A brief note on colimits

A cocone in a category \(\mathcal{C}\)

• A cone in \(\mathcal{C}^{op}\)

• Given a diagram \(\mathcal{J}\xrightarrow{D}\mathcal{C}\), we may take the limit of the functor \(\mathcal{J}^{op}\xrightarrow{D^{op}}\mathcal{C}^{op}\) is a cocone in \(\mathcal{C}\) - the colimit of \(D\) is this cocone.

Let \(\mathcal{C}\xrightarrow{F}\mathcal{D}\) be a functor. How should we define its opposite: \(\mathcal{C}^{op}\xrightarrow{F^{op}}\mathcal{D}^{op}\)

Solution(1)

• There is an isomorphism between a category and its opposite, meaning there are natural functors \(\overset{\cong}\rightarrow\) which alternate between them.

• Define \(\mathcal{C}^{op}\xrightarrow{F^{op}}\mathcal{D}^{op}\) as \(F ; \overset{\cong}\rightarrow\). This is a valid functor as it is the composition of two functors.