Categories equivalent to the category of rational H-spaces

Arkowitz, Martin

doi:10.1007/bf01170937

Cited by 4 publications

(4 citation statements)

References 18 publications

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“…Thus N is zero or singly generated by part 1 and Nakayama's lemma ( [7, pp.8-9]). Hence N is in S f t by Lemma 2.5 (1).…”

Section: ))mentioning

confidence: 79%

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Cogroups in the category of connected graded algebras whose inverse and antipode coincide

Kihara

2013

Preprint

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Let A be a cogroup in the category of connected graded algebras over a commutative ring R. Let ν denote the inverse of A and χ the antipode of the underlying Hopf algebra of A. We clarify the differences and similarities of ν and χ, and show that ν coincides with χ if and only if A is commutative as a graded algebra. Let A co CG be the category of cogroups satisfying these equivalent conditions. If R is a field, the category A co CG is completely determined. We also establish an equivalence of the full subcategory of A co CG consisting of objects of finite type with a full subcategory of the category of positively graded R-modules without any assumption on R. The results in the case of R = Q are applied to the theory of co-H-groups.

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“…Thus N is zero or singly generated by part 1 and Nakayama's lemma ( [7, pp.8-9]). Hence N is in S f t by Lemma 2.5 (1).…”

Section: ))mentioning

confidence: 79%

“…(⇒) We can easily see that N is zero or singly geenerated. Thus the implication follows from Lemma 2.5 (1).…”

Section: ))mentioning

confidence: 79%

Cogroups in the category of connected graded algebras whose inverse and antipode coincide

Kihara

2013

Preprint

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“…Let F denote the full subcategory of M consisting of objects isomorphic to one of 0, R [1], R [2], · · · (resp. 0, R [2], R [4] · · ·) if ch R = 2 (resp.…”

Section: Main Algebraic Resultsmentioning

confidence: 99%

“…The usual argument implies that g coincides with h, and hence that g is an inverse of f . 2 (1) and the construction in [7]). Then formula (2.6 ) in [7] implies the commutativity of the diagram…”

Section: Inverse and Antipodementioning

confidence: 97%

Commutativity and cocommutativity of cogroups in the category of connected graded algebras

Kihara

2015

Topology and its Applications

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Let A CG be the category of cogroups in the category A of connected graded algebras over a fixed commutative ring R. We study the full subcategory A co CG consisting of objects whose underlying algebras are graded commutative, together with the full subcategory A coCG consisting of cocommutative objects and the full subcategory co A CG consisting of objects whose underlying coalgebras are graded cocommutative. We establish categorical equivalences of these full subcategories with categories of simpler algebraic objects, and obtain the inclusion relation of the full subcategories. Since H * (ΩX; R) is a cogroup in A for a 1-connected co-H-group (under the assumption that R is a field), the algebraic results are applied to the theory of co-H-groups. We study when H * (ΩX; R) is in A coCG or A co CG , and generalize a theorem of Kachi.

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Minimal models of loop spaces and suspensions

Arkowitz

Lupton

1993

Manuscripta Math

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Categories equivalent to the category of rational H-spaces

Cited by 4 publications

References 18 publications

Cogroups in the category of connected graded algebras whose inverse and antipode coincide

Cogroups in the category of connected graded algebras whose inverse and antipode coincide

Commutativity and cocommutativity of cogroups in the category of connected graded algebras

Minimal models of loop spaces and suspensions

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