# Category Theory

Last updated: Mon, 10 Dec 2018

Universal property: $P_1$ along with $X$ and $Y$ defines a product. There exists a unique morphism between $P_1$ and $P_2$ preserving the product structure.

$\begin{xy} \xymatrix{ P_2 \ar@/_/[ddr]_y \ar@{.>}[dr]|{\exists!} \ar@/^/[drr]^x \\ & P_1 \ar[d]^q \ar[r]_p & X \\ & Y & } \end{xy}$

A functor $F$ from category $\mathcal{A}$ to category $\mathcal{B}$, $F : \mathcal{A} \rightarrow \mathcal{B}$ is defined to be covariant if $F(m \circ n) = F(m) \circ F(n)$, and contravariant if $F(m \circ n) = F(n) \circ F(m)$.

## Fibered product

The object $X \times_Z Y$ along with the morphisms $pr_X$ and $pr_Y$ is called fibered product.

$\begin{xy} \xymatrix{ U \ar@/_/[ddr] \ar@{.>}[dr]|{\exists!} \ar@/^/[drr] \\ & X \times_Z Y \ar[d]^{pr_Y} \ar[r]_{pr_X} & X \ar[d]_f \\ & Y \ar[r]^g & Z } \end{xy}$