Metrics on triangulated categories

JOURNAL OF PURE AND APPLIED ALGEBRA

卷:224

Article

Neeman, Amnon¹ 

[1] Australian Natl Univ, Math Sci Inst, Ctr Math & Its Applicat, Bldg 145, Canberra, ACT 2601, Australia

关键词: Triangulated categories;    Homological functors;    Metrics;   

DOI  :  10.1016/j.jpaa.2019.106206

来源: Elsevier

【 摘 要 】

In a 1973 article Lawvere defined (among many other things) metrics on categories- the article has been enormously influential over the years, spawning a huge literature. In recent work, which is surveyed in the current note, we pursue a largely-unexplored angle: we complete categories with respect to their Lawvere metrics. This turns out to be particularly interesting when the category is triangulated and the Lawvere metric is good; a metric is good if it is translation invariant and the balls of radius epsilon > 0 shrink rapidly enough as a decreases. The definitions are all made precise at the beginning of the note. And the main theorem is that a certain natural subcategory S(S), of the completion of S with respect to a good metric, is triangulated. There is also a theorem which, under restrictive conditions, gives a procedure for computing S(S). As examples we discuss the special cases (1) where S is the homotopy category of finite spectra, and (2) where S = D-b (R- mod), the derived category of bounded complexes of finitely generated R-modules over a noetherian ring R. (C) 2019 Elsevier B.V. All rights reserved.

【 授权许可】

Free