The Stacks project

4.15 Limits and colimits in the category of sets

Not only do limits and colimits exist in $\textit{Sets}$ but they are also easy to describe. Namely, let $M : \mathcal{I} \to \textit{Sets}$, $i \mapsto M_ i$ be a diagram of sets. Denote $I = \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$. The limit is described as

\[ \mathop{\mathrm{lim}}\nolimits _\mathcal {I} M = \{ (m_ i)_{i\in I} \in \prod \nolimits _{i\in I} M_ i \mid \forall \phi : i \to i' \text{ in }\mathcal{I}, M(\phi )(m_ i) = m_{i'} \} . \]

So we think of an element of the limit as a compatible system of elements of all the sets $M_ i$.

On the other hand, the colimit is

\[ \mathop{\mathrm{colim}}\nolimits _\mathcal {I} M = (\coprod \nolimits _{i\in I} M_ i)/\sim \]

where the equivalence relation $\sim $ is the equivalence relation generated by setting $m_ i \sim m_{i'}$ if $m_ i \in M_ i$, $m_{i'} \in M_{i'}$ and $M(\phi )(m_ i) = m_{i'}$ for some $\phi : i \to i'$. In other words, $m_ i \in M_ i$ and $m_{i'} \in M_{i'}$ are equivalent if there are a chain of morphisms in $\mathcal{I}$

\[ \xymatrix{ & i_1 \ar[ld] \ar[rd] & & i_3 \ar[ld] & & i_{2n-1} \ar[rd] & \\ i = i_0 & & i_2 & & \ldots & & i_{2n} = i' } \]

and elements $m_{i_ j} \in M_{i_ j}$ mapping to each other under the maps $M_{i_{2k-1}} \to M_{i_{2k-2}}$ and $M_{i_{2k-1}} \to M_{i_{2k}}$ induced from the maps in $\mathcal{I}$ above.

This is not a very pleasant type of object to work with. But if the diagram is filtered then it is much easier to describe. We will explain this in Section 4.19.

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