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Community Bot
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Let $\cal C$ be a model category which is also additive. Suppose that the homotopy category $\operatorname{Ho}\mathcal C$ is additive, for example this is true when the weak equivalences in $\cal C$ is closed under biproducts (see this questionthis question).

If we take a cofibrant object $X$ and a fibrant object $Y$ then there is a natural isomorphism $$ \operatorname{Ho}\mathcal C(X,Y) \cong \mathcal C(X,Y)/\sim $$ where $\sim$ is the homotopy relation. Is this always a group isomorphism?

Let $\cal C$ be a model category which is also additive. Suppose that the homotopy category $\operatorname{Ho}\mathcal C$ is additive, for example this is true when the weak equivalences in $\cal C$ is closed under biproducts (see this question).

If we take a cofibrant object $X$ and a fibrant object $Y$ then there is a natural isomorphism $$ \operatorname{Ho}\mathcal C(X,Y) \cong \mathcal C(X,Y)/\sim $$ where $\sim$ is the homotopy relation. Is this always a group isomorphism?

Let $\cal C$ be a model category which is also additive. Suppose that the homotopy category $\operatorname{Ho}\mathcal C$ is additive, for example this is true when the weak equivalences in $\cal C$ is closed under biproducts (see this question).

If we take a cofibrant object $X$ and a fibrant object $Y$ then there is a natural isomorphism $$ \operatorname{Ho}\mathcal C(X,Y) \cong \mathcal C(X,Y)/\sim $$ where $\sim$ is the homotopy relation. Is this always a group isomorphism?

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Paul Slevin
Paul Slevin
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A model category which is an additive category

Let $\cal C$ be a model category which is also additive. Suppose that the homotopy category $\operatorname{Ho}\mathcal C$ is additive, for example this is true when the weak equivalences in $\cal C$ is closed under biproducts (see this question).

If we take a cofibrant object $X$ and a fibrant object $Y$ then there is a natural isomorphism $$ \operatorname{Ho}\mathcal C(X,Y) \cong \mathcal C(X,Y)/\sim $$ where $\sim$ is the homotopy relation. Is this always a group isomorphism?