Categories – Exercise 1
[Two-out-of-three property] Let f: A→B and g: B→ C be two morphisms. If $$g$$ and $$gf$$ are isomorphisms, then so is f$$. Solution: There exists...
Category Theory
Products in a Category – Exercise 1
Demonstrate that A × B ≅ B × A. Solution: The product A × B satisfies the universal property, for any object X and arrows...
Universal Property of Quotient Vector Space
Describe the universal property of quotient vector spaces. Solution: Let $$V \in Obj(Vect_{K})$$. Any subspace $$U\subseteq V$$ and its projection $$\pi: V\longrightarrow V/U$$ form a...
Functors – Exercise 1
Show that functors preserve isomorphism. If $$a\sim a’$$ in $$\mathcal{C}$$, then $$F(a)\sim F(a’)$$ in $$\mathcal{H}$$, with $$F:\mathcal{C}\longrightarrow\mathcal{H}$$. Solution: Let $$f:a\longrightarrow a’$$ be a isomorphism with...