Documentation

Mathlib.CategoryTheory.ChosenFiniteProducts

Categories with chosen finite products #

We introduce a class, ChosenFiniteProducts, which bundles explicit choices for a terminal object and binary products in a category C. This is primarily useful for categories which have finite products with good definitional properties, such as the category of types.

Given a category with such an instance, we also provide the associated symmetric monoidal structure so that one can write X ⊗ Y for the explicit binary product and 𝟙_ C for the explicit terminal object.

Implementation notes #

For cartesian monoidal categories, the oplax-monoidal/monoidal/braided structure of a functor F preserving finite products is uniquely determined. See the ofChosenFiniteProducts declarations.

We however develop the theory for any F.OplaxMonoidal/F.Monoidal/F.Braided instance instead of requiring it to be the ofChosenFiniteProducts one. This is to avoid diamonds: Consider eg 𝟭 C and F ⋙ G.

In applications requiring a finite preserving functor to be oplax-monoidal/monoidal/braided, avoid attribute [local instance] ofChosenFiniteProducts but instead turn on the corresponding ofChosenFiniteProducts declaration for that functor only.

Projects #

An instance of ChosenFiniteProducts C bundles an explicit choice of a binary product of two objects of C, and a terminal object in C.

Users should use the monoidal notation: X ⊗ Y for the product and 𝟙_ C for the terminal object.

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    @[instance 100]
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    The unique map to the terminal object.

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      This lemma follows from the preexisting Unique instance, but it is often convenient to use it directly as apply toUnit_unique forcing lean to do the necessary elaboration.

      Construct a morphism to the product given its two components.

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        theorem CategoryTheory.ChosenFiniteProducts.lift_fst {C : Type u} [Category.{v, u} C] [ChosenFiniteProducts C] {T X Y : C} (f : T X) (g : T Y) :
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        theorem CategoryTheory.ChosenFiniteProducts.lift_snd {C : Type u} [Category.{v, u} C] [ChosenFiniteProducts C] {T X Y : C} (f : T X) (g : T Y) :
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        theorem CategoryTheory.ChosenFiniteProducts.tensorHom_fst {C : Type u} [Category.{v, u} C] [ChosenFiniteProducts C] {X₁ X₂ Y₁ Y₂ : C} (f : X₁ X₂) (g : Y₁ Y₂) :
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        theorem CategoryTheory.ChosenFiniteProducts.tensorHom_fst_assoc {C : Type u} [Category.{v, u} C] [ChosenFiniteProducts C] {X₁ X₂ Y₁ Y₂ : C} (f : X₁ X₂) (g : Y₁ Y₂) {Z : C} (h : X₂ Z) :
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        theorem CategoryTheory.ChosenFiniteProducts.tensorHom_snd {C : Type u} [Category.{v, u} C] [ChosenFiniteProducts C] {X₁ X₂ Y₁ Y₂ : C} (f : X₁ X₂) (g : Y₁ Y₂) :
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        theorem CategoryTheory.ChosenFiniteProducts.tensorHom_snd_assoc {C : Type u} [Category.{v, u} C] [ChosenFiniteProducts C] {X₁ X₂ Y₁ Y₂ : C} (f : X₁ X₂) (g : Y₁ Y₂) {Z : C} (h : Y₂ Z) :
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        Construct an instance of ChosenFiniteProducts C given an instance of HasFiniteProducts C.

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          @[deprecated CategoryTheory.ChosenFiniteProducts.map_toUnit_comp_terminalComparison (since := "2025-04-09")]

          Alias of CategoryTheory.ChosenFiniteProducts.map_toUnit_comp_terminalComparison.

          If F preserves terminal objects, then terminalComparison F is an isomorphism.

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            When C and D have chosen finite products and F : C ⥤ D is any functor, prodComparison F A B is the canonical comparison morphism from F (A ⊗ B) to F(A) ⊗ F(B).

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              If the product comparison morphism is an iso, its inverse is natural in both argument.

              If the product comparison morphism is an iso, its inverse is natural in the right argument.

              If the product comparison morphism is an iso, its inverse is natural in the left argument.

              The product comparison morphism from F(A ⊗ -) to FA ⊗ F-, whose components are given by prodComparison.

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                If F preserves the limit of the pair (A, B), then the binary fan given by (F.map fst A B, F.map (snd A B)) is a limit cone.

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                  If F preserves the limit of the pair (A, B), then prodComparison F A B is an isomorphism.

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                    If prodComparison F A B is an isomorphism, then F preserves the limit of pair A B.

                    The restriction of a cartesian-monoidal category along an object property that's closed under finite products is cartesian-monoidal.

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                      Any functor between cartesian-monoidal categories is oplax monoidal.

                      This is not made an instance because it would create a diamond for the oplax monoidal structure on the identity and composition of functors.

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                        Any functor between cartesian-monoidal categories is oplax monoidal in a unique way.

                        A finite-product-preserving functor between cartesian monoidal categories is monoidal.

                        This is not made an instance because it would create a diamond for the monoidal structure on the identity and composition of functors.

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                          A functor between cartesian monoidal categories is monoidal iff it preserves finite products.

                          A finite-product-preserving functor between cartesian monoidal categories is braided.

                          This is not made an instance because it would create a diamond for the braided structure on the identity and composition of functors.

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                            @[deprecated CategoryTheory.Functor.OplaxMonoidal.ofChosenFiniteProducts (since := "2025-04-24")]

                            Alias of CategoryTheory.Functor.OplaxMonoidal.ofChosenFiniteProducts.


                            Any functor between cartesian-monoidal categories is oplax monoidal.

                            This is not made an instance because it would create a diamond for the oplax monoidal structure on the identity and composition of functors.

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                              @[deprecated CategoryTheory.Functor.Monoidal.ofChosenFiniteProducts (since := "2025-04-24")]

                              Alias of CategoryTheory.Functor.Monoidal.ofChosenFiniteProducts.


                              A finite-product-preserving functor between cartesian monoidal categories is monoidal.

                              This is not made an instance because it would create a diamond for the monoidal structure on the identity and composition of functors.

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                                @[deprecated CategoryTheory.Functor.Braided.ofChosenFiniteProducts (since := "2025-04-24")]

                                Alias of CategoryTheory.Functor.Braided.ofChosenFiniteProducts.


                                A finite-product-preserving functor between cartesian monoidal categories is braided.

                                This is not made an instance because it would create a diamond for the braided structure on the identity and composition of functors.

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