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Extending to a model structure is not a first-order property. (English) Zbl 07319060

Summary: Let \(\mathcal{C}\) be a finitely bicomplete category and \(\mathcal{W}\) a subcategory. We prove that the existence of a model structure on \(\mathcal{C}\) with \(\mathcal{W}\) as the subcategory of weak equivalence is not first order expressible. Along the way we characterize all model structures where \(\mathcal{C}\) is a partial order and show that these are determined by the homotopy categories.

MSC:

55U35 Abstract and axiomatic homotopy theory in algebraic topology
03B10 Classical first-order logic
18B35 Preorders, orders, domains and lattices (viewed as categories)
06A07 Combinatorics of partially ordered sets
03C07 Basic properties of first-order languages and structures
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References:

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