This paper gives analogies of Hutton’s quasi-uniformities for complete lattices and extensions of Shi’s pointwise quasi-uniformities to closed set lattices.CLGOH denotes the category of all complete lattices with generalized order homomorphisms asmorphisms and CSL the category of all closed set lattices.HQUnif and SPQUnif denote the categories of Hutton quasi-uniform spaces on complete lattices...

The base category C is the full subcategory of the category of posets having for objects finite power of the poset 0 6 1. We write [1] the poset 0 6 1. We write I, J,K, . . . the object of C and 1I : I → I the identity map of I. If f : J → I and g : K → J we write fg : K → I their composition. If I is an object of C, we have two constant maps c0 : I → [1] and c1 : I → [1]. We write π1 : I × [1]...

a ≤ b iff a is the supremum of the subset {a} ∩ {b}. On any elementary topos E , one has the functor which to an object A associates the object TA = Ã which classifies partial maps into A (cf e.g. [J] 1.2). This functor T carries a monad structure T= (T, η, μ) ; it is a submonad of the power ”set” monad P= (P, η, μ), as described in, say, [AL],[Mi], or [J] 5.3. We shall analyze the category of ...

The category Rel of sets and relations has two natural traced monoidal structures: in Rel Tr , the tensor is given by disjoint union, and in Rel Tr by products of sets. Already in 1976, predating the definition of traced monoidal categories by 20 years, Bainbridge has shown how to model flowcharts and networks in these two respective settings. Bainbridge has also pointed out that one can move f...

We give here some examples of non pointed protomodular categories C satisfying a property similar to the property of representation of actions which holds for the pointed protomodular category Gp of groups: any slice category of Gp, any category of groupoids with a fixed set of objects, any essentially affine category. This property gives rise to an internal construction of the center of any ob...