## Modal Operators for Coequations (2001)

### BibTeX

@MISC{Awodey01modaloperators,

author = {Steve Awodey and Jesse Hughes},

title = {Modal Operators for Coequations},

year = {2001}

}

### OpenURL

### Abstract

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 coequations-as-predicates 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...