Six model structures for DG-modules over DGAs: model category theory in homological action
By T. Barthel, Jon P. May, Emily Riehl. 2013
66 citations recorded by the source corpus.
In Part 1, we describe six projective-type model structures on the category of dierential graded modules over a dierential graded algebra A over a commutative ring R. When R is a eld, the six collapse to three and are well-known, at least to folklore, but in the general case the new relative and mixed model structures oer interesting alterna- tives to the model structures in common use. The construction of some of these model structures requires two new variants of the small object argument, an enriched and an algebraic one, and we describe these more generally. In Part 2, we present a variety of theoretical and calculational co- brant approximations in these model categories. The classical bar con- struction gives cobrant approximations in the relative model structure, but generally not in the usual one. In the usual model structure, there are two quite dierent ways to lift cobrant approximations from the level of homology modules over homology algebras, where they are classi- cal projective resolutions, to the level of DG-modules over DG-algebras. The new theory makes model theoretic sense of earlier explicit calcu- lations based on one of these constructions. A novel phenomenon we encounter is isomorphic cobrant approximations with dierent combi- natorial structure such that things proven in one avatar are not readily proven in the other.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/6caa45a7ff2403e1b9feb517c0cdeba363f143aa
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Citations in the corpus
- Algebraic weak factorisation systems I: Accessible AWFS☆ (2014, unknown): “For applications that make serious use of the algebraicity so obtained, see [21, 23]; for applications utilising the extra generality of cofibrant generation by a small category, rather than a small set of morphisms, see [5, 7].”
- Categorical Homotopy Theory (2014, methodology): “Indeed, the same sets of generating cofibrations and trivial cofibrations produce both model structures! We find it particularly interesting to note that the Hurewicz-type model structure, which is not cofibrantly generated in the traditional sense, is cofibrantly generated when this notion is enric”