<-- Go Back Last Updated: 10/06/2025

Co-Cone Category**

Given a (typically small) category \(\mathcal{I}\), a category \(\mathcal{C}\), and a functor \(F:\mathcal{I}\to\mathcal{C}\), a co-cone under \(F\) is an object \(X\in\mathcal{C}_0\) with a collection of arrows to \(X\) from the image of each object of \(\mathcal{I}\) under \(F\), \(\{(f_i:F(i)\to X)\in\mathcal{C}_1|i\in \mathcal{I}_0\}\), which is compatible with arrows in \(\mathcal{I}\) i.e. \(\forall (g:i\to j)\in\mathcal{I}_1:f_i=F_j\circ F(g)\). Given two co-cones \((X,\{f_i\})\) and \((Y,\{g_i\})\), an arrow between them is some \((u:X\to Y)\in \mathcal{C}_0\) which respects the mappings from \(F\), i.e. \(\forall i\in\mathcal{I}_0:f_i=u\circ g_i\). The collection of all co-cones over \(F\) with arrows between them thus forms a category, \((F\downarrow \Delta\), an example of a comma category.