For a complete cartesian-closed category V with coproducts, and for any pointed endofunctor T of the category of sets satisfying a suitable Beck-Chevalley-type condition, it is shown that the category of lax reflexive (T,V)-algebras is a quasitopos. This result encompasses many known and new examples of quasitopoi. Mathematics Subject Classification: 18C20, 18D15, 18A05, 18B30, 18B35.

We introduce the new category VRel(H) consisting of H-fuzzy relation spaces and H-fuzzy mappings between them satisfying a certain condition, where the concept of H-fuzzy mapping is the modification of one of fuzzy mapping introduced by Demirci[6]. And we investigate VRel(H) in the sense of a topological universe and show that VRel(H) is Cartesian closed over Set. Moreover, we construct the cat...

We can regard operations that discard information, like specializing to a particular case or dropping the intermediate steps of a proof, as projections, and operations that reconstruct information as liftings. By working with several projections in parallel we can make sense of statements like “Set is the archetypal Cartesian Closed Category”, which means that proofs about CCCs can be done in t...