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The internal hom of topological modules

This file equips the category of topological modules with an internal hom TopModuleCat.linHom, the space of continuous linear maps carrying the topology induced from the compact-open topology, together with its functoriality (TopModuleCat.linHomMap) and a constructor (TopModuleCat.homOfBilinear) bundling a bilinear pairing with jointly continuous uncurried form into a morphism to the internal hom.

Material destined for Mathlib.Algebra.Category.ModuleCat.Topology.Basic.

Imports (2)

Imported by 4 files

abbrev TopModuleCat.linHom The internal hom of two topological modules: the space of continuous linear maps A →L[k] B carrying the topology induced from the compact-open topology on C(A, B) line 35 source
Source documentation
The internal hom of two topological modules: the space of continuous linear maps A →L[k] B carrying the topology induced from the compact-open topology on C(A, B).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, TopologicalSpace, TopModuleCat
Proof uses: TopModuleCat.of, ContinuousLinearMap, RingHom.id, TopModuleCat.toModuleCat
20 instances, coercions and auxiliary constants left out.
```
abbrev linHom (M1 M2 : TopModuleCat k) : TopModuleCat k := .of k (M1 →L[k] M2)
```
def TopModuleCat.linHomMap Pre- and post-composition induce a morphism between the internal homs of topological modules line 39 source
Source documentation
Pre- and post-composition induce a morphism between the internal homs of topological modules.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, TopologicalSpace, TopModuleCat, Quiver.Hom, TopModuleCat.linHom
Proof uses: TopModuleCat.ofHom, ContinuousLinearMap, RingHom.id, TopModuleCat.toModuleCat, ContinuousLinearMap.mk, LinearMap.mk, AddHom.mk, ContinuousLinearMap.comp, TopModuleCat.Hom.hom
34 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
def linHomMap {A A' B B' : TopModuleCat k} (a : A' ⟶ A) (b : B ⟶ B') :
    linHom A B ⟶ linHom A' B' :=
…
```
Mentions: TopModuleCat.linHom
lemma TopModuleCat.linHomMap_apply line 53 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, TopologicalSpace, TopModuleCat, Quiver.Hom, TopModuleCat.toModuleCat, TopModuleCat.linHom, Eq, RingHom.id, ContinuousLinearMap, CategoryTheory.ConcreteCategory.hom, TopModuleCat.linHomMap, TopModuleCat.Hom.hom
Proof uses: rfl
34 instances, coercions and auxiliary constants left out.
def TopModuleCat.homOfBilinear Bundle a bilinear pairing with jointly continuous uncurried form into a morphism to the internal hom line 60 source
Source documentation
Bundle a bilinear pairing with jointly continuous uncurried form into a morphism to the internal hom. Stating this for abstract topological modules keeps all instance searches on abstract carriers.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, TopologicalSpace, TopModuleCat, TopModuleCat.toModuleCat, ContinuousLinearMap, RingHom.id, Eq, HAdd.hAdd, HSMul.hSMul, Continuous, Prod, Prod.fst, Prod.snd, Quiver.Hom, TopModuleCat.linHom
Proof uses: TopModuleCat.ofHom, ContinuousLinearMap.mk, LinearMap.mk, AddHom.mk
75 instances, coercions and auxiliary constants left out.
```
def homOfBilinear {A B C : TopModuleCat k} (F : ↥A → (↥B →L[k] ↥C))
    (hadd : ∀ a a' b, F (a + a') b = F a b + F a' b)
    (hsmul : ∀ (c : k) a b, F (c • a) b = c • F a b)
    (hF : Continuous fun p : ↥A × ↥B ↦ F p.1 p.2) :
    A ⟶ linHom B C :=
…
```
Mentions: TopModuleCat.linHom
lemma TopModuleCat.homOfBilinear_apply line 73 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, TopologicalSpace, TopModuleCat, TopModuleCat.toModuleCat, ContinuousLinearMap, RingHom.id, Eq, HAdd.hAdd, HSMul.hSMul, Continuous, Prod, Prod.fst, Prod.snd, TopModuleCat.linHom, CategoryTheory.ConcreteCategory.hom, TopModuleCat.homOfBilinear
Proof uses: rfl
64 instances, coercions and auxiliary constants left out.