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.

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