nLab
proadjoint

Definition

Given a functor $F:C\to D$ we say that $F$ admits a proadjoint if the canonical extension $\mathrm{pro}(F):\mathrm{pro}(C)\to \mathrm{pro}(D)$ of $F$ to the categories of pro-objects has a left adjoint $G$. In other words, there is a functor $G:\mathrm{pro}(D)\to \mathrm{pro}(C)$ and a bijection

$$\mathrm{pro}(C)(\mathrm{GY}\prime ,X)\cong \mathrm{pro}(D)(Y\prime ,y(F)X)$$pro(C)(GY',X) \cong pro(D)(Y',y(F)X)

natural in $X$ and $Y\prime$, where $y:C\hookrightarrow \mathrm{pro}(C)$ is the Yoneda embedding into the category of proobjects $\mathrm{pro}(C)\subset {\mathrm{Set}}^{C^{\mathrm{op}}}$. Equivalently, for every prorepresentable functor $X:C{\prime }^{\mathrm{op}}\to \mathrm{Set}$, the functor $X\mapsto X\circ F$ is also prorepresentable.

References

• J.-M. Cordier, T. Porter, Shape theory : Categorical Methods of Approximation, (sec. 2.3), Mathematics and its Applications, Ellis Horwood Ltd., March 1989, 207 pages.Dover addition (2008) (Link to publishers here)