal categories satisfying a mild side condition: if, for an object H, the endofunctor H⊗_+I has a final coalgebra, T, then T is a monoid. This specializes to the above case for the monoidal category of all endofunctors.

TCS Journal 2003 Journal Article

On final coalgebras of continuous functors

• Jiřı́ Adámek

Continuous endofunctors F of locally finitely presentable categories carry a natural metric on their final coalgebra. Whenever F(0) has an element, this metric is proved to be a Cauchy completion of the initial algebra of F. This is illustrated on the poset of real numbers represented as a final coalgebra of an endofunctor of Pos by Pavlović and Pratt. Under additional assumptions on the locally finitely presentable category, all finitary endofunctors are proved to have a final coalgebra constructed in ω+ω steps of the natural iteration construction.

TCS Journal 2003 Journal Article

Preface

• Jiřı́ Adámek
• Martı́n Escardó
• Martin Hofmann