Cone (Category Theory)

Idea

In category theory, cones and cocones are used to construct Limit (Category Theory) and colimits. We an interpret them categorically by treating the diagram as a function from some indexing category $\mathcal{I}$.

Definitions

in category theory, a cone for a diagram $D:\mathcal{I}\Rightarrow \mathcal{C}$ is a $\mathcal{C}$ object $d$ paired with a Natural Transformation $N:\Delta d\Rightarrow D$ where $\Delta d:\mathcal{I}\Rightarrow \mathcal{C}$ is the constant functor. Naturality tells us that for all $f:a\to b$ in $D$, the cone commutes with $f$:

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
d \arrow[d, "\alpha_a"] \arrow[r, "id_d"] & d \arrow[d, "\alpha_b"] \\
a \arrow[r, "f"] & b
\end{tikzcd}
\end{document}

Remarks

A functor will map every cone into a cone in the destination category.