29 research outputs found
Operads, clones, and distributive laws
International audienceWe show how non-symmetric operads (or multicategories), symmetric operads, and clones, arise from three suitable monads on Cat, each extending to a (pseudo-)monad on the bicategory of categories and profunctors. We also explain how other previous categorical analyses of operads (via Day's tensor products, or via analytical functors) fit with the profunctor approach
Operads, clones, and distributive laws
International audienceWe show how non-symmetric operads (or multicategories), symmetric operads, and clones, arise from three suitable monads on Cat, each extending to a (pseudo-)monad on the bicategory of categories and profunctors. We also explain how other previous categorical analyses of operads (via Day's tensor products, or via analytical functors) fit with the profunctor approach
Monads of regular theories
We characterize the category of monads on and the category of Lawvere
theories that are equivalent to the category of regular equational theories.Comment: 36 page
Foundations of Algebraic Theories and Higher Dimensional Categories
Universal algebra uniformly captures various algebraic structures, by
expressing them as equational theories or abstract clones. The ubiquity of
algebraic structures in mathematics and related fields has given rise to
several variants of universal algebra, such as symmetric operads, non-symmetric
operads, generalised operads, and monads. These variants of universal algebra
are called notions of algebraic theory. In the first part of this thesis, we
develop a unified framework for notions of algebraic theory which includes all
of the above examples. Our key observation is that each notion of algebraic
theory can be identified with a monoidal category, in such a way that theories
correspond to monoid objects therein. We introduce a categorical structure
called metamodel, which underlies the definition of models of theories. We also
consider morphisms between notions of algebraic theory, which are a monoidal
version of profunctors. Every strong monoidal functor gives rise to an adjoint
pair of such morphisms, and provides a uniform way to establish isomorphisms
between categories of models in different notions of algebraic theory. A
general structure-semantics adjointness result and a double categorical
universal property of categories of models are also shown.
In the second part of this thesis, we shift from the general study of
algebraic structures, and focus on a particular algebraic structure: higher
dimensional categories. Among several existing definitions of higher
dimensional categories, we choose to look at the one proposed by Batanin and
later refined by Leinster. We show that the notion of extensive category plays
a central role in Batanin and Leinster's definition. Using this, we generalise
their definition by allowing enrichment over any locally presentable extensive
category.Comment: 134 pages, PhD thesi
On the formal theory of pseudomonads and pseudodistributive laws
We contribute to the formal theory of pseudomonads, i.e. the analogue for
pseudomonads of the formal theory of monads. In particular, we solve a problem
posed by Steve Lack by proving that, for every Gray-category K, there is a
Gray-category Psm(K) of pseudomonads, pseudomonad morphisms, pseudomonad
transformations and pseudomonad modifications in K. We then establish a
triequivalence between Psm(K) and the Gray-category of pseudomonads introduced
by Marmolejo. Finally, these results are applied to give a clear account of the
coherence conditions for pseudodistributive laws. 40 pages. Comments welcome.Comment: This submission replaces arXiv:0907:1359v1, titled "On the coherence
conditions for pseudo-distributive laws". 40 page
Truncation of Unitary Operads
We introduce truncation ideals of a -linear unitary symmetric operad
and use them to study ideal structure, growth property and to classify operads
of low Gelfand-Kirillov dimension
Strong pseudomonads and premonoidal bicategories
Strong monads and premonoidal categories play a central role in clarifying
the denotational semantics of effectful programming languages. Unfortunately,
this theory excludes many modern semantic models in which the associativity and
unit laws only hold up to coherent isomorphism: for instance, because
composition is defined using a universal property. This paper remedies the
situation. We define premonoidal bicategories and a notion of strength for
pseudomonads, and show that the Kleisli bicategory of a strong pseudomonad is
premonoidal. As often in 2-dimensional category theory, the main difficulty is
to find the correct coherence axioms on 2-cells. We therefore justify our
definitions with numerous examples and by proving a correspondence theorem
between actions and strengths, generalizing a well-known category-theoretic
result.Comment: Comments and feedback welcome