This is a trial post to test the blog.
Two years ago, Yuki Maehara and I observed that limits of constant functors are pointwise. See this note for more details: Terminals are pointwise. At that point, I could not find a reference for this fact.
Recently, I came across Bob Pare’s paper “Three easy pieces” (Section 3. The square root of adjoints). The primary statement is:
[!Theorem] 3.5 in the paper Let $F \colon\mathcal{C}\to \mathcal{D}$ be a functor, and suppose that $F\times F$ has a left adjoint $G$. Then, $F$ itself has a left adjoint $G’\colon \mathcal{D}\to \mathcal{C}$.
This is derived from the following general result about 2-monads:
[!Theorem] 3.4 Theorem Let $\mathbf{K}$ be a 2-category and $T$ be a 2-monad on it. TFAE for a 1-cell $f\colon x\to y$ in $\mathbf{K}$:
- $Tf$ has a left adjoint in $\mathbf{K}$.
- $Tf$ has a left adjoint in the Eilenberg-Moore 2-category of $T$ (b/w the free algebras on them).
- $Tf$ has a left adjoint in the Kleisli 2-category of $T$.
Taking $T=[\mathcal{A},-]$ on $\mathbf{Cat}$, Kleisli maps $F\colon\mathcal{C} \rightsquigarrow\mathcal{D}$ are $\mathcal{A}$-indexed families of functors $(F_{a}\colon \mathcal{C}\to \mathcal{D})_a$ , and the composition is the component-wise composition. As long as $\mathcal{A}$ is non-empty, you get the (component-wise) left adjoint of the original functor as a Kleisli left adjoint.
Still, the statement that “limits of diagrams taking values in constant functors are pointwise” does not follow from this:
[!Proposition] Limits of constant functors are pointwise If $[\mathcal{J},\Delta_{\mathcal{A}}]$ has an absolute left-lifting along $\Delta_{\mathcal{J}}\colon [\mathcal{A},\mathcal{C}]\to[\mathcal{J},[\mathcal{A},\mathcal{C}]]$ and $\mathcal{A}$ is non-empty, then $\Delta_{\mathcal{J}}\colon \mathcal{C}\to [\mathcal{J},\mathcal{C}]$ has a left adjoint.