# Some doubts in Category Theory

I am a beginner in Category Theory and have encountered two doubts in the subject:

Consider $\mathcal U$ to be a category and $X,Y$ be objects in $\mathcal U$. Then if $f:X\to Y$ is a morphism then a section of $f$ is defined to be a morphism $g:Y\to X$ such that $f\circ g=id_Y$.

Is it trivial that this morphism $g$ always exists?

Let $\mathcal U$ and $\mathcal U'$ be two categories and $Fun(\mathcal U,\mathcal U')$ be the (same variance) category of functors from $\mathcal U$ to $\mathcal U'$. Then the morphisms of $Fun(\mathcal U,\mathcal U')$ are defined as follows: if $\lambda,\mu$ are two (assume covariate) functors from $\mathcal U$ to $\mathcal U'$ then a morphism $t:\lambda\to\mu$ is the collection of morphisms $t_X:\lambda(X)\to \mu(X)$ where $X$ varies over all objects in $\mathcal U$.

This $t_X$ is chosen in a way such that for any morphism $f:X\to Y$, $\mu(f)\circ t_X=t_Y\circ \lambda(f)$.

What is the motivation for this last definition of morphisms on $Fun(\mathcal U,\mathcal U')$? An example would be really helpful.

• Here's an example I like : If $n \ge 1$, you can define a functor $\textrm{GL}_n$ from the category of commutative unitary rings to the category of groups (for any ring $A$, $\textrm{GL}_n(A)$ is the groups of $n \times n$ invertible matrices with coefficients in $A$. If $f : A \to B$ is a ring morphism, then $\textrm{GL}_n(f)$ is the morphism obtained by applying $f$ component by component). There is a natural transformation $\det : \textrm{GL}_n \to \textrm{GL}_1$ (where $\det_A : \textrm{GL}_n(A) \to \textrm{GL}_1(A)$ is simply the usual determinant map as you know it). – Joel Cohen Aug 22 '16 at 15:08
• This is a really nice example. Thank you for sharing it! – Landon Carter Aug 22 '16 at 15:28

2. Morphisms between functors are also called natural transformations, and are ubiquitous in mathematics. This notion captures the idea that some construction of an object $G(X)$ from an object $F(X)$ can be performed in a canonical, coordinate-free, simple and clean way. You will find many examples in the wikipedia article. In fact, Saunders Mac Lane once said