Examples of limits and colimits

 Lecture topic

Let $C$ be a category and $X,Y,Z\in \Obj(C)$. Choose $I$ to be a category with $\mathcal F:I\to C$ a functor as described below. Then we may consider the limit and colimit of $\mathcal F$, noting that they may not always exist, as there may be no suitable natural transformation $i$ or $\pi$.

The limit and colimit of the category $I$ with two points and two arrows going between the points in opposite directions, namely

are not interesting to consider. That is because as a category, it must satisfy compositions, so $f\circ g=\id$, which is a restrictive condition on $f$ and $g$. We may define a new map $h:X\to X$ with $h=f\circ g$, but then more maps, such as $h\circ f$ and so on need to be defined, which complicate the situation.

References: Borceux (Handbook of Categorical Algebra I, Chapter 2)