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Model theory of comodules

Published online by Cambridge University Press:  12 March 2014

Septimiu Crivei ,
Mike Prest  and
Geert Reynders
Septimiu Crivei
Affiliation:
Department of Mathematics, Babes-Bolyai University, Cluj, Romania, E-mail: crivei@math.ubbcluj.ro
Mike Prest
Affiliation:
Department of Mathematics, University of Manchester, Oxford Road, Manchester. M13 9PL., Great Britain, E-mail: mprest@maths.man.ac.uk
Geert Reynders
Affiliation:
Kapellestraat 10, 3080 Tervuren., Belgium, E-mail: geert.reyndersl@pandora.be
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Extract

The purpose of this paper is to establish some basic points in the model theory of comodules over a coalgebra. It is not even immediately apparent that there is a model theory of comodules since these are not structures in the usual sense of model theory. Let us give the definitions right away so that the reader can see what we mean.

Fix a field k. A k-coalgebra C is a k-vector space equipped with a k-linear map Δ: C → C ⊗ C, called the comultiplication (by ⊗ we always mean tensor product over k), and a k-linear map ε: C → k, called the counit, such that Δ⊗1C = 1C ⊗ Δ (coassociativity) and (1C ⊗ ε)Δ = 1C = (ε ⊗ 1C)Δ, where we identify C with both k ⊗ C and C ⊗ k. These definitions are literally the duals of those for a k-algebra: express the axioms for C′ to be a k-algebra in terms of the multiplication map μ: C′ ⊗ C′ → C′ and the “unit” (embedding of k into C′), δ: k → C′ in the form that certain diagrams commute and then just turn round all the arrows. See [5] or more recent references such as [7] for more.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 69 , Issue 1 , March 2004 , pp. 137 - 142
Copyright
Copyright © Association for Symbolic Logic 2004

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References

REFERENCES

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