# Tag Archives: group theory

## Group Theory ∞ (Epilogue)

That concludes the end of the series of notes on Group Theory. Has it been successful? I don’t know, but I’m reasonably pleased with the way the notes turn out, except the fact that there’s a huge disparity between the … Continue reading

## Group Theory XII.5 (More Category Theory)

Since a category is really a bunch of abstract objects and arrows between them, we can reverse them by duality. Let C be a category. The opposite category Cop is the category such that: Ob(Cop) = Ob(C); for any objects … Continue reading

## Group Theory XII.4 (Category Theory: Functors)

In this section, we will explore further concepts in category theory. First, we shall talk about “maps” between categories. Let C, D be categories. A (covariant) functor (written as F : C → D) is a map F : Ob(C) → … Continue reading

## Group Theory XII.3 (More Universal Properties)

In this section, we shall get more practice with universal properties for various algebraic constructions. First, take the following categories: Set = category of sets, with morphisms = set functions; Grp = category of groups, with morphisms = group homomorphisms; … Continue reading

## Group Theory XII.2 (Category Theory)

Let’s take a closer look at the proofs and definition in the previous section. What concepts have we used? We have considered groups, homomorphisms between them, composition of homomorphisms, identity homomorphisms, isomorphisms and inverse homomorphisms. But the last two items … Continue reading