Universal property

Roberto Alvarez´

2020 its Hom-functor

op HomC : C × C → Set

sends I an object (C, C0) ∈ C op × C to the set

0 0 HomC (C, C ) := {f : C → C | f is a C -morphism}

I a morphism (f , g) : (A, A0) → (B, B0), where f : B → A and g : A0 → B0 are C -morphisms, to the function

0 0 HomC (f , g) := h 7→ g ◦ h ◦ f : HomC (A, A ) → HomC (B, B )

Hom-functors

Let C be a locally small category I an object (C, C0) ∈ C op × C to the set

0 0 HomC (C, C ) := {f : C → C | f is a C -morphism}

I a morphism (f , g) : (A, A0) → (B, B0), where f : B → A and g : A0 → B0 are C -morphisms, to the function

0 0 HomC (f , g) := h 7→ g ◦ h ◦ f : HomC (A, A ) → HomC (B, B )

Hom-functors

Let C be a locally small category its Hom-functor

op HomC : C × C → Set

sends I a morphism (f , g) : (A, A0) → (B, B0), where f : B → A and g : A0 → B0 are C -morphisms, to the function

0 0 HomC (f , g) := h 7→ g ◦ h ◦ f : HomC (A, A ) → HomC (B, B )

Hom-functors

Let C be a locally small category its Hom-functor

op HomC : C × C → Set

sends I an object (C, C0) ∈ C op × C to the set

0 0 HomC (C, C ) := {f : C → C | f is a C -morphism} Hom-functors

Let C be a locally small category its Hom-functor

op HomC : C × C → Set

sends I an object (C, C0) ∈ C op × C to the set

0 0 HomC (C, C ) := {f : C → C | f is a C -morphism}

I a morphism (f , g) : (A, A0) → (B, B0), where f : B → A and g : A0 → B0 are C -morphisms, to the function

0 0 HomC (f , g) := h 7→ g ◦ h ◦ f : HomC (A, A ) → HomC (B, B ) I HomC (A, −) : C → Set which, for a C -morphisms f : B → C

HomC (A, f ) := HomC (idA, f ) : HomC (A, B) → HomC (A, C)

op I Dually, HomC (−, A) := HomC op (A, −) : C → Set

Hom-functors

We can also define the partially applied Hom-functors op I Dually, HomC (−, A) := HomC op (A, −) : C → Set

Hom-functors

We can also define the partially applied Hom-functors

I HomC (A, −) : C → Set which, for a C -morphisms f : B → C

HomC (A, f ) := HomC (idA, f ) : HomC (A, B) → HomC (A, C) Hom-functors

We can also define the partially applied Hom-functors

I HomC (A, −) : C → Set which, for a C -morphisms f : B → C

HomC (A, f ) := HomC (idA, f ) : HomC (A, B) → HomC (A, C)

op I Dually, HomC (−, A) := HomC op (A, −) : C → Set I for each natural transformation α : HomC (A, −) → F there is an element uα := αA(idA) ∈ F(A) I for each u ∈ F(A) there is an associated natural transformation α : HomC (A, −) → F defined as αB(f ) := F(f )(u)

(F(g) ◦ αB)(f ) = F(g)(F(f )(u)) = F(g ◦ f )(u)

= αC(g ◦ f ) = (αC ◦ Hom(A, g))(f )

Yoneda construction

Let F : C → Set be a functor I for each u ∈ F(A) there is an associated natural transformation α : HomC (A, −) → F defined as αB(f ) := F(f )(u)

(F(g) ◦ αB)(f ) = F(g)(F(f )(u)) = F(g ◦ f )(u)

= αC(g ◦ f ) = (αC ◦ Hom(A, g))(f )

Yoneda construction

Let F : C → Set be a functor

I for each natural transformation α : HomC (A, −) → F there is an element uα := αA(idA) ∈ F(A) (F(g) ◦ αB)(f ) = F(g)(F(f )(u)) = F(g ◦ f )(u)

= αC(g ◦ f ) = (αC ◦ Hom(A, g))(f )

Yoneda construction

Let F : C → Set be a functor

I for each natural transformation α : HomC (A, −) → F there is an element uα := αA(idA) ∈ F(A) I for each u ∈ F(A) there is an associated natural transformation α : HomC (A, −) → F defined as αB(f ) := F(f )(u) Yoneda construction

Let F : C → Set be a functor

I for each natural transformation α : HomC (A, −) → F there is an element uα := αA(idA) ∈ F(A) I for each u ∈ F(A) there is an associated natural transformation α : HomC (A, −) → F defined as αB(f ) := F(f )(u)

(F(g) ◦ αB)(f ) = F(g)(F(f )(u)) = F(g ◦ f )(u)

= αC(g ◦ f ) = (αC ◦ Hom(A, g))(f ) Proof. Let u ∈ F(A) and let α be its associated natural transformation

uα = αA(idA) = F(idA)(u) = u

Now, let α : Hom(A, −) → F and let α0 be the natural transformation associated to uα

0 αB(f ) = F(f )(uα) = F(f )(αA(idA)) = αB(Hom(A, f )(idA)) = αB(f )

so α0 = α

Yoneda lemma

Theorem (Yoneda) The construction is a bijection ∼ Hom[C ,Set](HomC (A, −), F) = F(A) Now, let α : Hom(A, −) → F and let α0 be the natural transformation associated to uα

0 αB(f ) = F(f )(uα) = F(f )(αA(idA)) = αB(Hom(A, f )(idA)) = αB(f )

so α0 = α

Yoneda lemma

Theorem (Yoneda) The construction is a bijection ∼ Hom[C ,Set](HomC (A, −), F) = F(A)

Proof. Let u ∈ F(A) and let α be its associated natural transformation

uα = αA(idA) = F(idA)(u) = u Yoneda lemma

Theorem (Yoneda) The construction is a bijection ∼ Hom[C ,Set](HomC (A, −), F) = F(A)

Proof. Let u ∈ F(A) and let α be its associated natural transformation

uα = αA(idA) = F(idA)(u) = u

Now, let α : Hom(A, −) → F and let α0 be the natural transformation associated to uα

0 αB(f ) = F(f )(uα) = F(f )(αA(idA)) = αB(Hom(A, f )(idA)) = αB(f )

so α0 = α The associated functor Y : C → [C op, Set] is fully faithful. op I Y(A) := HomC (−, A) : C → Set for A ∈ C

I Y(f )C := Hom(idC, f ) : Hom(C, A) → Hom(C, B) for a C -morphism f : A → B

Yoneda embedding

Corollary There is a bijection

HomC (A, B) → Hom[C op,Set](HomC op (A, −), HomC op (B, −)) Yoneda embedding

Corollary There is a bijection

HomC (A, B) → Hom[C op,Set](HomC op (A, −), HomC op (B, −))

The associated functor Y : C → [C op, Set] is fully faithful. op I Y(A) := HomC (−, A) : C → Set for A ∈ C

I Y(f )C := Hom(idC, f ) : Hom(C, A) → Hom(C, B) for a C -morphism f : A → B Definition (Universal element) An element u ∈ F(A) has the universal property iff for all B ∈ C and v ∈ F(B) there is a unique morphism f : A → B such that v = F(f )(u) The pair (A, u) is called a universal element of F.

Definition (Representation) A functor F : C → Set is called representable iff it is naturally isomorphic to Hom(A, −) for some A ∈ C . The pair (A, u) is called a universal element of F.

Definition (Representation) A functor F : C → Set is called representable iff it is naturally isomorphic to Hom(A, −) for some A ∈ C . Definition (Universal element) An element u ∈ F(A) has the universal property iff for all B ∈ C and v ∈ F(B) there is a unique morphism f : A → B such that v = F(f )(u) Definition (Representation) A functor F : C → Set is called representable iff it is naturally isomorphic to Hom(A, −) for some A ∈ C . Definition (Universal element) An element u ∈ F(A) has the universal property iff for all B ∈ C and v ∈ F(B) there is a unique morphism f : A → B such that v = F(f )(u) The pair (A, u) is called a universal element of F. Proof. Let (A, u) and (A0, u0) be universal elements; then there are unique f : A → A0 and f 0 : A0 → A such that u0 = F(f )(u) and u = F(f 0)(u0) so u = F(f 0 ◦ f )(u) and u0 = F(f ◦ f 0)(u), now there must be an unique h : A → A such that u = F(h)(u) i.e. idA, so 0 f ◦ f = idA 0 Similarly for f ◦ f = idA0

Definition (Universal element) An element u ∈ F(A) has the universal property iff for all B ∈ C and v ∈ F(B) there is a unique morphism f : A → B such that v = F(f )(u) The pair (A, u) is called a universal element of F. Corollary Universal elements are unique up to a unique isomorphism. now there must be an unique h : A → A such that u = F(h)(u) i.e. idA, so 0 f ◦ f = idA 0 Similarly for f ◦ f = idA0

Definition (Universal element) An element u ∈ F(A) has the universal property iff for all B ∈ C and v ∈ F(B) there is a unique morphism f : A → B such that v = F(f )(u) The pair (A, u) is called a universal element of F. Corollary Universal elements are unique up to a unique isomorphism. Proof. Let (A, u) and (A0, u0) be universal elements; then there are unique f : A → A0 and f 0 : A0 → A such that u0 = F(f )(u) and u = F(f 0)(u0) so u = F(f 0 ◦ f )(u) and u0 = F(f ◦ f 0)(u), Definition (Universal element) An element u ∈ F(A) has the universal property iff for all B ∈ C and v ∈ F(B) there is a unique morphism f : A → B such that v = F(f )(u) The pair (A, u) is called a universal element of F. Corollary Universal elements are unique up to a unique isomorphism. Proof. Let (A, u) and (A0, u0) be universal elements; then there are unique f : A → A0 and f 0 : A0 → A such that u0 = F(f )(u) and u = F(f 0)(u0) so u = F(f 0 ◦ f )(u) and u0 = F(f ◦ f 0)(u), now there must be an unique h : A → A such that u = F(h)(u) i.e. idA, so 0 f ◦ f = idA 0 Similarly for f ◦ f = idA0 Proof. I Assuming u has the universal property, then let the pointwise inverse of the associated natural transformation be −1 αB (v) := f : F(B) → Hom(A, B) where f is the morphism implied by the property at v ∈ F(B) I Assuming α : Hom(A, −) → F is a natural isomorphism, then for B ∈ C and v ∈ F(B) we have

−1 −1 F(αB (v))(u) = αB(αB (v)) = v

Theorem u has the universal property iff its associated natural transformation is an isomorphism. I Assuming α : Hom(A, −) → F is a natural isomorphism, then for B ∈ C and v ∈ F(B) we have

−1 −1 F(αB (v))(u) = αB(αB (v)) = v

Theorem u has the universal property iff its associated natural transformation is an isomorphism. Proof. I Assuming u has the universal property, then let the pointwise inverse of the associated natural transformation be −1 αB (v) := f : F(B) → Hom(A, B) where f is the morphism implied by the property at v ∈ F(B) Theorem u has the universal property iff its associated natural transformation is an isomorphism. Proof. I Assuming u has the universal property, then let the pointwise inverse of the associated natural transformation be −1 αB (v) := f : F(B) → Hom(A, B) where f is the morphism implied by the property at v ∈ F(B) I Assuming α : Hom(A, −) → F is a natural isomorphism, then for B ∈ C and v ∈ F(B) we have

−1 −1 F(αB (v))(u) = αB(αB (v)) = v its universal element is ({0, 1}, {1}); for any set X and Y ∈ P(X) (i.e. Y ⊆ X) there is a f : X → {0, 1} such that Y = {x | f (x) ∈ {1}}

Example: subsets

Consider the contravariant powerset functor P : Setop → Set

P(f ) := Y 7→ {x | f (x) ∈ Y} Example: subsets

Consider the contravariant powerset functor P : Setop → Set

P(f ) := Y 7→ {x | f (x) ∈ Y}

its universal element is ({0, 1}, {1}); for any set X and Y ∈ P(X) (i.e. Y ⊆ X) there is a f : X → {0, 1} such that Y = {x | f (x) ∈ {1}} its universal element (if any) is (X × Y, (π1, π2)); for Z ∈ C and (f1, f2) ∈ HomC (Z, X) × HomC (Z, Y) there is f : Z → X × Y such that (f1, f2) = (π1 ◦ f , π2 ◦ f )

Dually, consider the covariant functor

HomC (X, −) × HomC (Y, −) : C → Set

its universal element (if any) is (X + Y, (ι1, ι2)); for Z ∈ C and (f1, f2) ∈ HomC (X, Z) × HomC (Y, Z) there is f : X + Y → Z such that (f1, f2) = (f ◦ ι1, f ◦ ι2)

Example: products and co-products

Consider the contravariant functor

op HomC (−, X) × HomC (−, Y) : C → Set for Z ∈ C and (f1, f2) ∈ HomC (Z, X) × HomC (Z, Y) there is f : Z → X × Y such that (f1, f2) = (π1 ◦ f , π2 ◦ f )

Dually, consider the covariant functor

HomC (X, −) × HomC (Y, −) : C → Set

its universal element (if any) is (X + Y, (ι1, ι2)); for Z ∈ C and (f1, f2) ∈ HomC (X, Z) × HomC (Y, Z) there is f : X + Y → Z such that (f1, f2) = (f ◦ ι1, f ◦ ι2)

Example: products and co-products

Consider the contravariant functor

op HomC (−, X) × HomC (−, Y) : C → Set

its universal element (if any) is (X × Y, (π1, π2)); Dually, consider the covariant functor

HomC (X, −) × HomC (Y, −) : C → Set

its universal element (if any) is (X + Y, (ι1, ι2)); for Z ∈ C and (f1, f2) ∈ HomC (X, Z) × HomC (Y, Z) there is f : X + Y → Z such that (f1, f2) = (f ◦ ι1, f ◦ ι2)

Example: products and co-products

Consider the contravariant functor

op HomC (−, X) × HomC (−, Y) : C → Set

its universal element (if any) is (X × Y, (π1, π2)); for Z ∈ C and (f1, f2) ∈ HomC (Z, X) × HomC (Z, Y) there is f : Z → X × Y such that (f1, f2) = (π1 ◦ f , π2 ◦ f ) its universal element (if any) is (X + Y, (ι1, ι2)); for Z ∈ C and (f1, f2) ∈ HomC (X, Z) × HomC (Y, Z) there is f : X + Y → Z such that (f1, f2) = (f ◦ ι1, f ◦ ι2)

Example: products and co-products

Consider the contravariant functor

op HomC (−, X) × HomC (−, Y) : C → Set

its universal element (if any) is (X × Y, (π1, π2)); for Z ∈ C and (f1, f2) ∈ HomC (Z, X) × HomC (Z, Y) there is f : Z → X × Y such that (f1, f2) = (π1 ◦ f , π2 ◦ f )

Dually, consider the covariant functor

HomC (X, −) × HomC (Y, −) : C → Set for Z ∈ C and (f1, f2) ∈ HomC (X, Z) × HomC (Y, Z) there is f : X + Y → Z such that (f1, f2) = (f ◦ ι1, f ◦ ι2)

Example: products and co-products

Consider the contravariant functor

op HomC (−, X) × HomC (−, Y) : C → Set

its universal element (if any) is (X × Y, (π1, π2)); for Z ∈ C and (f1, f2) ∈ HomC (Z, X) × HomC (Z, Y) there is f : Z → X × Y such that (f1, f2) = (π1 ◦ f , π2 ◦ f )

Dually, consider the covariant functor

HomC (X, −) × HomC (Y, −) : C → Set

its universal element (if any) is (X + Y, (ι1, ι2)); Example: products and co-products

Consider the contravariant functor

op HomC (−, X) × HomC (−, Y) : C → Set

its universal element (if any) is (X × Y, (π1, π2)); for Z ∈ C and (f1, f2) ∈ HomC (Z, X) × HomC (Z, Y) there is f : Z → X × Y such that (f1, f2) = (π1 ◦ f , π2 ◦ f )

Dually, consider the covariant functor

HomC (X, −) × HomC (Y, −) : C → Set

its universal element (if any) is (X + Y, (ι1, ι2)); for Z ∈ C and (f1, f2) ∈ HomC (X, Z) × HomC (Y, Z) there is f : X + Y → Z such that (f1, f2) = (f ◦ ι1, f ◦ ι2) For a set X, consider the functor HomSet(X, U(−)) : C → Set

its universal element (if any) is (Free(X), ηX : X → U(Free(X))); for all B ∈ C and η0 : X → U(B) there is an f : Free(X) → B 0 such that η = Hom(X, U(f ))(ηX) = U(f ) ◦ ηX

If Free(X) exists for all X, then Free is an adjoint to U and η : idSet → U ◦ Free is called its unit.

Example: Free objects

Assuming C is the category of some algebraic structure, let U : C → Set be the forgetful functor. its universal element (if any) is (Free(X), ηX : X → U(Free(X))); for all B ∈ C and η0 : X → U(B) there is an f : Free(X) → B 0 such that η = Hom(X, U(f ))(ηX) = U(f ) ◦ ηX

If Free(X) exists for all X, then Free is an adjoint to U and η : idSet → U ◦ Free is called its unit.

Example: Free objects

Assuming C is the category of some algebraic structure, let U : C → Set be the forgetful functor. For a set X, consider the functor HomSet(X, U(−)) : C → Set for all B ∈ C and η0 : X → U(B) there is an f : Free(X) → B 0 such that η = Hom(X, U(f ))(ηX) = U(f ) ◦ ηX

If Free(X) exists for all X, then Free is an adjoint to U and η : idSet → U ◦ Free is called its unit.

Example: Free objects

Assuming C is the category of some algebraic structure, let U : C → Set be the forgetful functor. For a set X, consider the functor HomSet(X, U(−)) : C → Set

its universal element (if any) is (Free(X), ηX : X → U(Free(X))); If Free(X) exists for all X, then Free is an adjoint to U and η : idSet → U ◦ Free is called its unit.

Example: Free objects

Assuming C is the category of some algebraic structure, let U : C → Set be the forgetful functor. For a set X, consider the functor HomSet(X, U(−)) : C → Set

its universal element (if any) is (Free(X), ηX : X → U(Free(X))); for all B ∈ C and η0 : X → U(B) there is an f : Free(X) → B 0 such that η = Hom(X, U(f ))(ηX) = U(f ) ◦ ηX Example: Free objects

Assuming C is the category of some algebraic structure, let U : C → Set be the forgetful functor. For a set X, consider the functor HomSet(X, U(−)) : C → Set

its universal element (if any) is (Free(X), ηX : X → U(Free(X))); for all B ∈ C and η0 : X → U(B) there is an f : Free(X) → B 0 such that η = Hom(X, U(f ))(ηX) = U(f ) ◦ ηX

If Free(X) exists for all X, then Free is an adjoint to U and η : idSet → U ◦ Free is called its unit. Definition (Bilinear mapping) A mapping β : U(A) × U(B) → U(C) is called bilinear when for all a ∈ U(A) and b ∈ U(B) both β(a, −) : U(B) → U(C) and β(−, b) : U(A) → U(B) are group homomorphisms.

Example: tensor product

Let Ab be the category of abelian groups, A, B, C ∈ Ab and U : Ab → Set the forgetful functor. Example: tensor product

Let Ab be the category of abelian groups, A, B, C ∈ Ab and U : Ab → Set the forgetful functor. Definition (Bilinear mapping) A mapping β : U(A) × U(B) → U(C) is called bilinear when for all a ∈ U(A) and b ∈ U(B) both β(a, −) : U(B) → U(C) and β(−, b) : U(A) → U(B) are group homomorphisms. I Bilin(A, B; C) := {β : U(A) × U(B) → U(C) | β is bilinear} I Bilin(A, B; f ) := h 7→ U(f ) ◦ h Its universal element is (A ⊗ B, (a, b) 7→ a ⊗ b); for any abelian group G and bilinear g : U(A) × U(B) → U(G) there is an f : A ⊗ B → G such that g = (a, b) 7→ U(f )(a ⊗ b)

Bonus: this universal definition works for R-Modules for a ring R, replacing bilinearity with R-bilinearity.

Example: tensor product

Let Ab be the category of abelian groups, A, B, C ∈ Ab and U : Ab → Set the forgetful functor. Consider the functor Bilin(A, B; −) : Ab → Set defined as I Bilin(A, B; f ) := h 7→ U(f ) ◦ h Its universal element is (A ⊗ B, (a, b) 7→ a ⊗ b); for any abelian group G and bilinear g : U(A) × U(B) → U(G) there is an f : A ⊗ B → G such that g = (a, b) 7→ U(f )(a ⊗ b)

Bonus: this universal definition works for R-Modules for a ring R, replacing bilinearity with R-bilinearity.

Example: tensor product

Let Ab be the category of abelian groups, A, B, C ∈ Ab and U : Ab → Set the forgetful functor. Consider the functor Bilin(A, B; −) : Ab → Set defined as I Bilin(A, B; C) := {β : U(A) × U(B) → U(C) | β is bilinear} Its universal element is (A ⊗ B, (a, b) 7→ a ⊗ b); for any abelian group G and bilinear g : U(A) × U(B) → U(G) there is an f : A ⊗ B → G such that g = (a, b) 7→ U(f )(a ⊗ b)

Bonus: this universal definition works for R-Modules for a ring R, replacing bilinearity with R-bilinearity.

Example: tensor product

Let Ab be the category of abelian groups, A, B, C ∈ Ab and U : Ab → Set the forgetful functor. Consider the functor Bilin(A, B; −) : Ab → Set defined as I Bilin(A, B; C) := {β : U(A) × U(B) → U(C) | β is bilinear} I Bilin(A, B; f ) := h 7→ U(f ) ◦ h Bonus: this universal definition works for R-Modules for a ring R, replacing bilinearity with R-bilinearity.

Example: tensor product

Let Ab be the category of abelian groups, A, B, C ∈ Ab and U : Ab → Set the forgetful functor. Consider the functor Bilin(A, B; −) : Ab → Set defined as I Bilin(A, B; C) := {β : U(A) × U(B) → U(C) | β is bilinear} I Bilin(A, B; f ) := h 7→ U(f ) ◦ h Its universal element is (A ⊗ B, (a, b) 7→ a ⊗ b); for any abelian group G and bilinear g : U(A) × U(B) → U(G) there is an f : A ⊗ B → G such that g = (a, b) 7→ U(f )(a ⊗ b) Example: tensor product

Let Ab be the category of abelian groups, A, B, C ∈ Ab and U : Ab → Set the forgetful functor. Consider the functor Bilin(A, B; −) : Ab → Set defined as I Bilin(A, B; C) := {β : U(A) × U(B) → U(C) | β is bilinear} I Bilin(A, B; f ) := h 7→ U(f ) ◦ h Its universal element is (A ⊗ B, (a, b) 7→ a ⊗ b); for any abelian group G and bilinear g : U(A) × U(B) → U(G) there is an f : A ⊗ B → G such that g = (a, b) 7→ U(f )(a ⊗ b)

Bonus: this universal definition works for R-Modules for a ring R, replacing bilinearity with R-bilinearity. I In Set generalized elements are T indexed families I points are g.e. with form {∗} I lines are g.e. with form (0, 1) I paths are g.e. with form [0, 1] I if 1 ∈ C is a final object, then the generalized elements with form 1 are called global elements

Generalized elements

Definition We call a C -morphism a : T → A a generalized element of A with form T. Abusing notation we write a ∈ A and, for f : A → B, f (a) := f ◦ a ∈ B I points are g.e. with form {∗} I lines are g.e. with form (0, 1) I paths are g.e. with form [0, 1] I if 1 ∈ C is a final object, then the generalized elements with form 1 are called global elements

Generalized elements

Definition We call a C -morphism a : T → A a generalized element of A with form T. Abusing notation we write a ∈ A and, for f : A → B, f (a) := f ◦ a ∈ B I In Set generalized elements are T indexed families I lines are g.e. with form (0, 1) I paths are g.e. with form [0, 1] I if 1 ∈ C is a final object, then the generalized elements with form 1 are called global elements

Generalized elements

Definition We call a C -morphism a : T → A a generalized element of A with form T. Abusing notation we write a ∈ A and, for f : A → B, f (a) := f ◦ a ∈ B I In Set generalized elements are T indexed families I points are g.e. with form {∗} I paths are g.e. with form [0, 1] I if 1 ∈ C is a final object, then the generalized elements with form 1 are called global elements

Generalized elements

Definition We call a C -morphism a : T → A a generalized element of A with form T. Abusing notation we write a ∈ A and, for f : A → B, f (a) := f ◦ a ∈ B I In Set generalized elements are T indexed families I points are g.e. with form {∗} I lines are g.e. with form (0, 1) I if 1 ∈ C is a final object, then the generalized elements with form 1 are called global elements

Generalized elements

Definition We call a C -morphism a : T → A a generalized element of A with form T. Abusing notation we write a ∈ A and, for f : A → B, f (a) := f ◦ a ∈ B I In Set generalized elements are T indexed families I points are g.e. with form {∗} I lines are g.e. with form (0, 1) I paths are g.e. with form [0, 1] Generalized elements

Definition We call a C -morphism a : T → A a generalized element of A with form T. Abusing notation we write a ∈ A and, for f : A → B, f (a) := f ◦ a ∈ B I In Set generalized elements are T indexed families I points are g.e. with form {∗} I lines are g.e. with form (0, 1) I paths are g.e. with form [0, 1] I if 1 ∈ C is a final object, then the generalized elements with form 1 are called global elements Proof. From representability of Hom(A, −), its universal element is (A, idA), so for idA ∈ A then

f = f (idA) = g(idA) = g

Generalized elements

Corollary Let f , g : A → B be two C -morphisms, if for all a ∈ A we have f (a) = g(a) then f = g Generalized elements

Corollary Let f , g : A → B be two C -morphisms, if for all a ∈ A we have f (a) = g(a) then f = g Proof. From representability of Hom(A, −), its universal element is (A, idA), so for idA ∈ A then

f = f (idA) = g(idA) = g