Corpus ID: 118505122

Sweedler Theory for (co)algebras and the bar-cobar constructions

@article{Anel2013SweedlerTF,
  title={Sweedler Theory for (co)algebras and the bar-cobar constructions},
  author={M. Anel and A. Joyal},
  journal={arXiv: Category Theory},
  year={2013}
}
We prove that the category of dg-coalgebras is symmetric monoidal closed and that the category of dg-algebras is enriched, tensored, cotensored and strongly monoidal over that of coalgebras. We apply this formalism to reconstruct several known adjunctions, notably the bar-cobar adjunction. 
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References

SHOWING 1-10 OF 56 REFERENCES
A-infinity algebras, modules and functor categories
In this survey, we first present basic facts on A-infinity algebras and modules including their use in describing triangulated categories. Then we describe the Quillen model approach to A-infinityExpand
On Categories of Monoids, Comonoids, and Bimonoids
The categories of monoids, comonoids and bimonoids over a symmetric monoidal category C are investigated. It is shown that all of them are locally presentable provided C's underlying category is. AsExpand
Cyclic homology, comodules, and mixed complexes
relating cyclic and Hochschild homology, one can see that any formula for HC,(A @A’) will not only involve HC,(A) and HC,(A’), but also the “periodicity” operator S and the Hochschild groups.Expand
Lie theory for nilpotent L∞-algebras
The Deligne groupoid is a functor from nilpotent differential graded Lie algebras concentrated in positive degrees to groupoids; in the special case of Lie algebras over a field of characteristicExpand
Coalgebras over a commutative ring
By a coalgebra over the commutative ring K or a K-coalgebra, we understand a cocommutative, coassociative K-coalgebra with counit. More explicitly we mean a K-module C equipped with maps &C+C&C,Expand
Toposes, Triples and Theories
1. Categories.- 2. Toposes.- 3. Triples.- 4. Theories.- 5. Properties of Toposes.- 6. Permanence Properties of Toposes.- 7. Representation Theorems.- 8. Cocone Theories.- 9. More on Triples.- IndexExpand
Free Lie algebras
My principal references are [Serre:1965], [Reutenauer:1993], and [de Graaf:2000]. My interest in free Lie algebras has been motivated by the well known conjecture that Kac-Moody algebras can beExpand
Lectures on Exceptional Lie Groups
J. Frank Adams was internationally known and respected as one of the great algebraic topologists. Adams had long been fascinated with exceptional Lie groups, about which he published several papers,Expand
On the group H(Π
  • n), I, Annals of Mathematics, Vol 58, No. 1, pp. 55-106, July
  • 1953
Algèbres simpliciales S1-équivariantes et théorie de de Rham
  • Compos. Math
  • 2011
...
1
2
3
4
5
...