Homologies of Basic Algebraic Structures via Braidings and Quantum Shuffles 

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Categorifying braided differentials Basic categorical notions

We start with recalling some classical definitions from category theory.
Definition 5.1.1. ú A strict monoidal (or tensor) category is a category C endowed with 3 a bifunctor ⊗ : C × C → C satisfying the associativity condition; 3 an object I which is a left and right identity for ⊗. ú A strict monoidal category C is called pre-braided if it is endowed with a pre-braiding (or a commutativity constraint), i.e. a natural family of morphisms c = {cV,W : V ⊗W → W ⊗ V } ∀V,W ∈ Ob(C), satisfying cV,W⊗U = (IdW ⊗cV,U ) ◦ (cV,W ⊗ IdU), (5.1) cV ⊗W,U = (cV,U ⊗ IdW) ◦ (IdV ⊗cW,U ) (5.2) for any triple of objects V,W,U. “Natural” means here cV ′,W′ ◦ (f ⊗ g) = (g ⊗ f) ◦ cV,W (5.3) for all V,W, V ′,W′ ∈ Ob(C), f ∈ HomC(V, V ′), g ∈ HomC(W,W′). One talks about braidings and braided categories if the cV,W ’s are moreover isomorphisms. ú A braided category C is called symmetric if its braiding is symmetric: cV,W ◦ cW,V = IdW⊗V , ∀V,W ∈ Ob(C).

Basic examples revisited

According to lemma 4.2.1, every shelf S ∈ Ob(Set) is endowed with a pre-braiding σ⊳ : (a, b) 7→ (b, a ⊳ b). Since the one-element set I is a final object in Set, the unique morphism S → I is necessarily a braided character. The diagonal map D : a 7→ (a, a) gives a semi-pre-braided coalgebra structure on S, σ⊳-cocommutative if S is a spindle. Given a commutative unital ring R, the monoidal functor LinR provides then the linearization RS of our shelf in the additive category ModR. The induced semi-pre-braided coalgebra structure and braided character on RS give, according to theorem 6, a pre-bisimplicial and a weakly simplicial (in the spindle case) structures, and thus a bidegree −1 bidifferential. Associative and Leibniz algebras Next we consider more complicated “structural” braidings. We categorify theorem 5 and study several related questions.
1. Take a right-unital associative algebra (V, μ, ν) in a monoidal category (C,⊗, I).
(a) V can be endowed with a pre-braiding σAss := ν ⊗ μ : V ⊗ V = I ⊗ V ⊗ V → V ⊗ V.
(b) Comultiplication Ass := ν ⊗ IdV : V = I ⊗ V → V ⊗ V completes this pre-braiding into a σAss-cocommutative pre-braided coalgebra structure if ν is moreover a two-sided unit.
(c) Any algebra character ǫ ∈ HomUAlg(C)(V, I) is a braided character for (V, σAss).
(d) The pre-braiding σAss is demi-natural with respect to the unit ν. Moreover, for any algebra character ǫ, the pair (ν, ǫ) is normalized.
2. Take a unital Leibniz algebra (V, [, ], ν) in a symmetric preadditive category (C,⊗, I, c). (a) V can be endowed with an invertible braiding σLei := cV,V + ν ⊗ [, ].

The super trick

The first bonus one generally gains when passing to abstract symmetric categories is the possibility to derive graded and super versions of algebraic results for free, thanks to the Koszul flip τKoszul from (2.1). One clearly sees where to put signs, which is otherwise quite difficult to guess. Here is a typical example. Take a graded unital Leibniz algebra (V, [, ], ν), i.e. an object of ULei(ModGradR). Recall that the category ModGradR comes with the symmetric braiding τKoszul. Leibniz condition in this setting is [v, [w, u]] = [[v,w], u] − (−1)deg u degw[[v, u],w] for any homogeneous elements v,w, u ∈ V. On the figure 4.7 illustrating (Lei), the crossing on the right corresponds to the “internal” braiding cV,V = τKoszul. Theorem 5cat gives a braiding for V : σV : v ⊗ w 7−→ (−1)deg v degww ⊗ v + 1 ⊗ [v,w].

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Co-world, or the world upside down

One more nice feature of the categorical approach is an automatic treatment of du- alities. The most common notion of duality, the “upside-down” one, is described here, with the cobar complex for coalgebras (first defined by Cartier in [12]; cf. also [19] and [77]) providing an example. In the monoidal context, one has two more dualities, the “right-left” and the combined ones, treated in the next section.
Definition 5.4.1. Given a category C, its dual (or opposite) category Cop is constructed by keeping the objects of C and reversing all the arrows. In other words, the domain and codomain of any morphism change places. One writes fop ∈ HomCop(W, V ) for the morphism in Cop corresponding to an f ∈ HomC(V,W). We sometimes call Cop a co-category in order to avoid confusion with other notions of duality. Observe that this construction is involutive: (Cop)op = C. Example 5.4.2. A well-known example comes from the full subcategory vectk of Vectk consisting of finite dimensional vector spaces. The usual duality functor sending V to V ∗ := Homk(V, k) and f to f∗ gives an equivalence of symmetric preadditive categories vectk and (vectk)op. The duality principle (cf. [49], section II.2) tells that a “categorical” theorem for C implies a dual theorem for Cop by reversing all arrows and the order of arrows in every composition. Our aim here is to apply this principle to theorems 6 and 5cat.

Table of contents :

Introduction
Notations and conventions
I Homologies of Basic Algebraic Structures via Braidings and Quantum Shuffles 
2 Braided world: a short reminder
3 (Co)homologies of braided vector spaces
3.1 Pre-braiding + character 7−→ homology
3.2 Comultiplication 7−→ degeneracies
3.3 Loday’s hyper-boundaries
4 Basic examples: familiar (co)homologies recovered
4.1 Koszul complex
4.2 Rack complex
4.3 Bar complex
4.4 Leibniz complex
5 An upper world: categories
5.1 Categorifying braided differentials
5.2 Basic examples revisited
5.3 The super trick
5.4 Co-world, or the world upside down
5.5 Right-left duality
6 Braided modules and homologies with coefficients
6.1 Modules and bimodules over braided objects
6.2 Structure mixing techniques
II Hopf and Yetter-Drinfel′d Structures via Braided Systems 
7 Braided systems: general theory and examples
7.1 General recipe
7.2 A protoexample: pre-braided systems of algebras
7.3 A toy example: algebra bimodules
7.4 The first real example: two-sided crossed products
7.5 Yetter-Drinfel′d systems
7.6 Bialgebras
7.7 Yetter-Drinfel′d modules
7.8 Hopf (bi)modules
III A Categorification of Virtuality and Self-distributivity 
8 A survey of braid and virtual braid theories
8.1 Different avatars of braids
8.2 Virtual braids and virtual racks
9 Free virtual self-distributive structures
9.1 Adding virtual copies of elements
9.2 Free virtual shelves and P.Dehornoy’s methods
9.3 Free virtual quandles and a conjecture of V.O.Manturov
10 Categorical aspects of virtuality
10.1 A categorical counterpart of virtual braids
10.2 Flexibility of the categorical construction
11 Categorical aspects of self-distributivity
11.1 A categorified version of self-distributivity
11.2 Associative, Leibniz and Hopf algebras are shelves
11.3 Homologies of categorical shelves and spindles
Bibliography 

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