feat(Geometry/Manifold): The Tangent Functor on MfldCat

[Deicyde]

A simple motivating example for PR #38223. We define the tangentFunctor : MfldCat 𝕜 (n + 1) ⥤ MfldCat 𝕜 n which takes any manifold to its tangent bundle and any smooth map to its pushforward.

This is a standard construction in differential geometry. See: J. Lee, Smooth Manifolds, pg. 75

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• depends on: feat(Geometry/Manifold): add MfldCat, the category of C^n manifolds #38223

[github-actions]

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[github-actions]

PR summary caf1dee8e5

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Geometry.Manifold.Category.MfldCat.Basic (new file)	2216
Mathlib.Geometry.Manifold.Category.MfldCat.TangentFunctor (new file)	2217

---

Declarations diff

+ Hom
+ Hom.Simps.hom
+ Hom.equivContMDiffMap
+ Hom.hom
+ MfldCat
+ coe_of
+ coe_of_of
+ comp_app
+ const
+ const_apply
+ diffeomorphOfIso
+ ext
+ hom_comp
+ hom_ext
+ hom_id
+ hom_inv_id_apply
+ hom_ofHom
+ id_app
+ id_apply
+ inhabited
+ instance : Category (MfldCat 𝕜 n)
+ instance : CoeSort (MfldCat 𝕜 n) (Type u₂) := ⟨MfldCat.carrier⟩
+ instance : ConcreteCategory (MfldCat 𝕜 n)
+ instance : HasForget₂ (MfldCat 𝕜 n) TopCat.{u₂}
+ inv_hom_id_apply
+ isoOfDiffeomorph
+ ofHom
+ ofHom_apply
+ ofHom_comp
+ ofHom_hom
+ ofHom_id
+ ofNormedSpace
+ of_carrier
+ of_diffeomorphOfIso
+ of_isoOfDiffeomorph
+ tangentFunctor
++ coe_comp
++ coe_id

You can run this locally as follows

## summary with just the declaration names:
./scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh contains some details about this script.

---

No changes to technical debt.

You can run this locally as

./scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[mathlib-dependent-issues]

This PR/issue depends on:

• feat(Geometry/Manifold): add MfldCat, the category of C^n manifolds #38223
  By Dependent Issues (🤖). Happy coding!