Let $\mathbf{C}$ be a complete and cocomplete category. Let $\mathcal{M}_\mathbf{C}$ denote the set of model structures on $\mathbf{C}$. For example, as I gather from Tom Goodwillie's answer here, $\mathcal{M}_\mathbf{Set}$ is finite, and has nine elements. Is there a metric defined on $\mathcal{M}_\mathbf{C}$ that in some precise sense measures how "different" two model structures on $\mathbf{C}$ are?
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$\begingroup$ Well, $\mathcal{M}_\mathbf{C}$ is a poset in several different ways -- by inclusion of cofibrations or acyclic cofibrations (or their duals, which are just opposite to these posets) or inclusion of weak equivalences. If I have two model structures on a category and I want to think about how they're related, I usually start by thinking about how they're related in these poset structures... $\endgroup$– Tim Campion ♦Aug 14, 2016 at 12:55
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1$\begingroup$ Note that your set can be a proper class. Using Vopenka's principle, in the category of simplicial sets, there is one left determined model structure with respect to $\Delta[0]\to K$ with $K$ nonempty. $\endgroup$– Philippe GaucherAug 14, 2016 at 15:19
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One thing that might work is the following. Let $G = (\mathcal{M}_\mathbf{C},E)$ denote the graph whose vertices are elements of $\mathcal{M}_\mathbf{C}$, and two model structures $v$ and $v^\prime$ are connected by an edge $v\to v^\prime$ if $v^\prime$ is a (left or right) Bousfield localization of $v$. Then define $d(v,v^\prime)$ to be the minimum number of edges in a path connecting $v$ and $v^\prime$. If there's no path from $v$ to $v^\prime$, i.e., $v^\prime$ cannot be obtained by a sequence of (left or right) Bousfield localizations of $v$, then $d(v,v^\prime) = \infty$. I'm not sure if this is a metric, though, because I don't think it is symmetric.