feat(AlgebraicTopology/SimplicialSet): define isomorphisms in simplicial sets, and the coherent isomorphism simplicial set

[arnoudvanderleer]

show that any edge in a simplicial set, that is the image of the forward edge of the coherent isomorphism under a simplicial set morphism, is an isomorphism.

---

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PR summary eb5df47f87

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.AlgebraicTopology.SimplicialSet.CoherentIso (new file)	864

---

Declarations diff

+ IsIso
+ WalkingIso
+ WalkingIso.one
+ WalkingIso.zero
+ coev
+ coherentIso
+ equiv
+ equivFun
+ fromIso
+ hom
+ homInvId
+ idCompId
+ id_comp_id_aux
+ instance (n : ℕ) : DecidableEq (coherentIso _⦋n⦌)
+ instance : Category (WalkingIso)
+ inv
+ invHomId
+ isIsoHom
+ isIsoId
+ isIsoInv
+ isIsoMapHom
+ isIsoOfEqMapHom
+ map
+ toIso
+ x₀
+ x₁
+ yonedaEquiv_naturality
+ yonedaEquiv_symm_naturality
+++ ofEq

You can run this locally as follows

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

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

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

---

No changes to technical debt.

You can run this locally as

./scripts/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).

[arnoudvanderleer]

infinity-cosmos

[robin-carlier]

Not a full review, but here are already some comments/suggestions

[robin-carlier]

Here and below, you should use a universe level w, and use WalkingIso.{w} instead of WalkingIso:
using WalkingIso with an unspecified universe works but will create a universe-level metavariable, and we have observed in the library that these can be a cause of bad performances, so the good practice for now is to add an extra universe like this.

[robin-carlier]

You should also consider defining

def iso : zero.{u} ≅ one where
  hom := ()
  inv := ()

and then

@[simps!]
def toIso (F : WalkingIso.{w} ⥤ C) : F.obj zero ≅ F.obj one := F.mapIso iso

(here, I’m using the extra universe w to not make unintentional collisions with the levels in C. alternatively, you could use

variable (C : Type*) [Category* C]

instead of

variable (C : Type u) [Category.{v} C]

to leave the levels in C "irrelevant".)

[robin-carlier]

Suggested change

	def WalkingIso : Type u := ULift (Fin 2)
	@[match_pattern]
	def WalkingIso.zero : WalkingIso := ULift.up (0 : Fin 2)
	@[match_pattern]
	def WalkingIso.one : WalkingIso := ULift.up (1 : Fin 2)
	inductive WalkingIso : Type u where
	| zero : WalkingIso
	| one : WalkingIso
	attribute [aesop safe cases (rule_sets := [CategoryTheory])] WalkingIso
	attribute [grind cases] WalkingIso

Semireducible type synonyms (that’s the jargon for def … : Type _ := …) are usually brittle and prone to defeq abuse, and I think the current trend in mathlib is to move away from them in favor of custom inductives or structures. Here, since you are tagging zero and one to match on it, it’s probably better to directly make it
an inductive. The type annotation Type u makes sure it lives in the right universes.

The two line below helps automation understanding how to work with the object.

[robin-carlier]

Suggested change

	universe u v
	universe w v u

(see in other comments as to why I am suggesting adding a w).

[robin-carlier]

Additionally, you could consider adding extra instances like

instance {x y : WalkingIso.{u}} : Unique (x ⟶ y) := inferInstanceAs (Unique Unit)

[robin-carlier]

I think the name should be changed: usually, Is is a prefix for Prop-valued definition.
As quasicategory theory develops in mathlib, we will want it to be as backwards compatible as possible and to have the same names as the corresponding objects in ordinary category theory when possible: The current CategoryTheory.IsIso is the mere existence of such a structure.

Since it’s the structure of an isomorphism in the homotopy category, maybe hIso would be a suitable name for the structure?

[robin-carlier]

Suggested change

	{X : SSet} {m n : SimplexCategory} (f : m ⟶ n) (g : stdSimplex.obj n ⟶ X)
	: X.map f.op (yonedaEquiv g) = yonedaEquiv (stdSimplex.map f ≫ g)
	:= uliftYonedaEquiv_naturality _ _
	{X : SSet} {m n : SimplexCategory} (f : m ⟶ n) (g : stdSimplex.obj n ⟶ X) :
	X.map f.op (yonedaEquiv g) = yonedaEquiv (stdSimplex.map f ≫ g) :=
	uliftYonedaEquiv_naturality _ _

similarly below

[robin-carlier]

Please don’t change the copyright line of a file, only the Authors line.