Corpus ID: 118505122

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

  title={Sweedler Theory for (co)algebras and the bar-cobar constructions},
  author={M. Anel and A. Joyal},
  journal={arXiv: Category Theory},
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. 
On Sweedler's cofree cocommutative coalgebra
We give a direct proof of a result of Sweedler describing the cofree cocommutative coalgebra over a vector space, and use our approach to give an explicit construction of liftings of maps into thisExpand
Mapping Coalgebras I: Comonads
In this article we describe properties of the 2-functor from the 2-category of comonads to the 2-category of functors that sends a comonad to its forgetful functor. This allows us to describeExpand
Koszul duality and ∞-categories by Julian Holstein
In this paper we establish Koszul duality for dg categories and a class of curved coalgebras, generalizing the corresponding result for dg algebras and conilpotent curved coalgebras. We show that theExpand
Dg analogues of the Zuckerman functors and the dual Zuckerman functors I
In the first part of this series of papers we constructed dg analogues of the Zuckerman functors over commutative rings and the dual Zuckerman functors over the field of complex numbers. In thisExpand
Noncommutative deformation theory, the derived quotient, and DG singularity categories
We show that Braun-Chuang-Lazarev's derived quotient prorepresents a naturally defined noncommutative derived deformation functor. Given a noncommutative partial resolution of a Gorenstein algebra,Expand
D-structures and derived Koszul duality for unital operad algebras
Generalizing a concept of Lipshitz, Ozsv\'ath and Thurs-ton from Bordered Floer homology, we define $D$-structures on algebras of unital operads, which can also be interpreted as a generalization ofExpand
Monads on Higher Monoidal Categories
The structure and conditions that guarantee that the higher monoidal structure is inherited by the category of algebras over the monad are identified. Expand
Measuring Comodules and Enrichment
We study the existence of universal measuring comodules Q(M,N) for a pair of modules M,N in a braided monoidal closed category, and the associated enrichment of the global category of modules overExpand
Logic and linear algebra: an introduction
We give an introduction to logic tailored for algebraists, explaining how proofs in linear logic can be viewed as algorithms for constructing morphisms in symmetric closed monoidal categories withExpand
Integral models of Harish-Chandra modules of the finite covering groups of PU(1,1)
We compute the Harish-Chandra modules produced by the functor $I$ in the cases of integral models of principal and discrete series representations of finite covering groups of PU(1,1). In particular,Expand


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