nLab
pro-object

Contents

Idea

A pro-object of a category CC is a “formal cofiltered limit” of objects of CC.

The category of pro-objects of CC is written propro-CC. Such a category is sometimes called a ‘pro-category’, but notice that that is not the same thing as a pro-object in Cat.

“Pro” is short for “projective”. ( Projective limit is an older term for limit.) It is in contrast to “ind” in the dual notion of ind-object, standing for “inductive”, (and corresponding to inductive limit, the old term for colimit). In some (often older) sources, the term ‘projective system’ is used more or less synonymously for pro-object.

Definition

The objects of the category propro-CC are diagrams F:DCF:D\to C where DD is a small cofiltered category. The hom set of morphisms between F:DCF:D\to C and G:ECG:E\to C is

(1)pro-C(F,G)=lim eEcolim dDC(Fd,Ge)pro\text{-}C(F,G) = lim_{e\in E} colim_{d\in D} C(F d, G e)

The limit and colimit is taken in the category Set of sets.

Cofiltered limits in Set are given by sets of threads and filtered colimits by germs (classes of equivalences), thus a representative of sproC(F,G)s\in\mathrm{pro}C(F,G) is a thread whose each component is a germ:
s=(germ e(s)) eEs = (germ_e(s))_{e\in E} which can be more concretely written as ([s d e,e]) e([s_{d_e,e}])_e; thus [s d e,e]colim dDC(Fd,Ge)[s_{d_e,e}]\in colim_{d\in D} C(F d, G e) where s d e,eC(Fd e,Ge)s_{d_e,e}\in C(F d_e, G e) is some representative of the class; there is at least one d ed_e for each ee; if the domain EE is infinite, we seem to need an axiom of choice in general to find a function ed ee\mapsto d_e which will choose one representative in each class germ e(s)germ_e(s). Thus ss is given by the (equivalence class) of the following data

  • function ed ee\mapsto d_e

  • correspondence es d e,eC(Fd e,Ge)e\mapsto s_{d_e,e}\in C(F d_e, G e)

such that ([s d e,e]) e([s_{d_e,e}])_e is a thread, i.e. for any γ:ee\gamma: e\to e' we have an equality of classes (germs) [G(γ)s d e,e]=[s d e,e][G(\gamma)\circ s_{d_e,e}] = [s_{d_{e'},e'}]. This equality holds if there is a dd' and morphisms δ e:dd e\delta_e: d'\to d_e, δ e:dd e\delta_{e'}: d'\to d_{e'} such that G(γ)s d e,eFδ e=s d e,eFδ eG(\gamma)\circ s_{d_e,e}\circ F\delta_e = s_{d_{e'},e'}\circ F\delta_{e'}. (Usually in fact people consider the dual of DD and the dual of CC as filtered domains). Now if we chose a different function ed˜ ee\mapsto\tilde{d}_e instead then, ([s d e,e]) e=([s d˜ e,e]) e([s_{d_e,e}])_e = ([s_{\tilde{d}_e,e}])_e, hence by the definition od classes, for every ee there is a dDd''\in D and morphisms σ e:dd e\sigma_e : d''\to d_e, σ˜ e:dd˜ e\tilde\sigma_e:d''\to \tilde{d}_e such that s d˜ e,eF(σ˜ e)=s d e,eF(σ e)s_{\tilde{d}_e,e}\circ F(\tilde\sigma_e) = s_{d_e,e}\circ F(\sigma_e).

This definition is perhaps more intuitive in the dual case of ind-objects (pro-objects in C opC^{op}), where it can be seen as stipulating that the objects of CC are finitely presentable in indind-CC.

Definition of proC\mathrm{pro}C as a subcategory of functors

Another, equivalent, definition is to let propro-CC be the full subcategory of the opposite functor category/presheaf category [C,Set] op[C,Set]^{op} determined by those functors which are cofiltered limits of representables. This is reasonable since the copresheaf category [C,Set] op[C,Set]^{op} is the free completion of CC, so propro-CC is the “free completion of CC under cofiltered limits.”

Examples

Applications

Étale homotopy theory.

Procategories were used by Artin and Mazur in their work on étale homotopy theory. They associated to a scheme a ‘pro-homotopy type’. (This is discussed briefly at étale homotopy.) The important thing to note is that this was a pro-object in the homotopy category of simplicial sets, i.e., in the pro-homotopy category. Friedlander rigidified their construction to get an object in the pro-category of simplicial sets, and this opened the door to use of ‘homotopy pro-categories’.

Shape theory.

The form of shape theory developed by Mardešić and Segal, at about the same time as the work in algebraic geometry, again used the pro-homotopy category. Strong shape, developed by Edwards and Hastings, Porter and also in further work by Mardešić and Segal, used various forms of rigidification to get to the pro-category of spaces, or of simplicial sets. There methods of model category theory could be used.

References

    • A. Grothendieck, Techniques de déscente et théorèmes d’existence en géométrie algébrique, II: le théorème d’existence en théorie formelle des modules, Seminaire Bourbaki 195, 1960, (pdf).
  • (SGA4-1) Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4). Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. Lecture Notes in Mathematics 269, Springer 1972.

  • Michael Artin and Barry Mazur, Étale homotopy theory, 1969, No. 100 in Lecture Notes in Maths., Springer-Verlag, Berlin.

  • Jean-Marc Cordier, Tim Porter, Shape Theory , categorical methods of approximation, Dover (2008) (This is a reprint of the 1989 edition without amendments.)

  • Masaki Kashiwara, Pierre Schapira, Categories and Sheaves

  • Peter Johnstone, Stone Spaces.

  • S. Mardešić, J. Segal, Shape theory, North Holland 1982

Revised on February 4, 2014 21:59:18 by Tim Porter (2.26.27.187)