154 research outputs found
The homotopy theory of bialgebras over pairs of operads
We endow the category of bialgebras over a pair of operads in distribution
with a cofibrantly generated model category structure. We work in the category
of chain complexes over a field of characteristic zero. We split our
construction in two steps. In the first step, we equip coalgebras over an
operad with a cofibrantly generated model category structure. In the second one
we use the adjunction between bialgebras and coalgebras via the free algebra
functor. This result allows us to do classical homotopical algebra in various
categories such as associative bialgebras, Lie bialgebras or Poisson bialgebras
in chain complexes.Comment: 27 pages, final version, to appear in the Journal of Pure and Applied
Algebr
The Recursion Scheme from the Cofree Recursive Comonad
AbstractWe instantiate the general comonad-based construction of recursion schemes for the initial algebra of a functor F to the cofree recursive comonad on F. Differently from the scheme based on the cofree comonad on F in a similar fashion, this scheme allows not only recursive calls on elements structurally smaller than the given argument, but also subsidiary recursions. We develop a Mendler formulation of the scheme via a generalized Yoneda lemma for initial algebras involving strong dinaturality and hint a relation to circular proofs Ă la Cockett, Santocanale
The homotopy theory of coalgebras over a comonad
Let K be a comonad on a model category M. We provide conditions under which
the associated category of K-coalgebras admits a model category structure such
that the forgetful functor to M creates both cofibrations and weak
equivalences.
We provide concrete examples that satisfy our conditions and are relevant in
descent theory and in the theory of Hopf-Galois extensions. These examples are
specific instances of the following categories of comodules over a coring. For
any semihereditary commutative ring R, let A be a dg R-algebra that is
homologically simply connected. Let V be an A-coring that is semifree as a left
A-module on a degreewise R-free, homologically simply connected graded module
of finite type. We show that there is a model category structure on the
category of right A-modules satisfying the conditions of our existence theorem
with respect to the comonad given by tensoring over A with V and conclude that
the category of V-comodules in the category of right A-modules admits a model
category structure of the desired type. Finally, under extra conditions on R,
A, and V, we describe fibrant replacements in this category of comodules in
terms of a generalized cobar construction.Comment: 34 pages, minor corrections. To appear in the Proceedings of the
London Mathematical Societ
Unifying Structured Recursion Schemes
AbstractFolds and unfolds have been understood as fundamental building blocks for total programming, and have been extended to form an entire zoo of specialised structured recursion schemes. A great number of these schemes were unified by the introduction of adjoint folds, but more exotic beasts such as recursion schemes from comonads proved to be elusive. In this paper, we show how the two canonical derivations of adjunctions from (co)monads yield recursion schemes of significant computational importance: monadic catamorphisms come from the Kleisli construction, and more astonishingly, the elusive recursion schemes from comonads come from the Eilenberg–Moore construction. Thus, we demonstrate that adjoint folds are more unifying than previously believed.</jats:p
Structured general corecursion and coinductive graphs [extended abstract]
Bove and Capretta's popular method for justifying function definitions by
general recursive equations is based on the observation that any structured
general recursion equation defines an inductive subset of the intended domain
(the "domain of definedness") for which the equation has a unique solution. To
accept the definition, it is hence enough to prove that this subset contains
the whole intended domain.
This approach works very well for "terminating" definitions. But it fails to
account for "productive" definitions, such as typical definitions of
stream-valued functions. We argue that such definitions can be treated in a
similar spirit, proceeding from a different unique solvability criterion. Any
structured recursive equation defines a coinductive relation between the
intended domain and intended codomain (the "coinductive graph"). This relation
in turn determines a subset of the intended domain and a quotient of the
intended codomain with the property that the equation is uniquely solved for
the subset and quotient. The equation is therefore guaranteed to have a unique
solution for the intended domain and intended codomain whenever the subset is
the full set and the quotient is by equality.Comment: In Proceedings FICS 2012, arXiv:1202.317
- …