The aim of this paper is to relate initial algebra semantics and final coalgebra semantics. It is shown how these two approaches to the semantics of programming languages are each others dual, and some conditions are given under which they coincide. More precisely, it is shown how to derive initial semantics from final semantics, using the initiality and finality to ensure their equality. Moreo...

We de ne the continuum up to order isomorphism, and hence up to homeomorphism via the order topology, in terms of the nal coalgebra of either the functor N×X , product with the set of natural numbers, or the functor 1+N×X . This makes an attractive analogy with the de nition of N itself as the initial algebra of the functor 1+X , disjoint union with a singleton. We similarly specify Baire space...

Many recursive functions can be defined elegantly as the unique homomorphisms, between two algebras, two coalgebras, or one each, that are induced by some universal property of a distinguished structure. Besides the well-known applications in recursive functional programming, several basic modes of reasoning about scientific models have been demonstrated to admit such an exact meta-theory. Here...

Motivated by the success of classical random walks and chaotic dynamical systems, we study the quantisation of the random walk over Z and its relationships with a classical chaotic system x → 2xmod1, x ∈ [0,1]. In the physics literature, quantum random walks have been studied, for instance, by Ambainis et al. [1] and Konno et al. [8, 9]. In [8], Konno shows that the quantum random walk over Z c...

We give a short and streamlined proof of the following statement recently proven by author M. Zeinalian: cobar construction dg coassociative coalgebra normalized singular chains on path-connected pointed space is naturally quasi-isomorphic as associative algebra to based loop space. This extends classical theorem F. Adams originally for simply connected spaces. Our relating functor left adjoint...

we find a criterion for a morphism of coalgebras over a barr-exact category to be effective descent and determine (effective) descent morphisms for coalgebras over toposes in some cases. also, we study some exactness properties of endofunctors of arbitrary categories in connection with natural transformations between them as well as those of functors that these transformations induce between co...

The purpose of this dissertation is to further previous work on coalgebras as infinite statebased transition systems and their logical characterisation with particular focus on infinite regular behaviour and branching. Finite trace semantics is well understood [DR95] for nondeterministic labelled transition systems, and has recently [Jac04, HJS06] been generalised to a coalgebraic level where m...

Using results of Trnková, we first show that subcoalgebras are always closed under finite intersections. Assuming that the type functor F is bounded, we obtain a concrete representation of the terminal F-coalgebra. Several equivalent characterizations of boundedness are provided.

Braided bialgebras were defined by mimicking the definition of bialgebras in a braided category; see [Ta]. In this paper we are interested in those braided bialgebras that are connected as a coalgebra, and such that, up to multiplication by a certain scalar, their braiding restricted to the primitive part is a Hecke operator. To every braided bialgebra as above we associate a braided Lie algebr...

In this paper we show that it is possible to model observable behaviour of coalgebras independently from their internal dynamics, but within the general framework of representing behaviour by a map into a “final” coalgebra. In the first part of the paper we characterise Set-endofunctors F with the property that bisimilarity of elements of F -coalgebras coincides with having the same observable ...