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Let $C$ be a $V$-model category, and $\mathcal{K}$ a set of objects of $C$. Let me denote (derived) simplicial homotopy function complexes by $\text{Dmap}$ and derived $V$-function complexes by $\text{DMap}_V$.

Edit: By the latter, I just mean that $\text{DMap}_V$ is obtained from the $V$-enrichment $\text{Map}_V$ on $M$ by using cofibrant/fibrant replacements with respect to the given model structure on $C$, i.e., $\text{DMap}_V (X,Y) = \text{Map}_V (X^{\text{cof}},Y^{\text{fib}})$.

Under some technical assumptions (combinatoriality and right properness of $C$ are sufficient), one can form the $V$-enriched colocalization of $C$ at $\mathcal{K}$, denoted by $R^V_{\mathcal{K}} C$ whose weak equivalences are called $\mathcal{K}$-colocal equivalences.

Let us call the cofibrant objects of $R^V_{\mathcal{K}} C$ colocal objects. There are two possible characterizations of colocal objects, namely

  1. $A$ is colocal if and only if
    1. $A$ is cofibrant with respect to the original model structure, and
    2. For every $\mathcal{K}$-colocal equivalence $h: X \to Y$, the induced map $\text{Dmap}(A,h): \text{Dmap}(A,X) \to \text{Dmap}(A,Y)$ is a weak equivalence of simplicial sets

or

  1. $A$ is colocal if and only if
    1. $A$ is cofibrant with respect to the original model structure, and
    2. For every $\mathcal{K}$-colocal equivalence $h: X \to Y$, the induced map $\text{DMap}_V (A,h): \text{DMap}_V (A,X) \to \text{DMap}_V (A,Y)$ is a weak equivalence in $V$.

Edit: Made 1. and 2. more precise.

Question: Why is 1. equivalent to 2.?

In more detail:

One construction of $R^V_{\mathcal{K}} C$ is given as the classical Bousfield colocalization $R_{K \otimes \mathcal{G}_V} C$ of $C$ at the set $K \otimes \mathcal{G}_V$ where $\otimes: C \times V \to C$ is part of the $V$-enrichment on $C$ and $\mathcal{G}_V$ is a set of cofibrant homotopy generators of $V$.

This is Definition 2.10 of https://arxiv.org/abs/1411.0500, and the weak equivalences in $R^V_{\mathcal{K}} C$ are those maps $h: X \to Y$ such that the induced maps $\text{Dmap}(K \otimes G, X) \to \text{Dmap} (K \otimes G,Y)$ are weak equivalences of simplicial sets for $K \in \mathcal{K}, G \in \mathcal{G}_V$ by definition. However, Theorem 2.12 shows that the $\mathcal{K}$-colocal equivalences are precisely those maps $h: X \to Y$ that induce weak equivalences $\text{DMap}_V (K,X) \to \text{DMap}_V (K,Y)$ in $V$ for all $K \in \mathcal{K}$.

Thus, I would like to have a result similar to Theorem 2.12 for the two characterizations of cofibrant objects 1. and 2. from before. Characterization 1. is immediate from the construction and results in Hirschhorn's book.

Edit: Removed a lot of text concerning another article. I have now understood the notation in this article correctly, and my former elaborations do not make sense anymore.

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  • $\begingroup$ 1 and 2 are not equivalent. Not every Bousfield (co)localization of an enriched model category gives you an enriched model category. In fact, for a Bousfield colocalization to be enriched one must require that K⊗G_V consists of K-colocal objects, e.g., K⊗G_V⊂K. 2 implies 1 by applying the functor RMap(1_V,−) to the map in (2). The opposite is not true, e.g., for stacks the mapping space is the evaluation of the internal hom on the point, and there are morphisms of stacks that induced a weak equivalence when evaluated on the point but are not weak equivalences. $\endgroup$ Oct 15 '16 at 21:50
  • $\begingroup$ Thank you for your comment. I did not mean to suggest that the colocalization is enriched again. The derived $V$-function complexes are taken with respect to the original model structure on $C$. I will edit my questions to make this more precise. $\endgroup$ Oct 17 '16 at 18:20
  • $\begingroup$ The characterizations 1. and 2. only do have to hold for colocal equivalences. I agree that they are not equivalent for weak equivalences in the original model structure. I will review my proof of 1.=>2. in the light of your example. $\endgroup$ Oct 17 '16 at 18:28
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    $\begingroup$ I do now see that my alleged proof of 1.=>2. was wrong while your arguments show that 2.=>1.. However, it seems to me that the class of ordinary colocal equivalences is larger than the class of enriched colocal equivalences, so the ordinary colocalization inverts all maps that are inverted by the enriched colocalization. $\endgroup$ Oct 19 '16 at 15:43
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    $\begingroup$ Sorry, I meant to refer to the dual characterization of colocal equivalences. K-colocal equivalences are maps f such that RMap(K,f) is a weak equivalence for all K. K-colocal enriched equivalences are maps f such that RMap_V(K,f) is a weak equivalence for all K. The latter class is strictly larger than the former class because RMap(1_V,RMap_V(K,f))=RMap(K,f), i.e., there are more K-colocal enriched equivalences than there are K-colocal equivalences. $\endgroup$ Oct 19 '16 at 18:26
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It is not true that 1. is equivalent to 2.

We have that 2. implies 1. by one of Dmitri's comments to the question.

As Dmitri pointed out in the same comment, a counterexample to 1.=>2. can be constructed by a map of stacks that is not a weak equivalence but a weak equivalence at the point.

More precisely, we consider the enriched colocalization of the category of stacks at the object $\ast$. This procedure does not change the model category structure. The unique map from $\Omega^0_{\text{cl}}$, the levelwise discrete stack of closed $0$-forms, to the stack $\ast$ represented by a one-point manifold, which is the terminal object in stacks, is an equivalence at the point because there is a unique $0$-form on the point, which is closed. But the map $\Omega^0_{\text{cl}} \to \ast$ cannot be a weak equivalence of stacks because the underlying homotopy type of $\Omega^0_{\text{cl}}$ is a $K(\mathbb{R}_\delta,0)$ which is not trivial.

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