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Monad interleaving: a construction of the operad for Leinster’s weak ωcategories, Preprint, 2003, available at http://arxiv.org/abs/math/0309336
"... We show how to “interleave ” the monad for operads and the monad for contractions on the category Coll of collections, to construct the monad for the operadswithcontraction of Leinster. We first decompose the adjunction for operads and the adjunction for contractions into a chain of adjunctions ea ..."
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We show how to “interleave ” the monad for operads and the monad for contractions on the category Coll of collections, to construct the monad for the operadswithcontraction of Leinster. We first decompose the adjunction for operads and the adjunction for contractions into a chain of adjunctions each of which acts on only one dimension of the underlying globular sets at a time. We then exhibit mutual stability conditions that enable us to alternate the dimensionbydimension free functors. Hence we give an explicit construction of a left adjoint
Monad Transformers as Monoid Transformers
"... 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 preexisting monad need to be lifted to the new monad. In a compa ..."
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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 preexisting 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 F ω. Here, we recast and extend those results in a categorytheoretic setting. We abstract and generalize from monads to monoids (in a monoidal category), and from monad transformers to monoid transformers. The generalization brings more simplicity and clarity, and opens the way for lifting of operations with applicability beyond monads. Key words: Monad, Monoid, Monoidal Category
Computads and slices of operads.
, 2002
"... For a given ωoperad A on globular sets we introduce a sequence of symmetric operads on Set called slices of A and show how the connected limit preserving properties of slices are related to the property of the ..."
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For a given ωoperad A on globular sets we introduce a sequence of symmetric operads on Set called slices of A and show how the connected limit preserving properties of slices are related to the property of the
Coalgebraic Semantics for Parallel Derivation Strategies in Logic Programming
"... Abstract. Logic programming, a class of programming languages based on firstorder logic, provides simple and efficient tools for goaloriented proofsearch. Logic programming supports recursive computations, and some logic programs resemble the inductive or coinductive definitions written in functi ..."
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Abstract. Logic programming, a class of programming languages based on firstorder logic, provides simple and efficient tools for goaloriented proofsearch. Logic programming supports recursive computations, and some logic programs resemble the inductive or coinductive definitions written in functional programming languages. In this paper, we give a coalgebraic semantics to logic programming. We show that ground logic programs can be modelled by either Pf Pfcoalgebras or Pf Listcoalgebras on Set. We analyse different kinds of derivation strategies and derivation trees (prooftrees, SLDtrees, andor parallel trees) used in logic programming, and show how they can be modelled coalgebraically.
Multitensor lifting and strictly unital higher category theory
"... Abstract. In this article we extend the theory of lax monoidal structures, also known as multitensors, and the monads on categories of enriched graphs that they give rise to. Our first principal result – the lifting theorem for multitensors – enables us to see the Gray tensor product of 2categories ..."
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Abstract. In this article we extend the theory of lax monoidal structures, also known as multitensors, and the monads on categories of enriched graphs that they give rise to. Our first principal result – the lifting theorem for multitensors – enables us to see the Gray tensor product of 2categories and the Crans tensor product of Gray categories as part of this framework. We define weak ncategories with strict units by means of a notion of reduced higher operad, using the theory of algebraic weak factorisation systems. Our second principal result is to establish a lax tensor product on the category of weak ncategories with strict units, so that enriched categories with respect to this tensor product are exactly weak (n+1)categories with strict units. 1.
Internal monotonelight factorization for categories via preorders
 Theory Appl. Categories
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DESCENT FOR MONADS
"... Abstract. Motivated by a desire to gain a better understanding of the “dimensionbydimension” 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 describ ..."
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Abstract. Motivated by a desire to gain a better understanding of the “dimensionbydimension” 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 ↦ → Mnd(C) as an indexed category; on this base category, there is a natural topology. Then we single out a class of monads which are wellbehaved with respect to reindexing. The main result is now, that such monads form a stack. Using this, we can shed some light on the free strict ωcategory monad on globular sets and the free operadwithcontraction monad on the category of collections.