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Algebras, Coalgebras, Monads and Comonads
, 2001
"... Whilst the relationship between initial algebras and monads is wellunderstood, the relationship between nal coalgebras and comonads is less well explored. This paper shows that the problem is more subtle and that final coalgebras can just as easily form monads as comonads and dually, that initial a ..."
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Cited by 6 (3 self)
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Whilst the relationship between initial algebras and monads is wellunderstood, the relationship between nal coalgebras and comonads is less well explored. This paper shows that the problem is more subtle and that final coalgebras can just as easily form monads as comonads and dually, that initial algebras form both monads and comonads. In developing these theories we strive to provide them with an associated notion of syntax. In the case of initial algebras and monads this corresponds to the standard notion of algebraic theories consisting of signatures and equations: models of such algebraic theories are precisely the algebras of the representing monad. We attempt to emulate this result for the coalgebraic case by defining a notion cosignature and coequation and then proving the models of this syntax are precisely the coalgebras of the representing comonad.
Factorization systems and fibrations: Toward a fibred Birkhoff variety theorem
, 2002
"... It is wellknown that a factorization system on a category (with sufficient pullbacks) gives rise to a fibration. This paper characterizes the fibrations that arise in such a way, by making precise the logical structure that is given by factorization systems. The underlying motivation is to obtain g ..."
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Cited by 4 (0 self)
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It is wellknown that a factorization system on a category (with sufficient pullbacks) gives rise to a fibration. This paper characterizes the fibrations that arise in such a way, by making precise the logical structure that is given by factorization systems. The underlying motivation is to obtain general Birkho results in a fibred setting.
Modal Operators for Coequations
, 2001
"... this paper, we develop the theory of coequations from a logical viewpoint. To clarify, let G = #G, #, ## be a comonad on E , where G preserves regular monos and E is "coBirkho #" (see Definition 2.1). A coequation # over a set C of colors is a regular subobject of GC, the carrier of the cofree ..."
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this paper, we develop the theory of coequations from a logical viewpoint. To clarify, let G = #G, #, ## be a comonad on E , where G preserves regular monos and E is "coBirkho #" (see Definition 2.1). A coequation # over a set C of colors is a regular subobject of GC, the carrier of the cofree coalgebra # C : GC ## G 2 C over C. Hence, we can view # as a predicate over GC. In particular, we can form new coequations out of old by means of the logical connectives #, #, etc. Furthermore, we have available a modal operator taking a coequation # to the (carrier of the) largest subcoalgebra # contained in the coequation. As we will see, arises as the formal dual of a familiar operation on sets of equations in categories of algebras. Explicitly, the operator is dual to the closure operation taking a set E of equations over X (i.e., E # UFX UFX , where UFX is the carrier of the free algebra over X) to the least congruence containing E. Hence, is dual to the closure of sets of equations under the first four rules of inference of Birkho#'s equational logic (Birkho#, 1935). Thus, we see that closure under these rules of inferences is dual to the "coalgebra interior" of a set of elements. We introduce a modal operator that is dual to closure under Birkho#'s fifth rule of inference, i.e., substitution of terms for variables. We confirm that is an S4 operator and show that, under certain conditions, commutes with . We then prove the invariance theorem in terms of and . In this way, we develop the coequationsaspredicates view by augmenting the predicates over GC with two modal operators and and show that the partial order of covarieties definable by arbitrary coequations over C is isomorphic to the partial order of predicates # over GC such th...