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According to Hovey's Model categories (around Theorem 2.3.11), a chain homotopy can equivalently be described as a right homotopy wrt to the standard model structure of the category of chain complexes.

On the other hand, according to this question, it says that chain homotopy can equivalently be described as a left homotopy.

My understanding is that left and right homotopies, in general, do not coincide (when the domain object is cofibrant and the codomain object is fibrant, they do), and I'm confusing about the above two equivalent descriptions of chain homotopies.

Yuta
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1 Answers1

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A chain homotopy between chain maps $f,g\colon X→Y$ can be defined as a chain map $h\colon I⊗X→Y$, where $I$ denotes the chain complex of simplicial chains on a 1-simplex, i.e., ${\bf Z}⊕{\bf Z}←{\bf Z}$ in chain degrees 0 and 1, with the differential $1⊕-1$. This is a special case of a left homotopy with a cylinder object $I⊗X$.

By the hom-tensor adjunction, $h$ is adjoint to a map $H\colon X→{\rm Hom}(I,Y)$. Here ${\rm Hom}(I,Y)$ is a path object for $Y$, so the map $H$ is a right homotopy.

Dmitri P.
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  • My understanding is that left and right homotopies, in general, do not coincide. However, it seems that they do coincide in the category of chain complexes. Does that mean the category of chain complexes has a special property? – Yuta Dec 24 '20 at 07:08
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    @Yuta: The three notions of homotopy (chain homotopy, left homotopy, and right homotopy) are different for chain complexes. A chain homotopy is a special case of left and right homotopy. But being left or right homotopic does not imply being chain homotopic. Furthermore, left and right homotopies depend on the particular choice of a model structure on chain complexes (projective, injective, flat, ...). – Dmitri P. Dec 24 '20 at 07:25
  • I was misunderstanding the definition of being left (right) homotopic. I was thinking that the definition of, say, left homotopic depends on the choice of a cylinder object. But I realize that it means that there is a left homotopy for some cylinder object, thus the definition does not depend on a specific choice.

    Now I realize what "chain homotopy is a special case of a left homotopy" means. We have that for two parallel morphisms, they are

    1. (chain homotopic) => (left homotopic)
    2. (chain homotopic) => (right homotopic)

    but both converse does not hold. Is that right?

     – Yuta Dec 25 '20 at 04:42
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    @Yuta: Yes. In general, one needs additional conditions like cofibrance of the source and fibrancy of the target in order to freely change the cylinder object in a given homotopy. – Dmitri P. Dec 25 '20 at 04:55