No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
The Tannaka Representation Theorem for Separable Frobenius Functors
Abstract
A weak bialgebra is known to be a special case of a bialgebroid. In this paper we study the relationship of this fact with the Tannaka theory of bialgebroids as developed in [4]. We obtain a Tannaka representation theorem with respect to a separable Frobenius fiber functor. Comment: 7 pages
... Since the theorem is known for the Galois context, this yields, in particular, a proof of the fundamental (recognition) theorem for a new Tannakian context. This example is different from the additive cases [15], [13], [2], or their generalization [24], where the theorem is known to hold, and where the unit of the tensor product is always an object of finite presentation (that is, filtered colimits in the tensor category are constructed as in the category of sets), which is not the case in our context. ...
... In fact, from axioms ed) and su) for µ on Z it follows (2) 1 = y, z ∈Z µ a|y ⊗ µ z|b . Then, the claim follows by taking the infimum in both sides of equations (1) and (2), and then using the axioms uv) and in) for µ on X. ...
... ιµ a|b (1) = ιµ a|b ∧ y µ b|y = y ιµ a|b ∧ µ b|y (2) = ιµ a|b ∧ µ b|a , since all the other terms in the supremum are 0. Then ιµ a|b ≤ µ b|a = µι a|b . ...
The main result of this thesis is the construction of a tannakian context
over the category of sup-lattices, associated with an arbitrary Grothendieck
topos, and the attainment of new results in tannakian representation theory
from it.
Although many results were obtained and published historically linking Galois
and Tannaka theory (see our introduction), these are different and less general
since they assume the existence of Galois closures and work on Galois topos
rather than on arbitrary topos. Instead we, when talking about Galois theory,
mean the extension to arbitrary topos of Joyal and Tierney, critical to get the
results of this thesis.
The tannakian context associated with a Grothendieck topos is obtained
through the process of taking relations of its localic cover. Then, through an
investigation and exhaustive comparison of the constructions of the Galois and
Tannaka theories, we prove the equivalence of their fundamental recognition
theorems (see section 8).
Since the (bi)categories of relations of a Grothendieck topos were
characterized by Carboni and Walters, a new recognition-type tannakian theorem
(theorem 8.12) is obtained, essentially different from those known so far.
... In this paper we construct an explicit Tannakian context for Galois theory, and prove the equivalence between its fundamental theorems. Since the theorem is known for the Galois context, this brings, in particular, a proof of the fundamental (recognition) theorem for a Tannakian context different than the known additive cases [5], [4], [2], or their generalization [8], where it is assumed that the unit of the tensor product is an object of finite presentation (that is, filtered colimits in the tensor category are constructed as in the category of sets). ...
... ιµ a|b (1) = ιµ a|b ∧ y µ b|y = y ιµ a|b ∧µ b|y (2) = ιµ a|b ∧µ b|a , since all the other terms in the supremum are 0. Then ιµ a|b ≤ µ b|a = µι a|b . ...
Strong similarities have been long observed between the Galois (Categories
Galoisiennes) and the Tannaka (Categories Tannakiennes) theories of
representation of groups. In this paper we construct an explicit (neutral)
Tannakian context for the Galois theory of atomic topoi, and prove the
equivalence between its fundamental theorems. Since the theorem is known for
the Galois context, this yields, in particular, a proof of the fundamental
(recognition) theorem for a new Tannakian context. This example is different
from the additive cases or their generalization, where the theorem is known to
hold, and where the unit of the tensor product is always an object of finite
presentation, which is not the case in our context.
... where ∆ (2) (1) = 1 (1) ⊗ 1 (2) ⊗ 1 (3) and ∆(1) = 1 [1] ⊗ 1 [2] . (1) ε t (xε t (y)) = ε t (xy), ε s (ε s (x)y) = ε s (xy). ...
In this note we give a precise statement and a detailed proof for reconstruction problem of weak bialgebra maps. As an application we characterize indecomposability of weak algebras in categorical setting.
Every monoidal functor G: C --> M has a canonical factorization through the category of bimodules over some monoid R in M such that the factor U: C -->_R M_R is strongly unital. Using this result and the characterization of the forgetful functors M_A -->_R M_R of bialgebroids A over R given by Schauenburg together with their bimonad description given by the author recently here we characterize the "long" forgetful functors M_A -->_R M_R --> M of both bialgebroids and weak bialgebras. Comment: 16 pp, AMS Latex, Talk given at "Hopf Algebras in Noncommutative Geometry and Physics", Brussels, June, 2002
We develop the Tannaka-Krein duality for monoidal functors with target in the categories of bimodules over a ring. The \coend of such a functor turns out to be a Hopf algebroid over this ring. Using the result of a previous paper we characterize a small abelian, locally finite rigid monoidal category as the category of rigid comodules over a transitive Hopf algebroid.
Bialgebroids, separable bialgebroids, and weak Hopf algebras are compared from a categorical point of view. Then properties of weak Hopf algebras and their applications to finite index and finite depth inclusions of von Neumann algebras are shortly reviewed. A hint is given at a duality between bialgebroid actions and abstract inclusions in 2-categories.
This paper defines and proves the correctness of the appropriate string diagrams for various kinds of monoidal categories with duals.
We introduce a variant on the graphical calculus of Cockett and Seely for monoidal functors and illustrate it with a discussion of Tannaka reconstruction, some of which is known and some of which is new. The new portion is: given a separable Frobenius functor F: A --> B from a monoidal category A to a suitably complete or cocomplete braided autonomous category B, the usual formula for Tannaka reconstruction gives a weak bialgebra in B; if, moreover, A is autonomous, this weak bialgebra is in fact a weak Hopf algebra.
We give a detailed comparison between the notion of a weak Hopf algebra (also called a quantum groupoid by Nikshych and Vainerman), and that of a -bialgebra due to Takeuchi (and also called a bialgebroid or quantum (semi)groupoid by Lu and Xu). A weak bialgebra is the same thing as a -bialgebra in which R is Frobenius-separable. We extend the comparison to cover module and comodule theory, duality, and the question when a bialgebroid should be called a Hopf algebroid.
It is well known that strong monoidal functors preserve duals. In this short note we show that a slightly weaker version of functor, which we call "Frobenius monoidal", is sufficient. Comment: 8 pages; added new material
Weak Hopf algebras and quantum groupoids. Noncommutative geometry and quantum groups
- P Schauenburg
- M Hajac
- W Pusz
M. Hajac and W. Pusz, Banach Center Publ. 61, Polish Acad. Sci., Warsaw,
2003.
An introduction to Tannaka duality and quantum groups, Part II of category theory
- A Joyal