# Terminal object as a limit

$n$-lab says that that a limit (resp colimit) over the empty diagram in a category $\mathcal{C}$ is the terminal (resp initial) object of $\mathcal{C}$.

Is this true because if we are to give a morphism to each object in the empty diagram, we must factor through the limit over the empty diagram - where each object vacuuously has a morphism to the objects of the empty diagram, and hence necessarily has a unique morphism to this limit.

• It seems you do not need the empty category in your question. You should also formulate more precisely: If the empty diagram has a limit, then the limit is a terminal object (no morphisms are required because the diagram is empty), and conversely if a terminal object exists, then the empty diagram has a limit. – Paul Frost Aug 4 '18 at 23:21
• @PaulFrost The empty diagram is just the image of the empty category under some functor to $\mathcal{C}$ right? (it's true it was unnecessary to denote it at all though I guess) – Heaven Decays Aug 4 '18 at 23:35
• You are right, a diagram in $\mathcal{C}$ is a functor from a small category $I$ to $\mathcal{C}$. But I believe that is clear what the empty diagram is without explicitly defining the empty category ;-) – Paul Frost Aug 4 '18 at 23:41
• @PaulFrost Sure :) – Heaven Decays Aug 4 '18 at 23:42
• $\mathsf{Hom}(-,\prod_{s\in S}A_s)\cong\prod_{s\in S}\mathsf{Hom}(-,A_s)$ assuming $S$-fold products exist in the category. The product on the right is in $\mathbf{Set}$. Now, if $S$ is empty... – Derek Elkins Aug 5 '18 at 1:28

The limit of a diagram $\Delta$ is a cone over $\Delta$ with a certain universal property (see https://en.wikipedia.org/wiki/Limit_(category_theory).
A cone over the empty diagram $\Delta_\emptyset$ is nothing else than an object $X$ of $\mathcal{C}$, and the universal property says that $X$ is a limit of $\Delta_\emptyset$ if and only if each object $Y$ admits a unique morphism $Y \to X$.
Given a small category $J$, a category $\mathcal C$ admits all limits of shape $J$ whenever the "constant diagram" functor $$\mathcal C \to \mathcal C^J,\quad X\mapsto \{j \mapsto X, (i\overset f \to j) \mapsto \mathrm{id}_X\}$$ admits a right adjoint. When $J=\varnothing$, then $\mathcal C^J = 1$ (the terminal category). But a right adjoint of the unique functor $\mathcal C \to 1$ is precisely the selection of a terminal object in $\mathcal C$.