CatDat

Implication Details

Claim: Given a functor whose codomain is trivial, then it is continuous.

Proof: We need to show that for every category $\C$ the unique functor $!_{\C} : \C \to 1$ into the trivial category is continuous. This easy to verify directly because in the trivial category every limit is, well, trivial. More generally, for every category $\C$ and every set $S$ the diagonal functor $\Delta : \C \to \C^S$ is continuous. Here we apply this to $S = \varnothing$ so that $\C^S$ is the trivial category.