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feat(CategoryTheory/Enriched/Ordinary/Limits): Add conical limits for enriched ordinary categories #20904

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feat(CategoryTheory/Enriched/Ordinary/Limits): Add conical limits for enriched ordinary categories #20904

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4b410b8
chore(CategoryTheory/Enriched/Ordinary): create folder to allow API d…
joneugster Jan 21, 2025
f961132
update Mathlib.lean
joneugster Jan 21, 2025
83b4fe1
fix
joneugster Jan 21, 2025
071662b
add Enriched.IsConicalLimit
joneugster Jan 21, 2025
ab4188f
update Mathlib.lean
joneugster Jan 21, 2025
db68bab
remove unecessary import
joneugster Jan 21, 2025
dcb10f7
add docs; cleanup
joneugster Jan 21, 2025
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3 changes: 2 additions & 1 deletion Mathlib.lean
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Expand Up @@ -1774,7 +1774,8 @@ import Mathlib.CategoryTheory.Enriched.Basic
import Mathlib.CategoryTheory.Enriched.FunctorCategory
import Mathlib.CategoryTheory.Enriched.HomCongr
import Mathlib.CategoryTheory.Enriched.Opposite
import Mathlib.CategoryTheory.Enriched.Ordinary
import Mathlib.CategoryTheory.Enriched.Ordinary.Basic
import Mathlib.CategoryTheory.Enriched.Ordinary.Limits.IsConicalLimit
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Equivalence
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2 changes: 1 addition & 1 deletion Mathlib/AlgebraicTopology/SimplicialCategory/Basic.lean
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Expand Up @@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.SimplicialSet.Monoidal
import Mathlib.CategoryTheory.Enriched.Ordinary
import Mathlib.CategoryTheory.Enriched.Ordinary.Basic

/-!
# Simplicial categories
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2 changes: 1 addition & 1 deletion Mathlib/CategoryTheory/Enriched/FunctorCategory.lean
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Expand Up @@ -3,7 +3,7 @@ Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Enriched.Ordinary
import Mathlib.CategoryTheory.Enriched.Ordinary.Basic
import Mathlib.CategoryTheory.Functor.Category
import Mathlib.CategoryTheory.Limits.Shapes.End

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2 changes: 1 addition & 1 deletion Mathlib/CategoryTheory/Enriched/HomCongr.lean
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Expand Up @@ -3,7 +3,7 @@ Copyright (c) 2024 Nick Ward. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nick Ward
-/
import Mathlib.CategoryTheory.Enriched.Ordinary
import Mathlib.CategoryTheory.Enriched.Ordinary.Basic

/-!
# Congruence of enriched homs
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2 changes: 1 addition & 1 deletion Mathlib/CategoryTheory/Enriched/Opposite.lean
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Expand Up @@ -3,7 +3,7 @@ Copyright (c) 2024 Daniel Carranza. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Daniel Carranza
-/
import Mathlib.CategoryTheory.Enriched.Ordinary
import Mathlib.CategoryTheory.Enriched.Ordinary.Basic
import Mathlib.CategoryTheory.Monoidal.Braided.Basic

/-!
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5 changes: 4 additions & 1 deletion ...lib/CategoryTheory/Enriched/Ordinary.lean → ...tegoryTheory/Enriched/Ordinary/Basic.lean
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Expand Up @@ -26,7 +26,7 @@ using an abbreviation for `EnrichedOrdinaryCategory SSet C`.

universe v' v u u'

open CategoryTheory Category MonoidalCategory
open CategoryTheory Category MonoidalCategory Opposite

namespace CategoryTheory

Expand Down Expand Up @@ -173,4 +173,7 @@ instance ForgetEnrichment.EnrichedOrdinaryCategory {D : Type*} [EnrichedCategory
homEquiv_id _ := Category.id_comp _
homEquiv_comp _ _ := Category.assoc _ _ _

/-- enriched coyoneda functor `(X ⟶[V] _) : C ⥤ V`. -/
abbrev eCoyoneda (X : C) := (eHomFunctor V C).obj (op X)

end CategoryTheory
41 changes: 41 additions & 0 deletions Mathlib/CategoryTheory/Enriched/Ordinary/Limits/IsConicalLimit.lean
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@@ -0,0 +1,41 @@
/-
Copyright (c) 2025 Jon Eugster. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emily Riehl, Dagur Asgeirsson, Jon Eugster
-/
import Mathlib.CategoryTheory.Enriched.Ordinary.Basic

/-!
# Conical limits / enriched limits

If `V` is a monoidal category and `C` a `V`-enriched ordinary category,
a `V`-enriched limit, or "conical limit", is a limit cone `c` in `C` with
the property that for every `X : C`, the cone obtained by applying the coyoneda
functor `(X ⟶[V] -)` to `c` is a limit cone in `V`.
-/

universe v₁ u₁ w v' v u u'

namespace CategoryTheory.Enriched

open Limits

variable {J : Type u₁} [Category.{v₁} J]
variable (V : outParam <| Type u') [Category.{v'} V] [MonoidalCategory V]
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@joelriou joelriou Jan 21, 2025 •
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I think V should be an explicit parameter to IsConicalLimit. (This is what we have in the rest of the enriched category API, and I believe it should not change.)
Could you also add references to the mathematical literature? and also some minimal API (like constructors which under suitable assumptions may take as input only the isLimit field, or only the isConicalLimit fields?)

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@joelriou joelriou Jan 21, 2025 •
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To be more precise, for the API I am suggesting, I do not think we would have to assume HasLimitsOfShape J V as in #20905.

variable {C : Type u} [Category.{v} C] [EnrichedOrdinaryCategory V C]
variable {F : J ⥤ C} (c : Cone F)

/--
A limit cone `c` in a `V`-enriched ordinary category `C` is a *`V`-enriched limit*
(or *conical limit*) if for every `X : C`, the cone obtained by applying the coyoneda
functor `(X ⟶[V] -)` to `c` is a limit cone in `V`.
-/
structure IsConicalLimit where
/-- A conical limit cone is a limit cone. -/
isLimit : IsLimit c
/--
The cone obtained by applying the coyoneda functor `(X ⟶[V] -)` to `c` is a limit cone in `V`.
-/
isConicalLimit (X : C) : IsLimit <| (eCoyoneda V X).mapCone c

end CategoryTheory.Enriched
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