230 research outputs found
Algebraic Theories over Nominal Sets
We investigate the foundations of a theory of algebraic data types with
variable binding inside classical universal algebra. In the first part, a
category-theoretic study of monads over the nominal sets of Gabbay and Pitts
leads us to introduce new notions of finitary based monads and uniform monads.
In a second part we spell out these notions in the language of universal
algebra, show how to recover the logics of Gabbay-Mathijssen and
Clouston-Pitts, and apply classical results from universal algebra.Comment: 16 page
Theories of analytic monads
We characterize the equational theories and Lawvere theories that correspond
to the categories of analytic and polynomial monads on Set, and hence also the
categories of the symmetric and rigid operads in Set. We show that the category
of analytic monads is equivalent to the category of regular-linear theories.
The category of polynomial monads is equivalent to the category of rigid
theories, i.e. regular-linear theories satisfying an additional global
condition. This solves a problem A. Carboni and P. T. Johnstone. The Lawvere
theories corresponding to these monads are identified via some factorization
systems.Comment: 29 pages. v2: minor correction
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
Second-Order Algebraic Theories
Fiore and Hur recently introduced a conservative extension of universal
algebra and equational logic from first to second order. Second-order universal
algebra and second-order equational logic respectively provide a model theory
and a formal deductive system for languages with variable binding and
parameterised metavariables. This work completes the foundations of the subject
from the viewpoint of categorical algebra. Specifically, the paper introduces
the notion of second-order algebraic theory and develops its basic theory. Two
categorical equivalences are established: at the syntactic level, that of
second-order equational presentations and second-order algebraic theories; at
the semantic level, that of second-order algebras and second-order functorial
models. Our development includes a mathematical definition of syntactic
translation between second-order equational presentations. This gives the first
formalisation of notions such as encodings and transforms in the context of
languages with variable binding
Corecursive Algebras, Corecursive Monads and Bloom Monads
An algebra is called corecursive if from every coalgebra a unique
coalgebra-to-algebra homomorphism exists into it. We prove that free
corecursive algebras are obtained as coproducts of the terminal coalgebra
(considered as an algebra) and free algebras. The monad of free corecursive
algebras is proved to be the free corecursive monad, where the concept of
corecursive monad is a generalization of Elgot's iterative monads, analogous to
corecursive algebras generalizing completely iterative algebras. We also
characterize the Eilenberg-Moore algebras for the free corecursive monad and
call them Bloom algebras
Variations on Algebra: monadicity and generalisations of equational theories
Dedicated to Rod Burstal
Initial Semantics for Strengthened Signatures
We give a new general definition of arity, yielding the companion notions of
signature and associated syntax. This setting is modular in the sense requested
by Ghani and Uustalu: merging two extensions of syntax corresponds to building
an amalgamated sum. These signatures are too general in the sense that we are
not able to prove the existence of an associated syntax in this general
context. So we have to select arities and signatures for which there exists the
desired initial monad. For this, we follow a track opened by Matthes and
Uustalu: we introduce a notion of strengthened arity and prove that the
corresponding signatures have initial semantics (i.e. associated syntax). Our
strengthened arities admit colimits, which allows the treatment of the
\lambda-calculus with explicit substitution.Comment: In Proceedings FICS 2012, arXiv:1202.317
Generic Trace Logics
We combine previous work on coalgebraic logic with the coalgebraic traces
semantics of Hasuo, Jacobs, and Sokolova
Diagrammatic presentations of enriched monads and varieties for a subcategory of arities
The theory of presentations of enriched monads was developed by Kelly, Power,
and Lack, following classic work of Lawvere, and has been generalized to apply
to subcategories of arities in recent work of Bourke-Garner and the authors. We
argue that, while theoretically elegant and structurally fundamental, such
presentations of enriched monads can be inconvenient to construct directly in
practice, as they do not directly match the definitional procedures used in
constructing many categories of enriched algebraic structures via operations
and equations.
Retaining the above approach to presentations as a key technical
underpinning, we establish a flexible formalism for directly describing
enriched algebraic structure borne by an object of a -category in terms
of parametrized -ary operations and diagrammatic equations for a suitable
subcategory of arities . On this basis we introduce the
notions of diagrammatic -presentation and -ary variety, and we show that
the category of -ary varieties is dually equivalent to the category of
-ary -monads. We establish several examples of diagrammatic
-presentations and -ary varieties relevant in both mathematics and
theoretical computer science, and we define the sum and tensor product of
diagrammatic -presentations. We show that both -relative monads and
-pretheories give rise to diagrammatic -presentations that directly
describe their algebras. Using diagrammatic -presentations as a method of
proof, we generalize the pretheories-monads adjunction of Bourke and Garner
beyond the locally presentable setting. Lastly, we generalize Birkhoff's Galois
connection between classes of algebras and sets of equations to the above
setting
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