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To my knowledge the notion of a t-category was first introduced Beilinson, Bernstein and Deligne's Faiseaux Pervers. But while they explain the name "perverse sheaf", they don't give any indication how they came up with the name t-category.

Does anyone know whether the "t" in "t-category"/"t-structure" stands for something specific?

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I had always assumed that $t$ stood for "truncation". –  Donu Arapura Feb 5 '12 at 22:54
    
@Donu: That might be a good rationale for the terminology, but in any case the section starts out with the definition of "t-category" as a special type of triangulated category which is natural for the theory of perverse sheaves. It would be interesting to know the viewpoints (possibly divergent) of the three authors, one of whom probably did the major writing here (in French). –  Jim Humphreys Feb 5 '12 at 23:12
    
A little late, but someone once told me that Bernstein told them that it stands for "truncation". (Now you just have to decide how much weight you give to third-hand accounts.) –  Artie Prendergast-Smith Jun 6 '13 at 20:03

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up vote 8 down vote accepted

Though I don't have inside information, it's clear that the notes in Asterisque 100 (1982) by BBD gave the first formal definition of t-category and t-structure on a triangulated category (section 1.3). Since the nearest t-word involved is "triangulated" and no direct rationale is offered for the language introduced, that probably suggested the terminology here. Of course, someone else may have a more subtle theory.

P.S. Reviewing the original text further, I see in their Example 1.3.2(1) the suggestive word tronque, which converts me to Donu's viewpoint. It would have simplified things for the reader of BBD just to start with a definition of categorie tronque (abbreviated t-category).

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