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

- $A$ is colocal if and only if
- $A$ is cofibrant with respect to the original model structure, and
- 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**

- $A$ is colocal if and only if
- $A$ is cofibrant with respect to the original model structure, and
- 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.*

largerthan the class of enriched colocal equivalences, so the ordinary colocalization inverts all maps that are inverted by the enriched colocalization. $\endgroup$4more comments