5,164 research outputs found
Initial algebra for a system of right-linear functors
In 2003 we showed that right-linear systems of equations over regular expressions, when interpreted in a category of trees, have a solution when ever they enjoy a specific property that we called hierarchicity and that is instrumental to avoid critical mutual recursive definitions. In this note, we prove that a right-linear system of polynomial endofunctors on a cocartesian monoidal closed category which enjoys parameterized left list arithmeticity, has an initial algebra, provided it satisfies a property similar to hierarchicity
Initial algebra for a system of right-linear functors
In 2003 we showed that right-linear systems of equations over regular expressions, when interpreted in a category of trees, have a solution whenever they enjoy a specific property that we called hierarchicity and that is instrumental to avoid critical mutual recursive definitions. In this note, we prove that a right-linear system of polynomial endofunctors on a cocartesian monoidal closed category which enjoys parameterized left list arithmeticity, has an initial algebra, provided it satisfies a property similar to hierarchicity
Squared Hopf algebras and reconstruction theorems
Given an abelian k-linear rigid monoidal category V, where k is a perfect
field, we define squared coalgebras as objects of cocompleted V tensor V
(Deligne's tensor product of categories) equipped with the appropriate notion
of comultiplication. Based on this, (squared) bialgebras and Hopf algebras are
defined without use of braiding.
If V is the category of k-vector spaces, squared (co)algebras coincide with
conventional ones. If V is braided, a braided Hopf algebra can be obtained from
a squared one.
Reconstruction theorems give equivalence of squared co- (bi-, Hopf) algebras
in V and corresponding fibre functors to V (which is not the case with other
definitions). Finally, squared quasitriangular Hopf coalgebra is a solution to
the problem of defining quantum groups in braided categories.Comment: Latex2e, 31 pages, to appear in the Proceedings of Banach Center
Minisemester on Quantum Groups, November 199
The balanced tensor product of module categories
The balanced tensor product M (x)_A N of two modules over an algebra A is the
vector space corepresenting A-balanced bilinear maps out of the product M x N.
The balanced tensor product M [x]_C N of two module categories over a monoidal
linear category C is the linear category corepresenting C-balanced right-exact
bilinear functors out of the product category M x N. We show that the balanced
tensor product can be realized as a category of bimodule objects in C, provided
the monoidal linear category is finite and rigid.Comment: 19 pages; v3 is author-final versio
N-complexes as functors, amplitude cohomology and fusion rules
We consider N-complexes as functors over an appropriate linear category in
order to show first that the Krull-Schmidt Theorem holds, then to prove that
amplitude cohomology only vanishes on injective functors providing a well
defined functor on the stable category. For left truncated N-complexes, we show
that amplitude cohomology discriminates the isomorphism class up to a
projective functor summand. Moreover amplitude cohomology of positive
N-complexes is proved to be isomorphic to an Ext functor of an indecomposable
N-complex inside the abelian functor category. Finally we show that for the
monoidal structure of N-complexes a Clebsch-Gordan formula holds, in other
words the fusion rules for N-complexes can be determined.Comment: Final versio
- …