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In the projective model structure on the category of chain complexes, is the homotopicness between $f:A\to B$ and $g:A\to B$ is equivalent to null-homotopicness of $g-f:A\to B$ ?

I think it's intuitively true but can not find the proof.

Yuta
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  • Since you mention a model structure I guess by "homotopic" you mean the model category definition. But what do you mean by "null-homotopic"? The algebra definition? – Zhen Lin Jan 28 '21 at 07:22
  • @ZhenLin By "null-homotopic", I meant "homotopic to the $0$ map". – Yuta Jan 28 '21 at 09:42

1 Answers1

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The claim is true for any model structure on any additive category.

Suppose we have a (weak) path object for $B$, i.e. an object $P$ and weak equivalences $j : B \to P$ and $q_0, q_1: P \to B$ such that $q_0 \circ j = q_1 \circ j = \textrm{id}_B$, and a homotopy from $f$ to $g$ w.r.t. this path object, i.e. a morphism $h : A \to P$ such that $q_0 \circ h = f$ and $q_1 \circ h = g$. Then $h - j \circ f$ is a homotopy from $0$ to $g - f$: indeed, $q_0 \circ (h - j \circ f) = q_0 \circ h - q \circ j \circ f = f - f = 0$ and similarly $q_1 \circ (h - j \circ f) = q_0 \circ h - q \circ j \circ f = g - f$, as required.

Actually, more is true: even without a model structure, the localisation of an additive category is always additive and the localisation functor is additive.

Zhen Lin
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