The first aim of this thesis is to give a unified and self-contained treatment of a number of known results related to the theory of herds. This gives us the technical tools to deal with the second aim of our work which is to obtain some new results about herds and coherds in the pure categorical setting. A herd at this level is a pretorsor τ : Q → QPQ satisfying associativity and unitality properties with respect to a formal dual structure M = (A, B, P,Q, σᴬ, σᴮ) ...

Bruni, Viola

English

BRUNI, Viola

Archivio istituzionale della ricerca - Università di Ferrara

Herds and Coherds

EprintsUnife

1. Abstract
The first aim of this thesis is to give a unified and self-contained treatment of a
number of known results related to the theory of herds. This gives us the technical
tools to deal with the second aim of our work which is to obtain some new results
about herds and coherds in the pure categorical setting.
A herd at this level is a pretorsor τ : Q → QPQ satisfying associativity and
unitality properties with respect to a formal dual structureM = (A,B, P,Q, σA, σB)
where A and B are monads over two different categories A and B respectively, P and
Q are bimodule functors with respect to the monads and σA and σB are functorial
morphisms satisfying linearity and compatibility conditions. In particular we study
the special class of tame herds defined by the following extra assumptions.
A formal dual structure M is called a Morita context if it satisfies also the balanced
conditions of compatibility with the bimodule structures. Moreover, we will say that
a Morita context M is tame if the lifted functorial morphisms ABσABA : AQBBPA →
IdAA and BAσ
B
AB : BPAAQB → IdBB are isomorphisms so that the lifted functors
AQB : BB → AA and BPA : AA → BB yield a category equivalence.
We can prove that such tame herds yield a correspondence with Galois functors.
A formal dual structureM is called regular whenever (A, uA) = EquFun (uAA,AuA)
and (B, uB) = EquFun (uBB,BuB) and the herd related to this formal dual structure
is also called regular.
Under the regularity assumption starting with a herd, one can construct a coherd.
Similarly, one can define the dual notions of copretorsor, coherd and (regular)
formal codual structure. Then, conversely, beginning with a coherd over a regular
formal codual structure, a herd can be obtained.
Furthermore, we analyze what happens starting with a regular formal dual structure
and a herd, constructing a formal codual structure and a coherd and then computing
a new formal dual structure. Under the extra assumption that the starting herd is
tame, the final formal dual structure computed comes out to be closely related to
the starting one.
A few examples illustrate the theory we develop about herds and coherds.
In the last part of this thesis we investigate the bicategory of balanced bimodule
functors which are one of the most useful tools in this work and it is inspired by the
balanced bimodules in the classical sense.
1


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Herds and Coherds

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Archivio istituzionale della ricerca - Università di Ferrara

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