Let $I$ be a small category and $\mathcal{C}$ an $I$-complete category. Denote $\iota : I \rightarrow \hat I$ the inclusion into the category obtained from $I$ by “adding an initial object”. A cone over some diagram $D: I \to \mathcal{C}$ can be considered as a diagram $\hat D : \hat I \to \mathcal{C}$ such that $[\iota, \mathcal{C}](\hat D) = D$, where $[ \iota, \mathcal{C}] : [\hat I,\mathcal{C}] \to [I, \mathcal{C}]$ is the obvious functor associated to $\iota$.

Now it is easy to see that the functor $lim_I :[I, \mathcal{C}] \to [\hat I,\mathcal{C}]$ associating to each diagram its limit-cone is a right adjoint of $[ \iota, \mathcal{C}] : [\hat I,\mathcal{C}] \to [I, \mathcal{C}]$ (the counit is just the identity). Thus, $lim_I$ could be actually defined in this way up to isomorphism (because of unicity of adjoints up to isomorphism).

My question is wether this works even if we don't require $\mathcal{C}$ to be $I$-complete, more precisely: is any category $\mathcal{C}$, for which $[ \iota, \mathcal{C}] : [\hat I,\mathcal{C}] \to [I, \mathcal{C}]$ has a right adjoint, $I$-complete (so that we can then define $lim_I$ to be this right adjoint)?

It seems to me that we might have to require the counit of the adjunction to be an isomorphism. In that case $-$ have such adjunctions been studied (and if yes, where can I read about it)? are they closed under composition for example?


1 Answer 1


Note that the diagonal functor $\Delta\colon \mathcal C \to [I,\mathcal C]$ factorizes as $\mathcal C \to [\hat I,\mathcal C]\to [I,\mathcal C]$ where the first functor is the diagonal functor for $\hat I$ and the second functor is the restriction functor you describe. Further, the right adjoint $[\hat I,\mathcal C]\to \mathcal C$ certainly exists, since $\hat I$ has an initial object. So, if a right adjoint $[I,\mathcal C]\to [\hat I,\mathcal C]$ exists, then the composition $[I,\mathcal C]\to [\hat I,\mathcal C]\to C$, which maps a diagram to its limiting object, must be a right adjoint to the diagonal functor $\mathcal C \to [I,\mathcal C]$, and so is, up to isomorphism, the limits of shape $I$ functor.

For generalities about adjunctions you may wish to read Mac Lane's chapter on adjunctions. There are many other texts of course that will also treat adjunctions elementarily.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .