nLab ind-pro-object

An ind-pro-object in a category $C$ is a formal colimit of formal limits of objects in category $C$. In other words, it is an ind-object in the category $pro(C)$ of pro-objects in $C$. Ind-pro-objects form a category $ind(pro(C))$. Similarly, there is a category $pro(ind(C))$ of pro-ind-objects and its iterated versions. For example, higher local fields can be considered as ring objects in (iterated versions of) $pro(ind(C))$.

Regarding that there is a natural map from colimit of a limit to the limit of a colimit, but in general not the other way around, every ind-pro-object provides a pro-ind-object: there is a canonical functor $ind(pro(C))\to pro(ind(C))$.

In some prominent cases in applications, say if $C$ is a Quillen exact category, then certain subcategory of locally compact objects $ind(pro(C))$ is defined and studied by Beilinson and Kato.

• A. A. Beĭlinson, How to glue perverse sheaves, in: K-theory, arithmetic and geometry (Moscow, 1984–1986), 42–51, Lecture Notes in Math. 1289, Springer 1987.

• Kazuya Kato, Existence theorem for higher local fields, Geometry & Topology Monographs 3 (2000) 165–195 doi

• Luigi Previdi, Locally compact objects in exact categories, arxiv/0710.2509; Sato Grassmannians for generalized Tate spaces, Tohoku Math. J. (2) 64:4 (2012), 489-538, arxiv/1002.4863 euclid; The homotopy determinantal torsor, poster at Newton Institute, pdf; Generalized Tate spaces, PhD thesis, Yale 2010 record

• Benjamin Hennion, Formal loops, Tate objects and tangent Lie algebras, arxiv/1412.0053

• Oliver Braunling, Michael Groechenig, Jesse Wolfson, Tate objects in exact categories (with appendix by Jan Šťovíček and Jan Trlifaj), arxiv/1402.4969

• M. Kapranov, E. Vasserot, Vertex algebras and the formal loop space, Publications Mathématiques de l’IHÉS 100, 209-269 (2004) eudml math.AG/0107143

• M. Kapranov, Infinite-dimensional objects in algebraic geometry, Pathways lectures, Keio university 2007,

  real audio videos at link

• D. Gaitsgory, D. Kazhdan, Representations of algebraic groups over a 2-dimensional local field, GAFA 14 (2004), 535–574, math.RT/0302174

• D. V. Osipov, A. N. Parshin, Harmonic analysis on local fields and adelic spaces I, Izvestiya: Mathematics 2008, 72:5, pp. 915-976 arxiv/0707.1766

The following work of Drinfeld does not treat ind-pro-objects (only ind-schemes and some pro-modules over them), but has influenced many of the works above on Tate objects:

• Vladimir Drinfeld, Infinite-dimensional vector bundles in algebraic geometry: an introduction, in: The unity of mathematics, 263–304, Progr. Math., 244, Birkhäuser Boston 2006, math.AG/0309155

A category $C_1$ somewhat similar to that of Beilinson and Kato, and its higher analogues $C_n$ were studied in

• D.V. Osipov, Adeles on $n$-dimensional schemes and categories $C_n$, Intern. J. Math. 18:3 (2007) 269-279, math.AG/0509189