nLab proadjoint

Definition

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

$$pro(C)(GY',X) \cong pro(D)(Y',y(F)X)$$

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

Examples

Shape theory, étale homotopy and Galois theory

If an (∞,1)-topos $\mathbf{H}$ has a genuine left adjoint (∞,1)-functor $\Pi$ to its constant ∞-stack functor $\Delta$, then that may be interpreted as sending each object to its fundamental ∞-groupoid, see at fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos.

In general such a left adjoint $\Pi$ does not exist, but the pro-left adjoint $\Pi_{pro}$ to $\Delta$ always exists. This hence produces a pro-version of the fundamental ∞-groupoid-construction known generally as the étale homotopy type. In algebraic geometry and arithmetic geometry this reproduces the content of Galois theory (the pro-etale fundamental group is the Galois group/algebraic fundamental group) while in topology this reproduces the concept of shape. (Whence the term shape modality for $\Pi$).

Related concepts

• pro-homotopy theory

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)

• Marc Hoyois, A note on Étale homotopy, 2013 (pdf)