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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 herehere, $\mathcal{M}_\mathbf{Set}$ 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?

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?

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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The space of model category structures

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?