Cocommutative coalgebras

• Applicant: Professor Dr. Ulrich Krähmer
• Subject Area: Mathematics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 496523143

Category theory provides a unified language for the formulation of concepts and results of various areas of mathematical research. In geometry, objects such as sets or topological spaces typically form Cartesian categories, i.e., in simple terms, categories in which two given objects have a natural product. Instead, abelian categories play a central role in classical algebra and representation theory. In these, morphisms have for example kernels, and one may form the quotient of an object by a subobject.Cartesian categories can also be characterised as symmetric monoidal categories in which all objects are naturally cocommutative coalgebras (comonoids). Starting from a category C such as that of vector spaces over a field k, we may thus study the Cartesian category of cocommutative coalgebras in C (or C^op). For example, one obtains in this way the Cartesian category of affine schemes over k.The main aim of this first-time proposal (Erstantrag) is to revisit some concepts and results from algebra and geometry from this perspective, such as the following:1. Racks provide an abstraction of the adjoint action of a group on itself in the absence of an associative product and have applications e.g. in the theory of symmetric spaces and of knots. Carter, Crans, Elhamdadi and Saito defined deformations of racks; one of our goals is the development of a general cohomology theory of racks in categories like those mentioned above.2. Clones are abstract systems of n-ary operations studied in particular in complexity theory and universal algebra. One mostly considers their actions on finite sets, but Hyland and others also studied them in general Cartesian categories. As in 1. we are interested in questions of homological nature. For example, do deformations of clones fit into the framework of deformations of representations of PROPs as developed by Merkulov and Vallette?3. Hopf algebras are (certain) algebras in categories of coalgebras. A classical result is the theorem of Milnor and Moore that classifies cocommutative Hopf algebras. Its proof by Cartier employs techniques from algebraic topology (\Lambda-rings) which we feel should work in greater generality. One goal is to formulate a more general version of this proof. This should cover in particular the recent classification of cocommutative Hopf algebroids due to Moerdijk and Mrcun. All these points are expected to also relate to current research on pointed Hopf algebras and Nichols algebras; here cocommutative coalgebras in braided rather than symmetric monoidal categories are of crucial importance.

• DFG Programme: Research Grants