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Let G be a group and k some field. A representation of the group G over the field k is usually defined as a vector space V over k equipped with a linear action of the group G. One of important problems in Representation Theory is the study of the category Rep(G) of such representations.

In my talk I will explain that there is another natural way to describe this category, namely, it can be described as a category of sheaves on some "geometric" object - the basic groupoid BG of the group G.

I will explain that this more sophisticated categorical description is the more "correct" one. For example, this new description gives a more adequate description of the category Rep(G) in cases when we have additional structures (algebraic groups, some extra symmetries and so on).

I will show that in many cases the category Rep(G) has a natural extension - the category M(G) - that has better behavior.

My talk will partially follow my paper in arXiv:1410.0435.