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Let me first give an abstract definition of the homotopy colimit. Let $C$ be a cofibrantly generated model category and let $D$ be a small category. There is an adjunction $$ \operatorname{colim}\colon Functor(D,C)\leftrightarrows C\colon\operatorname{const} $$ defining the colimit as a left adjoint to the functor sending an object $X$ to the constant diagram $E:D\to C$, $E(d)=X$. The category $Functor(D,C)$ on the left-hand side carries the projective model structure where weak equivalences and fibrations are defined objectwise.

With this model structure, the adjunction above is a Quillen adjunction and has thus a derived adjunction $$ \operatorname{Hocolim}\colon Ho(Functor(D,C))\leftrightarrows Ho(C)\colon\operatorname{Hoconst} $$ whose left-adjoint is called the homotopy colimit by definition.

There is up to unique isomorphism just one adjoint functor to some given functor. Hence, if $\operatorname{Hoconst}$ is that given functor, the homotopy colimit functor is unique up to unique isomorphism. On the other hand I learned that the homotopy colimit is unique up to isomorphism in the homotopy category but it is not unique up to unique isomorphism. Could someone help me to clarify this confusing situation?

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The correct generalization of "unique up to unique isomorphism" in these higher settings is "the space of possibilities is contractible." – Qiaochu Yuan Feb 25 '14 at 18:35
up vote 3 down vote accepted

When people say "unique up to unique isomorphism" it must always be understood in the appropriate sense. This particular case is an instance of the uniqueness of objects defined by adjunctions. Suppose we have an adjunction $$F \dashv U : \mathcal{D} \to \mathcal{C}$$ where $F : \mathcal{C} \to \mathcal{D}$ is the left adjoint. Then, for each object $C$ in $\mathcal{C}$, we have the following natural bijection: $$\mathcal{D} (F C, D) \cong \mathcal{C} (C, U D)$$ Thus, for any other object $D'$ equipped with a natural bijection $$\mathcal{C} (C, U D) \cong \mathcal{D} (D', D)$$ there is a unique isomorphism $D' \to F C$ such that the induced map $\mathcal{D} (F C, D) \to \mathcal{D} (D', D)$ is the composite of the two bijections above.

More explicitly, homotopy colimits are unique up to unique isomorphism in the homotopy category, so long as you require that the candidate isomorphism be compatible with the colimiting cocones. This is exactly the same as in ordinary categories – except perhaps for a subtlety about what a diagram is and what a cocone is.

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Thank you. What means not unique up to unique isomorphism for a homotopy colimit then? I.e. for what appropriate sense is this statement made? – user8463524 Feb 26 '14 at 7:41
They are not unique up to unique isomorphism in the original category. In fact they are not even well-defined up to isomorphism – only up to weak equivalence. – Zhen Lin Feb 26 '14 at 8:00

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