A-infinity functors #
An A∞ functor ℱ : ℬ ⟶ 𝒞 between nonunital A∞ categories on graded linear quivers C and
D consists of a map of objects F : C → D and, for every composable string X₀, …, Xₙ of C
with n ≥ 1, a component
fₙ : Hom(Xₙ₋₁, Xₙ) ⊗ ⋯ ⊗ Hom(X₀, X₁) ⟶ Hom(F X₀, F Xₙ)
of degree 1 - n, subject to the A∞ functor equations on every composable string.
The components are stored as an A∞ morphism from the total algebra of morphisms of ℬ to that of
the A∞ category 𝒞.induced F induced along the map of objects
(TauCeti.AInfinityCategory.induced), whose components are path-compatible along
TauCeti.GradedLinearQuiver.InducedQuiver.mk: they send a composable string of C to a morphism
between the images of its endpoints, and every other string to zero. The target is the induced
category rather than 𝒞 itself because a string which is not composable in C can become
composable after applying F, and the operations of 𝒞 need not vanish on it, whereas the
operations of the induced category do. On a string which is not composable every term of the
A∞ morphism equation vanishes, so the A∞ morphism equation of the total algebras amounts to the
A∞ functor equations on composable strings.
The arity-one component gives a degree-preserving linear map ℱ.map X Y on each Hom module, and
it is a chain map for the differentials m₁. In general it preserves the composition m₂ only up
to the homotopy given by the arity-two component. A strict functor is an A∞ functor whose
components above arity one vanish (TauCeti.AInfinityStrictFunctor.toAInfinityFunctor); in
particular every A∞ category has an identity A∞ functor.
When the map of objects F is injective, the map of total modules of the induced category into
that of 𝒞 is injective and intertwines all operations. So an A∞ morphism of total algebras
into that of 𝒞 itself whose Taylor map is path-compatible along F factors uniquely through the
total algebra of 𝒞.induced F (TauCeti.AInfinityHom.liftInduced), and is an A∞ functor
(TauCeti.AInfinityFunctor.ofAInfinityHom). This is how A∞ morphisms built on total algebras,
such as those of homological transfer, become A∞ functors.
Main definitions #
TauCeti.AInfinityFunctor: anA∞functor between nonunitalA∞categories.TauCeti.AInfinityFunctor.map: the map of anA∞functor on a Hom module, its arity-one component.TauCeti.AInfinityStrictFunctor.totalHom: the strict morphism of total algebras of morphisms given by a strict functor.TauCeti.AInfinityStrictFunctor.toAInfinityFunctor: a strict functor as anA∞functor.TauCeti.AInfinityFunctor.id: the identityA∞functor.TauCeti.AInfinityFunctor.ofAInfinityHom: theA∞functor given by anA∞morphism of total algebras which is path-compatible along an injective map of objects.
Main results #
TauCeti.AInfinityFunctor.isPathCompatible_component: every component of anA∞functor is path-compatible along its map of objects.TauCeti.AInfinityFunctor.homInclusion_map: the map on a Hom module is the linear part of theA∞morphism of total algebras.TauCeti.AInfinityFunctor.homProjection_linearPart: each component of the linear part of theA∞morphism of total algebras is the map on the corresponding Hom module.TauCeti.AInfinityFunctor.map_mem: the map on a Hom module preserves degrees.TauCeti.AInfinityFunctor.map_homDifferential: the map on a Hom module is a chain map.TauCeti.AInfinityStrictFunctor.toAInfinityFunctor_map: the map on Hom modules of a strict functor, as anA∞functor, is its map as a strict functor.TauCeti.AInfinityFunctor.homInclusion_map_ofAInfinityHom: the functor of a path-compatibleA∞morphism acts on Hom modules by the linear part of the morphism.
References #
- B. Keller, Introduction to A-infinity algebras and modules, Sections 3.4 and 7.
- V. Lyubashenko, Category of A-infinity-categories, Homology, Homotopy and Applications 5
(2003), for
A∞functors as morphisms of bar constructions over a map of objects.
An A∞ functor between nonunital A∞ categories ℬ on C and 𝒞 on D: a map of
objects obj : C → D together with an A∞ morphism from the total algebra of morphisms of ℬ to
that of the induced category 𝒞.induced obj, whose components send a composable string of C to
a morphism between the images of its endpoints and every other string to zero. Its component of
arity n on a composable string X₀, …, Xₙ is the functor component
fₙ : Hom(Xₙ₋₁, Xₙ) ⊗ ⋯ ⊗ Hom(X₀, X₁) ⟶ Hom(obj X₀, obj Xₙ) of degree 1 - n.
- obj : C → D
The map on objects.
- toAInfinityHom : AInfinityHom ℬ.toAInfinityAlgebra (𝒞.induced self.obj).toAInfinityAlgebra
The components, as an
A∞morphism of total algebras of morphisms into the induced category. - isPathCompatible_component_of_pos (n : ℕ) (hn : 0 < n) : GradedLinearQuiver.IsPathCompatible GradedLinearQuiver.InducedQuiver.mk (self.toAInfinityHom.component n)
Every component of positive arity sends composable strings to morphisms between the images of their endpoints, and other strings to zero. The component of arity zero vanishes, so it is path-compatible automatically (
TauCeti.AInfinityFunctor.isPathCompatible_component).
Instances For
Path compatibility of the components of an A∞ morphism is path compatibility of its Taylor
map on words.
A∞ functors are determined by their object maps and their A∞ morphisms of total algebras.
The heterogeneous equality accounts for the target, which depends on the object map.
Every component of an A∞ functor sends composable strings to morphisms between the images of
their endpoints, and other strings to zero.
The Taylor map of an A∞ functor sends composable words from X to Y to morphisms from
⟨X⟩ to ⟨Y⟩ in the induced category, and non-composable words to zero.
The map on Hom modules #
The map of an A∞ functor on the Hom module X ⟶ Y: the component between the images of
X and Y of its linear part, that is, its arity-one component f₁.
Instances For
The map on a Hom module is the component of the linear part between the images of the objects.
The map on a Hom module is the linear part. Including the image of a morphism into the total module of the induced category gives the linear part of the included morphism.
The component between ℱ X and ℱ Y of the linear part of the A∞ morphism of total algebras
underlying ℱ is the map of ℱ on the component between X and Y.
The map of an A∞ functor on a Hom module preserves degrees.
The map of an A∞ functor on a Hom module commutes with taking homogeneous components.
The map of an A∞ functor on a Hom module is a chain map for the differentials m₁: this
is the arity-one A∞ functor equation.
The map of an A∞ functor on a Hom module is a chain map, as an equation of linear maps.
The map of an A∞ functor on a Hom module sends closed morphisms to closed morphisms.
Strict functors as A∞ functors #
The strict morphism of total algebras of morphisms given by a strict functor F, from the
total algebra of ℬ to that of the category induced from 𝒞 along the object map of F: on the
hom module of X and Y it is the map of F on that hom module.
Instances For
The strict morphism of total algebras of a strict functor sends a morphism X ⟶ Y to its
image under the functor, between ⟨X⟩ and ⟨Y⟩ in the induced quiver.
A strict functor as an A∞ functor: its A∞ morphism of total algebras is the strict
morphism TauCeti.AInfinityStrictFunctor.totalHom, so its components above arity one vanish.
Instances For
The map on Hom modules of a strict functor, regarded as an A∞ functor, is its map as a
strict functor.
The identity A∞ functor of an A∞ category: the identity strict functor, whose
components above arity one vanish.
Instances For
The A∞ morphism of total algebras of the identity A∞ functor is the strict morphism of the
identity strict functor, so its components above arity one vanish.
The identity A∞ functor fixes every morphism. Unlike TauCeti.AInfinityFunctor.id_map,
this can be used to rewrite where the codomain of LinearMap.id is only definitionally its
domain.
A∞ functors from A∞ morphisms of total algebras #
An A∞ morphism of total algebras which is path-compatible along an injective map of objects
is an A∞ functor with object map F, whose A∞ morphism is the factorization
TauCeti.AInfinityHom.liftInduced of f through the total algebra of 𝒞.induced F.
Instances For
The object map of the A∞ functor of a path-compatible A∞ morphism is the given map.
The A∞ morphism of the A∞ functor of a path-compatible A∞ morphism f is the
factorization of f through the induced category.
The map of the A∞ functor of a path-compatible A∞ morphism f on the Hom module X ⟶ Y is
the linear part of f, which sends morphisms X ⟶ Y to morphisms F X ⟶ F Y.