Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I know that left adjoints preserve colimits and right adjoints preserve limits. So clearly if the limits (resp. colimits) in both categories exist, the adjoints map them to each other.

My question is, whether it's garantueed that the limit exists?

To be more precise: Let $T\colon A\rightarrow B$ a functor that is a right (resp. left) adjoint of some functor and $D\colon I\rightarrow A$ a diagramn, such that it's limit (resp. colimit) (in $A$) exists. Does the limit (resp. colimt) $T\circ D$ always exist?

share|cite|improve this question

Yes, and this is precisely the statement that $T$ preserves limits! If $\{L \to D\}$ is a limit cone, and $T$ preserves limits, then $\{T(L) \to T \circ D\}$ is a limit cone.

share|cite|improve this answer
Thank you. I was only aware of the "preserveness" if both limits do exist. – John May 25 '13 at 12:57

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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