Is every dg-coalgebra the colimit of its finite dimensional dg-subcoalgebras?In which categories is every...
Is every dg-coalgebra the colimit of its finite dimensional dg-subcoalgebras?
In which categories is every coalgebra a sum of its finite-dimensional subcoalgebras?“Strøm-type” model structure on chain complexes?Is there a model category structure on non-negatively graded commutative chain algebras?Resolutions by Adapted Class of Objects and Model CategoriesMaps between sets and coalgebrasBounded dg algebra vs unbounded dg algebrasTensor product of coaugmented conilpotent coalgebrasBuilding conilpotent coalgebras from co-square-zero-extensionsLimits in subcategories of Powerset-coalgebrasIs $text{DGA}^{-}$ a monoidal model category?
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I saw this result in A Model Category Structure for Differential Graded Coalgebras by Getzler-Goerss, but when the coalgebra is non-negatively graded, is this property also satisfied when the dg coalgebra is $mathbb{Z}$-graded?.
Thanks.
at.algebraic-topology kt.k-theory-and-homology model-categories coalgebras dg-categories
$endgroup$
add a comment |
$begingroup$
I saw this result in A Model Category Structure for Differential Graded Coalgebras by Getzler-Goerss, but when the coalgebra is non-negatively graded, is this property also satisfied when the dg coalgebra is $mathbb{Z}$-graded?.
Thanks.
at.algebraic-topology kt.k-theory-and-homology model-categories coalgebras dg-categories
$endgroup$
$begingroup$
The result the OP is pointing to in Getzler-Goerss is Corollary 1.6.
$endgroup$
– David White
7 hours ago
add a comment |
$begingroup$
I saw this result in A Model Category Structure for Differential Graded Coalgebras by Getzler-Goerss, but when the coalgebra is non-negatively graded, is this property also satisfied when the dg coalgebra is $mathbb{Z}$-graded?.
Thanks.
at.algebraic-topology kt.k-theory-and-homology model-categories coalgebras dg-categories
$endgroup$
I saw this result in A Model Category Structure for Differential Graded Coalgebras by Getzler-Goerss, but when the coalgebra is non-negatively graded, is this property also satisfied when the dg coalgebra is $mathbb{Z}$-graded?.
Thanks.
at.algebraic-topology kt.k-theory-and-homology model-categories coalgebras dg-categories
at.algebraic-topology kt.k-theory-and-homology model-categories coalgebras dg-categories
edited 7 hours ago
David White
13.2k464105
13.2k464105
asked 8 hours ago
Victor TCVictor TC
962
962
$begingroup$
The result the OP is pointing to in Getzler-Goerss is Corollary 1.6.
$endgroup$
– David White
7 hours ago
add a comment |
$begingroup$
The result the OP is pointing to in Getzler-Goerss is Corollary 1.6.
$endgroup$
– David White
7 hours ago
$begingroup$
The result the OP is pointing to in Getzler-Goerss is Corollary 1.6.
$endgroup$
– David White
7 hours ago
$begingroup$
The result the OP is pointing to in Getzler-Goerss is Corollary 1.6.
$endgroup$
– David White
7 hours ago
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Yes. The category of $mathbb{Z}$-coalgebras is locally presentable, and objects are filtered colimits of finite dimensional subobjects. See the appendix to Coalgebraic models for combinatorial model categories by Ching and Riehl. See also Lemma 5.2 of Model Structures for Coalgebras by Drummond-Cole and Hirsh. This paper of Adamek and Porst might also be helpful.
By the way, the main result of the Getzler-Goerss paper you cite is generalized in Corollary 6.3.5 of A necessary and sufficient condition for induced model structures by Hess, Kedziorek, Riehl, and Shipley. It works for any $mathbb{Z}$-graded coalgebras over any commutative ring $R$.
$endgroup$
$begingroup$
Thank you very much, I will check the references.
$endgroup$
– Victor TC
6 hours ago
$begingroup$
@VictorTC As an additional remark, this is true more in general for coalgebras over cooperads, see Lemmas 4, 5, and Proposition 12 of the article Homotopy theory of unital algebras by B. Le Grignou.
$endgroup$
– Daniel Robert-Nicoud
4 hours ago
$begingroup$
N.B., we work explicitly with conilpotent coalgebras, and I do not know how to extend our method for proving presentability to the full category of coalgebras. So our paper may not be particularly useful for you.
$endgroup$
– Gabriel C. Drummond-Cole
1 hour ago
$begingroup$
@DanielRobert-Nicoud do you know how to resolve the seeming discrepancy between Lemma 5 of Le Grignou and Leonid's counterexample below?
$endgroup$
– Gabriel C. Drummond-Cole
1 hour ago
add a comment |
$begingroup$
For coassociative dg-coalgebras over any field $k$ the answer is positive, because:
Let $C$ be a $mathbb Z$-graded coalgebra and $Dsubset C$ a finite-dimensional ungraded subcoalgebra (of the underlying ungraded coalgebra) of $C$. Let $D^{gr}subset C$ denote the graded vector subspace spanned by all the grading components of the elements of $D$. Then $Dsubset D^{gr}$ and $D^{gr}$ is a finite-dimensional graded subcoalgebra of $C$.
Let $(C,d)$ be a dg-coalgebra and $Dsubset C$ be a finite-dimensional graded subcoalgebra of $C$. Set $D^{dg}=D+d(D)subset C$. Then $Dsubset D^{dg}$ and $D^{dg}$ is a finite-dimensional dg-subcoalgebra of $C$.
Using the observations 1. and 2. and the fact that any ungraded coassociative coalgebra is the union of its finite-dimensional subcoalgebras, one deduces the assertion that any $mathbb Z$-graded dg-coalgebra is the union of its finite-dimensional dg-subcoalgebras.
Possible generalizations: One can replace a field $k$ by a Noetherian commutative ring $k$ and speak about subcoalgebras that are finitely generated as $k$-modules (instead of "finite-dimensional"). All the assertions remain true.
One cannot drop the coassociativity condition. Indeed, even for ungraded coalgebras over a field of characteristic $0$, there is an example of a infinite-dimensional Lie coalgebra $L$ having no nonzero finite-dimensional subcoalgebras. The Lie coalgebra $L$ is simplest described in terms of its dual topological Lie algebra structure (on a pro-finite-dimensional topological vector space): $L^*=mathfrak g=k[[z]],d/dz$, the Lie algebra of vector fields on the formal disk.
$endgroup$
add a comment |
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2 Answers
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2 Answers
2
active
oldest
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$begingroup$
Yes. The category of $mathbb{Z}$-coalgebras is locally presentable, and objects are filtered colimits of finite dimensional subobjects. See the appendix to Coalgebraic models for combinatorial model categories by Ching and Riehl. See also Lemma 5.2 of Model Structures for Coalgebras by Drummond-Cole and Hirsh. This paper of Adamek and Porst might also be helpful.
By the way, the main result of the Getzler-Goerss paper you cite is generalized in Corollary 6.3.5 of A necessary and sufficient condition for induced model structures by Hess, Kedziorek, Riehl, and Shipley. It works for any $mathbb{Z}$-graded coalgebras over any commutative ring $R$.
$endgroup$
$begingroup$
Thank you very much, I will check the references.
$endgroup$
– Victor TC
6 hours ago
$begingroup$
@VictorTC As an additional remark, this is true more in general for coalgebras over cooperads, see Lemmas 4, 5, and Proposition 12 of the article Homotopy theory of unital algebras by B. Le Grignou.
$endgroup$
– Daniel Robert-Nicoud
4 hours ago
$begingroup$
N.B., we work explicitly with conilpotent coalgebras, and I do not know how to extend our method for proving presentability to the full category of coalgebras. So our paper may not be particularly useful for you.
$endgroup$
– Gabriel C. Drummond-Cole
1 hour ago
$begingroup$
@DanielRobert-Nicoud do you know how to resolve the seeming discrepancy between Lemma 5 of Le Grignou and Leonid's counterexample below?
$endgroup$
– Gabriel C. Drummond-Cole
1 hour ago
add a comment |
$begingroup$
Yes. The category of $mathbb{Z}$-coalgebras is locally presentable, and objects are filtered colimits of finite dimensional subobjects. See the appendix to Coalgebraic models for combinatorial model categories by Ching and Riehl. See also Lemma 5.2 of Model Structures for Coalgebras by Drummond-Cole and Hirsh. This paper of Adamek and Porst might also be helpful.
By the way, the main result of the Getzler-Goerss paper you cite is generalized in Corollary 6.3.5 of A necessary and sufficient condition for induced model structures by Hess, Kedziorek, Riehl, and Shipley. It works for any $mathbb{Z}$-graded coalgebras over any commutative ring $R$.
$endgroup$
$begingroup$
Thank you very much, I will check the references.
$endgroup$
– Victor TC
6 hours ago
$begingroup$
@VictorTC As an additional remark, this is true more in general for coalgebras over cooperads, see Lemmas 4, 5, and Proposition 12 of the article Homotopy theory of unital algebras by B. Le Grignou.
$endgroup$
– Daniel Robert-Nicoud
4 hours ago
$begingroup$
N.B., we work explicitly with conilpotent coalgebras, and I do not know how to extend our method for proving presentability to the full category of coalgebras. So our paper may not be particularly useful for you.
$endgroup$
– Gabriel C. Drummond-Cole
1 hour ago
$begingroup$
@DanielRobert-Nicoud do you know how to resolve the seeming discrepancy between Lemma 5 of Le Grignou and Leonid's counterexample below?
$endgroup$
– Gabriel C. Drummond-Cole
1 hour ago
add a comment |
$begingroup$
Yes. The category of $mathbb{Z}$-coalgebras is locally presentable, and objects are filtered colimits of finite dimensional subobjects. See the appendix to Coalgebraic models for combinatorial model categories by Ching and Riehl. See also Lemma 5.2 of Model Structures for Coalgebras by Drummond-Cole and Hirsh. This paper of Adamek and Porst might also be helpful.
By the way, the main result of the Getzler-Goerss paper you cite is generalized in Corollary 6.3.5 of A necessary and sufficient condition for induced model structures by Hess, Kedziorek, Riehl, and Shipley. It works for any $mathbb{Z}$-graded coalgebras over any commutative ring $R$.
$endgroup$
Yes. The category of $mathbb{Z}$-coalgebras is locally presentable, and objects are filtered colimits of finite dimensional subobjects. See the appendix to Coalgebraic models for combinatorial model categories by Ching and Riehl. See also Lemma 5.2 of Model Structures for Coalgebras by Drummond-Cole and Hirsh. This paper of Adamek and Porst might also be helpful.
By the way, the main result of the Getzler-Goerss paper you cite is generalized in Corollary 6.3.5 of A necessary and sufficient condition for induced model structures by Hess, Kedziorek, Riehl, and Shipley. It works for any $mathbb{Z}$-graded coalgebras over any commutative ring $R$.
edited 7 hours ago
answered 7 hours ago
David WhiteDavid White
13.2k464105
13.2k464105
$begingroup$
Thank you very much, I will check the references.
$endgroup$
– Victor TC
6 hours ago
$begingroup$
@VictorTC As an additional remark, this is true more in general for coalgebras over cooperads, see Lemmas 4, 5, and Proposition 12 of the article Homotopy theory of unital algebras by B. Le Grignou.
$endgroup$
– Daniel Robert-Nicoud
4 hours ago
$begingroup$
N.B., we work explicitly with conilpotent coalgebras, and I do not know how to extend our method for proving presentability to the full category of coalgebras. So our paper may not be particularly useful for you.
$endgroup$
– Gabriel C. Drummond-Cole
1 hour ago
$begingroup$
@DanielRobert-Nicoud do you know how to resolve the seeming discrepancy between Lemma 5 of Le Grignou and Leonid's counterexample below?
$endgroup$
– Gabriel C. Drummond-Cole
1 hour ago
add a comment |
$begingroup$
Thank you very much, I will check the references.
$endgroup$
– Victor TC
6 hours ago
$begingroup$
@VictorTC As an additional remark, this is true more in general for coalgebras over cooperads, see Lemmas 4, 5, and Proposition 12 of the article Homotopy theory of unital algebras by B. Le Grignou.
$endgroup$
– Daniel Robert-Nicoud
4 hours ago
$begingroup$
N.B., we work explicitly with conilpotent coalgebras, and I do not know how to extend our method for proving presentability to the full category of coalgebras. So our paper may not be particularly useful for you.
$endgroup$
– Gabriel C. Drummond-Cole
1 hour ago
$begingroup$
@DanielRobert-Nicoud do you know how to resolve the seeming discrepancy between Lemma 5 of Le Grignou and Leonid's counterexample below?
$endgroup$
– Gabriel C. Drummond-Cole
1 hour ago
$begingroup$
Thank you very much, I will check the references.
$endgroup$
– Victor TC
6 hours ago
$begingroup$
Thank you very much, I will check the references.
$endgroup$
– Victor TC
6 hours ago
$begingroup$
@VictorTC As an additional remark, this is true more in general for coalgebras over cooperads, see Lemmas 4, 5, and Proposition 12 of the article Homotopy theory of unital algebras by B. Le Grignou.
$endgroup$
– Daniel Robert-Nicoud
4 hours ago
$begingroup$
@VictorTC As an additional remark, this is true more in general for coalgebras over cooperads, see Lemmas 4, 5, and Proposition 12 of the article Homotopy theory of unital algebras by B. Le Grignou.
$endgroup$
– Daniel Robert-Nicoud
4 hours ago
$begingroup$
N.B., we work explicitly with conilpotent coalgebras, and I do not know how to extend our method for proving presentability to the full category of coalgebras. So our paper may not be particularly useful for you.
$endgroup$
– Gabriel C. Drummond-Cole
1 hour ago
$begingroup$
N.B., we work explicitly with conilpotent coalgebras, and I do not know how to extend our method for proving presentability to the full category of coalgebras. So our paper may not be particularly useful for you.
$endgroup$
– Gabriel C. Drummond-Cole
1 hour ago
$begingroup$
@DanielRobert-Nicoud do you know how to resolve the seeming discrepancy between Lemma 5 of Le Grignou and Leonid's counterexample below?
$endgroup$
– Gabriel C. Drummond-Cole
1 hour ago
$begingroup$
@DanielRobert-Nicoud do you know how to resolve the seeming discrepancy between Lemma 5 of Le Grignou and Leonid's counterexample below?
$endgroup$
– Gabriel C. Drummond-Cole
1 hour ago
add a comment |
$begingroup$
For coassociative dg-coalgebras over any field $k$ the answer is positive, because:
Let $C$ be a $mathbb Z$-graded coalgebra and $Dsubset C$ a finite-dimensional ungraded subcoalgebra (of the underlying ungraded coalgebra) of $C$. Let $D^{gr}subset C$ denote the graded vector subspace spanned by all the grading components of the elements of $D$. Then $Dsubset D^{gr}$ and $D^{gr}$ is a finite-dimensional graded subcoalgebra of $C$.
Let $(C,d)$ be a dg-coalgebra and $Dsubset C$ be a finite-dimensional graded subcoalgebra of $C$. Set $D^{dg}=D+d(D)subset C$. Then $Dsubset D^{dg}$ and $D^{dg}$ is a finite-dimensional dg-subcoalgebra of $C$.
Using the observations 1. and 2. and the fact that any ungraded coassociative coalgebra is the union of its finite-dimensional subcoalgebras, one deduces the assertion that any $mathbb Z$-graded dg-coalgebra is the union of its finite-dimensional dg-subcoalgebras.
Possible generalizations: One can replace a field $k$ by a Noetherian commutative ring $k$ and speak about subcoalgebras that are finitely generated as $k$-modules (instead of "finite-dimensional"). All the assertions remain true.
One cannot drop the coassociativity condition. Indeed, even for ungraded coalgebras over a field of characteristic $0$, there is an example of a infinite-dimensional Lie coalgebra $L$ having no nonzero finite-dimensional subcoalgebras. The Lie coalgebra $L$ is simplest described in terms of its dual topological Lie algebra structure (on a pro-finite-dimensional topological vector space): $L^*=mathfrak g=k[[z]],d/dz$, the Lie algebra of vector fields on the formal disk.
$endgroup$
add a comment |
$begingroup$
For coassociative dg-coalgebras over any field $k$ the answer is positive, because:
Let $C$ be a $mathbb Z$-graded coalgebra and $Dsubset C$ a finite-dimensional ungraded subcoalgebra (of the underlying ungraded coalgebra) of $C$. Let $D^{gr}subset C$ denote the graded vector subspace spanned by all the grading components of the elements of $D$. Then $Dsubset D^{gr}$ and $D^{gr}$ is a finite-dimensional graded subcoalgebra of $C$.
Let $(C,d)$ be a dg-coalgebra and $Dsubset C$ be a finite-dimensional graded subcoalgebra of $C$. Set $D^{dg}=D+d(D)subset C$. Then $Dsubset D^{dg}$ and $D^{dg}$ is a finite-dimensional dg-subcoalgebra of $C$.
Using the observations 1. and 2. and the fact that any ungraded coassociative coalgebra is the union of its finite-dimensional subcoalgebras, one deduces the assertion that any $mathbb Z$-graded dg-coalgebra is the union of its finite-dimensional dg-subcoalgebras.
Possible generalizations: One can replace a field $k$ by a Noetherian commutative ring $k$ and speak about subcoalgebras that are finitely generated as $k$-modules (instead of "finite-dimensional"). All the assertions remain true.
One cannot drop the coassociativity condition. Indeed, even for ungraded coalgebras over a field of characteristic $0$, there is an example of a infinite-dimensional Lie coalgebra $L$ having no nonzero finite-dimensional subcoalgebras. The Lie coalgebra $L$ is simplest described in terms of its dual topological Lie algebra structure (on a pro-finite-dimensional topological vector space): $L^*=mathfrak g=k[[z]],d/dz$, the Lie algebra of vector fields on the formal disk.
$endgroup$
add a comment |
$begingroup$
For coassociative dg-coalgebras over any field $k$ the answer is positive, because:
Let $C$ be a $mathbb Z$-graded coalgebra and $Dsubset C$ a finite-dimensional ungraded subcoalgebra (of the underlying ungraded coalgebra) of $C$. Let $D^{gr}subset C$ denote the graded vector subspace spanned by all the grading components of the elements of $D$. Then $Dsubset D^{gr}$ and $D^{gr}$ is a finite-dimensional graded subcoalgebra of $C$.
Let $(C,d)$ be a dg-coalgebra and $Dsubset C$ be a finite-dimensional graded subcoalgebra of $C$. Set $D^{dg}=D+d(D)subset C$. Then $Dsubset D^{dg}$ and $D^{dg}$ is a finite-dimensional dg-subcoalgebra of $C$.
Using the observations 1. and 2. and the fact that any ungraded coassociative coalgebra is the union of its finite-dimensional subcoalgebras, one deduces the assertion that any $mathbb Z$-graded dg-coalgebra is the union of its finite-dimensional dg-subcoalgebras.
Possible generalizations: One can replace a field $k$ by a Noetherian commutative ring $k$ and speak about subcoalgebras that are finitely generated as $k$-modules (instead of "finite-dimensional"). All the assertions remain true.
One cannot drop the coassociativity condition. Indeed, even for ungraded coalgebras over a field of characteristic $0$, there is an example of a infinite-dimensional Lie coalgebra $L$ having no nonzero finite-dimensional subcoalgebras. The Lie coalgebra $L$ is simplest described in terms of its dual topological Lie algebra structure (on a pro-finite-dimensional topological vector space): $L^*=mathfrak g=k[[z]],d/dz$, the Lie algebra of vector fields on the formal disk.
$endgroup$
For coassociative dg-coalgebras over any field $k$ the answer is positive, because:
Let $C$ be a $mathbb Z$-graded coalgebra and $Dsubset C$ a finite-dimensional ungraded subcoalgebra (of the underlying ungraded coalgebra) of $C$. Let $D^{gr}subset C$ denote the graded vector subspace spanned by all the grading components of the elements of $D$. Then $Dsubset D^{gr}$ and $D^{gr}$ is a finite-dimensional graded subcoalgebra of $C$.
Let $(C,d)$ be a dg-coalgebra and $Dsubset C$ be a finite-dimensional graded subcoalgebra of $C$. Set $D^{dg}=D+d(D)subset C$. Then $Dsubset D^{dg}$ and $D^{dg}$ is a finite-dimensional dg-subcoalgebra of $C$.
Using the observations 1. and 2. and the fact that any ungraded coassociative coalgebra is the union of its finite-dimensional subcoalgebras, one deduces the assertion that any $mathbb Z$-graded dg-coalgebra is the union of its finite-dimensional dg-subcoalgebras.
Possible generalizations: One can replace a field $k$ by a Noetherian commutative ring $k$ and speak about subcoalgebras that are finitely generated as $k$-modules (instead of "finite-dimensional"). All the assertions remain true.
One cannot drop the coassociativity condition. Indeed, even for ungraded coalgebras over a field of characteristic $0$, there is an example of a infinite-dimensional Lie coalgebra $L$ having no nonzero finite-dimensional subcoalgebras. The Lie coalgebra $L$ is simplest described in terms of its dual topological Lie algebra structure (on a pro-finite-dimensional topological vector space): $L^*=mathfrak g=k[[z]],d/dz$, the Lie algebra of vector fields on the formal disk.
edited 3 hours ago
answered 3 hours ago
Leonid PositselskiLeonid Positselski
11.2k13977
11.2k13977
add a comment |
add a comment |
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$begingroup$
The result the OP is pointing to in Getzler-Goerss is Corollary 1.6.
$endgroup$
– David White
7 hours ago