We give a Haskell implementation of Filinski’s normalisation by evaluation algorithm for the computational lambda-calculus with sums. Taking advantage of extensions to the GHC compiler, our implementation represents object language types as Haskell types and ensures that type errors are detected statically. Following Filinski, the implementation is parameterised over a residualising monad. The ...

Motivated by a desire to gain a better understanding of the “dimensionby-dimension” decompositions of certain prominent monads in higher category theory, we investigate descent theory for endofunctors and monads. After setting up a basic framework of indexed monoidal categories, we describe a suitable subcategory of Cat over which we can view the assignment C 7→ Mnd(C) as an indexed category; o...

For a quantale I, which is unit interval endowed with continuous triangular norm and the Barr extension β¯I of ultrafilter monad to I-Rel, characterization discrete presheaf associated given. It also proved that, when & Łucasiewicz norm, isomorphic saturated prefilter monad, product prime functional ideal submonad monad.

Introduction. This note is concerned with "categories with internal horn and | and we shall use the terminology from the paper [2] by EIL~.NBERG and Kv.Imy. The result proved may be stated briefly as follows : a Y/--monad ("strong monad") on a symmetric monoidal closed category ~ carries two canonical structures as closed functor. I f these agree (in which case we call the monad commutative), t...

Enrichment of ordinary monads over Cat or Gpd is fundamental to Max Kelly’s unified theory of coherence for categories with structure. So here, we investigate existence and unicity of enrichments of ordinary func­ tors, natural transformations, and hence also monads, over Cat and Gpd. We show that every ordinary natural transformation between 2-functors whose domain 2-category has either tensor...

Inspired by recent work of Batanin and Berger on the homotopy theory of operads, a general monad-theoretic context for speaking about structures within structures is presented, and the problem of constructing the universal ambient structure containing the prescribed internal structure is studied. Following the work of Lack, these universal objects must be constructed from simplicial objects ari...

The lift monad is the construction which to a poset freely adjoins a bottom element to it, or equivalently (from the classical viewpoint), the construction which freely adjoins suprema for subsets with at most one element. In constructive mathematics (i.e. inside a topos), these two constructions are no longer equivalent, since the equivalence is based on the boolean reasoning that a set with a...

The incremental approach to modular monadic semantics constructs complex monads by using monad transformers to add computational features to a preexisting monad. A complication of this approach is that the operations associated to the pre-existing monad need to be lifted to the new monad. In a companion paper by Jaskelioff, the lifting problem has been addressed in the setting of system Fω. Her...