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May, 2019
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Group Theory

In algebraic topology, we study invariants of spaces with values in algebraic objects. Homology groups and homotopy groups are invariants with values in Abelian groups . On the other hand, fundamental groups take values in not-necessarily Abelian groups. Groups are also used to construct spaces. For example, important spaces can be constructed as classifying spaces of groups.

Conversely, geometric techniques are useful when we study infinite groups.

It should be noted that categorical viewpoints are useful when we use groups. For example, a group is a small category with a single object in which all morphisms are invertible. Classifying spaces are constructed for groups first. And then the construction was extended to small categories. Furthermore group theoretic concepts such as center and semidirect product can be extended to categories. I don’t know who first found a generalization of center to categories but it can be found in Chirvasitu’s [ Chi ] . A categorical generalization of semidirect product is the Grothendieck construction .

Another categorical viewpoint is, although tautological, to regard a group as a group object in the category of sets. As a generalization, we consider group objects in other monoidal categories . For example, [ BW ] by Blohmann and Weinstein.

References