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Concept# Adjoint

Résumé

In mathematics, the term adjoint applies in several situations. Several of these share a similar formalism: if A is adjoint to B, then there is typically some formula of the type
:(Ax, y) = (x, By).
Specifically, adjoint or adjunction may mean:

- Adjoint of a linear map, also called its transpose
- Hermitian adjoint (adjoint of a linear operator) in functional analysis
- Adjoint endomorphism of a Lie algebra
- Adjoint representation of a Lie group
- Adjoint functors in category theory
- Adjunction (field theory)
- Adjunction formula (algebraic geometry)
- Adjunction space in topology
- Conjugate transpose of a matrix in linear algebra
- Adjugate matrix, related to its inverse
- Adjoint equation
- The upper and lower adjoints of a Galois connection in order theory
- The adjoint of a differential operator with general polynomial coefficients
- Kleisli adjunction
- Monoidal adjunction
- Quillen adjunction
- Axiom of adjunction in set theory
- Adjunction (rule of inference)

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Kathryn Hess Bellwald, Magdalena Kedziorek

A common technique for producing a new model category structure is to lift the fibrations and weak equivalences of an existing model structure along a right adjoint. Formally dual but technically much harder is to lift the cofibrations and weak equivalences along a left adjoint. For either technique to define a valid model category, there is a well-known necessary 'acyclicity' condition. We show that for a broad class of accessible model structures - a generalization introduced here of the well-known combinatorial model structures - this necessary condition is also sufficient in both the right-induced and left-induced contexts, and the resulting model category is again accessible. We develop new and old techniques for proving the acyclity condition and apply these observations to construct several new model structures, in particular on categories of differential graded bialgebras, of differential graded comodule algebras, and of comodules over corings in both the differential graded and the spectral setting. We observe moreover that (generalized) Reedy model category structures can also be understood as model categories of 'bialgebras' in the sense considered here.

Using Kadeishvili's formulas with appropriate signs, we show that the classical cobar construction from coalgebras to algebras \Omega: CoAlg -> Alg can be enhanced to a functor from Hopf algebras to E_2 algebras (for a certain choice of E_2 operad) \Omega : HopfAlg -> E_2 Alg, which, unlike its classical counterpart, is not strictly adjoint, but homotopically equivalent to the left adjoint of the enhanced bar construction B: E_2 Alg -> HopfAlg studied by Gerstenhaber--Voronov and Fresse.

2013The starting point for this project is the article of Kathryn Hess [11]. In this article, a homotopic version of monadic descent is developed. In the classical setting, one constructs a category D(𝕋) of coalgebras in the Eilenberg-Moore category of algebras D𝕋 for a given monad 𝕋 on a category D. There is a canonical functor Can𝕃𝕋 from D to D(𝕋), and if Can𝕃𝕋 is fully faithful, then 𝕋 satisfies descent, while if Can𝕃𝕋 is an equivalence of categories, then 𝕋 satisfies effective descent [19]. In [11], these two conditions are replaced by a weaker one, that these hold only up to homotopy. This is achieved by working with model categories that are enriched over simplicial sets. Homotopic descent is then defined by demanding that each component in (Can𝕃𝕋)A,B : MapD(A,B) → MapD(𝕋) (Can𝕃𝕋(A), Can𝕃𝕋 (B)) be a weak equivalence of simplicial sets. A similar but stronger condition involving the path components in D(𝕋) expresses effective homotopic descent. The first goal of this project is to develop a framework of homotopic descent for model categories that are enriched over model categories other than simplicial sets. The most important examples we have in mind are chain complexes and spectra. In order to achieve this goal, we tried to determine the most general conditions that are sufficient and necessary to make the theory work. To ease the formulation, let us say that we are working with a model category D that is enriched over a monoidal model category V. The crucial constructions we need are realization, respectively totalization, of (co)simplicial objects in D. These functors have to be Quillen functors to ensure that they have the correct homotopical behaviour. This implies that there must exist a Quillen adjunction between V and simplicial sets. Furthermore, we need to be able to transfer the enrichment and (co)tensoring over V to an enrichment and (co)tensoring over simplicial sets. This forces the Quillen adjunction to be monoidal. Another main point that has to be adressed is the question, of whether the enrichment of D carries over to an enrichment of D𝕋 and D(𝕋) and how this enrichment behaves. It turns out that this works well under mild assumptions on V. This leads then to the definition of homotopic descent by requiring that each component in (Can𝕃𝕋)A,B : MapD(A,B) → MapD(𝕋) (Can𝕃𝕋(A), Can𝕃𝕋 (B)) be a weak equivalence in M and similarly for effective homotopic descent. Using this definition, the theorems in [11] carry over to this more general context. Although the conditions on V are rather constraining regarding the relation with simplicial sets, the cases of chain complexes and spectra are included. For the time being we do not see how the constraints on V could be weakened. The second goal of this project is to apply the theory of homotopic descent to concrete examples. A good source of examples is homotopic Grothendieck descent in the category of spectra, i.e., S-modules. Classical Grothendieck descent deals with the adjunction induced by a morphism φ : B → A of monoids in a monoidal category (M,Λ, S), – BΛ A : ModB ⇄ ModA : φ*, which in turn induces a monad 𝕋φ := φ*(– ΛB A) on ModB. We consider in particular the case when the morphism in question is the unit of an S-algebra E, η : S → E There is a close relationship between comodules over a Hopf algebroid and objects in D(𝕋η). Associated to η we have the canonical co-ring Wη := E ΛS E and an isomorphism between D(𝕋η) and the category of comodules over Wη in the category of S-modules. This relationship is explored in an analysis of the stable Adams spectral sequence, the construction of which heavily relies on the monadic properties of the functor η*(E ΛS –) and can therefore be expressed in terms of D(𝕋η). We construct a spectral sequence that generalizes the stable Adams spectral sequence to any stable pointed model category such as unbounded chain complexes. One can give a description of the E2-term as an Ext in D(𝕋η), E2s,t = ExtD(𝕋η) (Can(A), Can(B)). If the spectral sequences converges, it abuts to π⁎MapD(A,B η^), where Bη^ is the derived 𝕋η-completion of B, which agrees with the usual derived completion in well-known special cases. Furthermore, Bη^ := Tot B^•, and B^• is kind of a fibrant cosimplicial resolution of B. Furthermore, the language of relative homological algebra for modules and comodules generalizes to definitions for algebras in D𝕋η and coalgebras in D(𝕋η). This shows that the construction of the Adams spectral sequence works in a more general setting, where one applies a functor to an abelian category, for example π⁎, only at the end, to be able to do computations in homological algebra. This general Adams spectral sequence is closely related to the descent spectral sequence of [11], and we have clarified this relationship.

Cours associés (2)

MATH-436: Homotopical algebra

This course will provide an introduction to model category theory, which is an abstract framework for generalizing homotopy theory beyond topological spaces and continuous maps. We will study numerous examples of model categories and their applications in algebra and topology.

PHYS-216: Mathematical methods for physicists

This course complements the Analysis and Linear Algebra courses by providing further mathematical background and practice required for 3rd year physics courses, in particular electrodynamics and quantum mechanics.

Séances de cours associées (10)