Let Mat be a category where the objets are the set of natural numbers $\mathbb{N}$ and for each pair $m,n \in \mathbb{N}$ the morphism $m \rightarrow n$ is a matriz $M_{nm}$ of size $n \times m$. Let $Vec_{\mathbb{R}}$ be the category of finite dimensional vector spaces over $\mathbb{R}$ where the morphisms are linear transformation.
It is a well known result that these categories are equivalent defining the functor $F: \boldsymbol{Mat} \rightarrow Vec_{\mathbb{R}}$ such that $Fn = \mathbb{R}^n$ and $FM_{nm}$ is the linear transformation that has the matrix $M_{nm}$ as the matrix representation.
My question is, are these categories isomorphic? I think the answer is no, of course, but in order to prove it, is there any result from which the statement follows directly? Or should I propose the existence of two functors $F,G$ such that $GF$ and $FG$ are the corresponding identities morhpisms and get a contradiction? Thanks
Best Answer
$\mathbf{Mat}$ has countably many objects, whereas $\mathbf{Vec}_{\mathbb{R}}$ has a proper class of objects. An isomorphism of categories is in particular a bijection on objects, so cannot exist between these.