I have two questions about Wikipedia's definition of triangulated categories.

  1. One of the axioms for distinguished triangles (TR 2) says that if $X\overset u\to Y \overset v\to Z \overset w\to X[1]$ is a distinguished triangle, then so are $Y \overset v\to Z \overset w\to X[1] \overset{-u[1]}\longrightarrow Y[1]$ and $Z[-1]\overset{-w[-1]}\longrightarrow X \overset u\to Y \overset v \to Z$. I get that the translation functor defines $u[1]$ and $w[-1]$, but what does the minus sign in $-u[1]$ and $-w[-1]$ mean?

  2. In TR 1, what does the final "$\to \cdot$" mean? Is it the same as a final "$\to $" as used in the definition of a triangle three lines further up?

Question 2 is really just a PS (I assume that the answer is an affirmative, because what else could it be?). It's question 1 that has me wondering.


1 Answer 1


In a triangulated category, the morphism sets are actually vector spaces over a base field (or modules over a base ring, depending on your level of generality). Hence $-u[1]$ is the morphism $u[1]$ multiplied by the scalar $-1$.

As for your second question, the arrow is what you expect in a triangle. If you wish to be precise, you could write $X\to X\to 0 \to X[1]$.

  • 1
    $\begingroup$ The hom-sets in an additive category are abelian groups. I'm just blind. $\endgroup$
    – Arthur
    Commented Aug 30, 2016 at 11:47
  • 4
    $\begingroup$ Hi, I'd like to add a small question to this answer: what is the "meaning" of such a minus sign? I mean, why is it there, and what would be unpleasant without it? Thanks in advance :) $\endgroup$
    – Nikio
    Commented Jul 25, 2020 at 16:13

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