1,273 research outputs found
Not every pseudoalgebra is equivalent to a strict one
We describe a finitary 2-monad on a locally finitely presentable 2-category
for which not every pseudoalgebra is equivalent to a strict one. This shows
that having rank is not a sufficient condition on a 2-monad for every
pseudoalgebra to be strictifiable. Our counterexample comes from higher
category theory: the strict algebras are strict 3-categories, and the
pseudoalgebras are a type of semi-strict 3-category lying in between
Gray-categories and tricategories. Thus, the result follows from the fact that
not every Gray-category is equivalent to a strict 3-category, connecting
2-categorical and higher-categorical coherence theory. In particular, any
nontrivially braided monoidal category gives an example of a pseudoalgebra that
is not equivalent to a strict one.Comment: 17 pages; added more explanation; final version, to appear in Adv.
Mat
Operads within monoidal pseudo algebras
A general notion of operad is given, which includes as instances, the operads
originally conceived to study loop spaces, as well as the higher operads that
arise in the globular approach to higher dimensional algebra. In the framework
of this paper, one can also describe symmetric and braided analogues of higher
operads, likely to be important to the study of weakly symmetric, higher
dimensional monoidal structures
Constructing applicative functors
Applicative functors define an interface to computation that is more general, and correspondingly weaker, than that of monads. First used in parser libraries, they are now seeing a wide range of applications. This paper sets out to explore the space of non-monadic applicative functors useful in programming. We work with a generalization, lax monoidal functors, and consider several methods of constructing useful functors of this type, just as transformers are used to construct computational monads. For example, coends, familiar to functional programmers as existential types, yield a range of useful applicative functors, including left Kan extensions. Other constructions are final fixed points, a limited sum construction, and a generalization of the semi-direct product of monoids. Implementations in Haskell are included where possible
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