Let's quote the definition of a weak limit for a diagram in a category $C$ that we can find in nLab:

A weak limit for a diagram in a category is a cone over that diagram which satisfies the existence property of a limit but not necessarily the uniqueness.

Let $C$ be a category that has weak limits for a diagram $D: I \to C$. Suppose that $C$ also has limits for $D$, are then all weak limits also limits? If this is not the case, can you provide a counterexample of a category that has both proper weak limits and limits for the same diagram $D$?

I tried to prove that a category with limits for a diagram $D$ cannot have proper weak limits for the same diagram: Let $W$ be a weak limit over $D$. As $C$ also has proper limits for $D$, let $L$ be such a limit cone. But then there must be a unique morphism of cones $\phi: W \to L$, and, by the existence property, a (not necessarily unique) morphism of cones $\psi: L \to W$. By the finality of $L$, we have that $\phi \circ \psi = \mathsf{id}_L$. However, I didn't manage to prove the other half of the isomorphism, namely $\psi \circ \phi = \mathsf{id}_W$.

In case a counterexample can be found, is it possible to find conditions on the diagram $D$ that ensure that weak limits and limits on $D$ coincide if they exist?


Actually, it is easy to see that every inhabited set is a weak terminal object in the category of sets, but only the singleton sets are terminal objects. Hence, there are proper weak terminal objects in the category of sets.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.