Return to Answer

replaced http://front.math.ucdavis.edu/ with https://arxiv.org/abs/

Source Link

edited Jun 22, 2022 at 7:16

1
2
3

There is a recent paper of Tobi Barthel, Emily Riehl, and myself that answers this question in a model categorical framework. We were lazy and only considered the one object case, although we believe our work generalizes to DG categories. In another respect we were not lazy: we work over a general commutative ring $R$, not just a field, and this introduces interesting subtleties. The reference is

http://front.math.ucdavis.edu/1310.1159 https://arxiv.org/abs/1310.1159

The standard model structure on (unbounded) DG modules over a DG $R$-algebra $A$ takes quasi-isomorphisms as weak equivalences. Cofibrant approximations are more general than semi-free resolutions, but they are retracts of semi-free resolutions given by model theoretic cellular DG modules. There is an early construction by my adviser, John Moore, in the 1959-60 Cartan seminar, which we modernize. This uses bicomplexes, as usual in differential homological algebra. There is another construction, originally due to Gugenheim and myself, dating from 1974, which we also modernize. Qiaochu, you will be interested that we drop surjectivity in that construction, meaning that we do not have model theoretic fibrations. We use multicomplexes, which are bigraded but have differentials that are sums of pieces that mimic differentials in spectral sequences. The drop of surjectivity and the use of multicomplexes gives a great gain of computability, as we illustrate by modernizing 1974 applications to the computation of the cohomology of many homogeneous spaces.

We also explain the role of the bar construction. When R is a field it constructs cofibrant approximations in the standard model structure. In general, it constructs cofibrant approximations in a relative model structure for which the weak equivalences are the maps of DG $A$-modules which are chain homotopy equivalences of underlying $R$-modules, and in that generality it is not semi-free.

There is a recent paper of Tobi Barthel, Emily Riehl, and myself that answers this question in a model categorical framework. We were lazy and only considered the one object case, although we believe our work generalizes to DG categories. In another respect we were not lazy: we work over a general commutative ring $R$, not just a field, and this introduces interesting subtleties. The reference is

http://front.math.ucdavis.edu/1310.1159

The standard model structure on (unbounded) DG modules over a DG $R$-algebra $A$ takes quasi-isomorphisms as weak equivalences. Cofibrant approximations are more general than semi-free resolutions, but they are retracts of semi-free resolutions given by model theoretic cellular DG modules. There is an early construction by my adviser, John Moore, in the 1959-60 Cartan seminar, which we modernize. This uses bicomplexes, as usual in differential homological algebra. There is another construction, originally due to Gugenheim and myself, dating from 1974, which we also modernize. Qiaochu, you will be interested that we drop surjectivity in that construction, meaning that we do not have model theoretic fibrations. We use multicomplexes, which are bigraded but have differentials that are sums of pieces that mimic differentials in spectral sequences. The drop of surjectivity and the use of multicomplexes gives a great gain of computability, as we illustrate by modernizing 1974 applications to the computation of the cohomology of many homogeneous spaces.

We also explain the role of the bar construction. When R is a field it constructs cofibrant approximations in the standard model structure. In general, it constructs cofibrant approximations in a relative model structure for which the weak equivalences are the maps of DG $A$-modules which are chain homotopy equivalences of underlying $R$-modules, and in that generality it is not semi-free.

There is a recent paper of Tobi Barthel, Emily Riehl, and myself that answers this question in a model categorical framework. We were lazy and only considered the one object case, although we believe our work generalizes to DG categories. In another respect we were not lazy: we work over a general commutative ring $R$, not just a field, and this introduces interesting subtleties. The reference is

https://arxiv.org/abs/1310.1159

The standard model structure on (unbounded) DG modules over a DG $R$-algebra $A$ takes quasi-isomorphisms as weak equivalences. Cofibrant approximations are more general than semi-free resolutions, but they are retracts of semi-free resolutions given by model theoretic cellular DG modules. There is an early construction by my adviser, John Moore, in the 1959-60 Cartan seminar, which we modernize. This uses bicomplexes, as usual in differential homological algebra. There is another construction, originally due to Gugenheim and myself, dating from 1974, which we also modernize. Qiaochu, you will be interested that we drop surjectivity in that construction, meaning that we do not have model theoretic fibrations. We use multicomplexes, which are bigraded but have differentials that are sums of pieces that mimic differentials in spectral sequences. The drop of surjectivity and the use of multicomplexes gives a great gain of computability, as we illustrate by modernizing 1974 applications to the computation of the cohomology of many homogeneous spaces.

We also explain the role of the bar construction. When R is a field it constructs cofibrant approximations in the standard model structure. In general, it constructs cofibrant approximations in a relative model structure for which the weak equivalences are the maps of DG $A$-modules which are chain homotopy equivalences of underlying $R$-modules, and in that generality it is not semi-free.

Source Link

created Dec 6, 2013 at 2:37

30.4k
3
96
140

There is a recent paper of Tobi Barthel, Emily Riehl, and myself that answers this question in a model categorical framework. We were lazy and only considered the one object case, although we believe our work generalizes to DG categories. In another respect we were not lazy: we work over a general commutative ring $R$, not just a field, and this introduces interesting subtleties. The reference is

http://front.math.ucdavis.edu/1310.1159

The standard model structure on (unbounded) DG modules over a DG $R$-algebra $A$ takes quasi-isomorphisms as weak equivalences. Cofibrant approximations are more general than semi-free resolutions, but they are retracts of semi-free resolutions given by model theoretic cellular DG modules. There is an early construction by my adviser, John Moore, in the 1959-60 Cartan seminar, which we modernize. This uses bicomplexes, as usual in differential homological algebra. There is another construction, originally due to Gugenheim and myself, dating from 1974, which we also modernize. Qiaochu, you will be interested that we drop surjectivity in that construction, meaning that we do not have model theoretic fibrations. We use multicomplexes, which are bigraded but have differentials that are sums of pieces that mimic differentials in spectral sequences. The drop of surjectivity and the use of multicomplexes gives a great gain of computability, as we illustrate by modernizing 1974 applications to the computation of the cohomology of many homogeneous spaces.

We also explain the role of the bar construction. When R is a field it constructs cofibrant approximations in the standard model structure. In general, it constructs cofibrant approximations in a relative model structure for which the weak equivalences are the maps of DG $A$-modules which are chain homotopy equivalences of underlying $R$-modules, and in that generality it is not semi-free.