I'm currently reading these short lecture notes and had a question regarding example 2.6(d) (also I think there is a typo in there, but I'm not sure. Anyway...)
In the given category $J$, consisting of three objects $A, B, C$ with morphisms $f, g$ from $C$ to $A, B$, respectively, the notes state that...
- A pullback is the limit of a $J$-shaped diagram
- A pushout is the colimit of a $J^{op}$-shaped diagram
But what about the limit of a $J^{op}$-shaped diagram? Or the colimit of a $J$-shaped diagram? Do these limits and colimits have a name? Are there any elementary examples in which these arise? I haven't seen them before! Thanks!
These are "uninteresting" in the following sense. The pullback of a pushout diagram $A \leftarrow B \rightarrow C$ is the identity arrow $B \to B$ (exercise). The dual statement for pullback diagrams is true by duality. The point is that these diagrams, in these directions, aren't imposing an interesting universal property.