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

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

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


Shape theory, étale homotopy and Galois theory

If an (∞,1)-topos H\mathbf{H} has a genuine left adjoint (∞,1)-functor Π\Pi to its constant ∞-stack functor Δ\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 left adjoint Π\Pi does not exist, but the pro-left adjoint Π pro\Pi_{pro} to Δ\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 gorup is the Galois group/algebraic fundamental group) while in topology this reproduces the concept of shape. (Whence the term shape modality for Π\Pi).


  • 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)

Revised on May 15, 2015 17:56:27 by Urs Schreiber (