Limits and Colimits – Part II

Limits and colimits – Part II

Category theory and its applications – ITI9200 https://compose.ioc.ee

15 March 2021 1 S = {finite sets} C has finite products, or is cartesian 2 S = {small sets} C has products 3 S = {Par} C has equalisers 4 S = {finite cats} C has finite limits, or is finitely complete 5 S = {small cats} C has (small) limits, or is complete

Categories having (co)limits

Category with S -limits Let C be a category. We say that C has S -limits if, for all categories I ∈ S and functors F : I → C, there is a limit cone over F in C. 2 S = {small sets} C has products 3 S = {Par} C has equalisers 4 S = {finite cats} C has finite limits, or is finitely complete 5 S = {small cats} C has (small) limits, or is complete

Categories having (co)limits

Category with S -limits Let C be a category. We say that C has S -limits if, for all categories I ∈ S and functors F : I → C, there is a limit cone over F in C.

1 S = {finite sets} C has finite products, or is cartesian 3 S = {Par} C has equalisers 4 S = {finite cats} C has finite limits, or is finitely complete 5 S = {small cats} C has (small) limits, or is complete

Categories having (co)limits

Category with S -limits Let C be a category. We say that C has S -limits if, for all categories I ∈ S and functors F : I → C, there is a limit cone over F in C.

1 S = {finite sets} C has finite products, or is cartesian 2 S = {small sets} C has products 4 S = {finite cats} C has finite limits, or is finitely complete 5 S = {small cats} C has (small) limits, or is complete

Categories having (co)limits

Category with S -limits Let C be a category. We say that C has S -limits if, for all categories I ∈ S and functors F : I → C, there is a limit cone over F in C.

1 S = {ﬁnite sets} C has ﬁnite products, or is cartesian 2 S = {small sets} C has products 3 S = {Par} C has equalisers 5 S = {small cats} C has (small) limits, or is complete

Categories having (co)limits

Category with S -limits Let C be a category. We say that C has S -limits if, for all categories I ∈ S and functors F : I → C, there is a limit cone over F in C.

1 S = {finite sets} C has finite products, or is cartesian 2 S = {small sets} C has products 3 S = {Par} C has equalisers 4 S = {finite cats} C has finite limits, or is finitely complete Categories having (co)limits

Category with S -limits Let C be a category. We say that C has S -limits if, for all categories I ∈ S and functors F : I → C, there is a limit cone over F in C.

1 S = {finite sets} C has finite products, or is cartesian 2 S = {small sets} C has products 3 S = {Par} C has equalisers 4 S = {finite cats} C has finite limits, or is finitely complete 5 S = {small cats} C has (small) limits, or is complete 1 S = {finite sets} C has finite coproducts, or is cocartesian 2 S = {small sets} C has coproducts 3 S = {Par} C has coequalisers 4 S = {finite cats} C has finite colimits, or is finitely cocomplete 5 S = {small cats} C has (small) colimits, or is cocomplete

Categories having (co)limits

Category with S -limits Let C be a category. We say that C has S -colimits if, for all categories I ∈ S and functors F : I → C, there is a colimit cone under F in C. 2 S = {small sets} C has coproducts 3 S = {Par} C has coequalisers 4 S = {finite cats} C has finite colimits, or is finitely cocomplete 5 S = {small cats} C has (small) colimits, or is cocomplete

Categories having (co)limits

Category with S -limits Let C be a category. We say that C has S -colimits if, for all categories I ∈ S and functors F : I → C, there is a colimit cone under F in C.

1 S = {finite sets} C has finite coproducts, or is cocartesian 3 S = {Par} C has coequalisers 4 S = {finite cats} C has finite colimits, or is finitely cocomplete 5 S = {small cats} C has (small) colimits, or is cocomplete

Categories having (co)limits

Category with S -limits Let C be a category. We say that C has S -colimits if, for all categories I ∈ S and functors F : I → C, there is a colimit cone under F in C.

1 S = {finite sets} C has finite coproducts, or is cocartesian 2 S = {small sets} C has coproducts 4 S = {finite cats} C has finite colimits, or is finitely cocomplete 5 S = {small cats} C has (small) colimits, or is cocomplete

Categories having (co)limits

Category with S -limits Let C be a category. We say that C has S -colimits if, for all categories I ∈ S and functors F : I → C, there is a colimit cone under F in C.

1 S = {ﬁnite sets} C has ﬁnite coproducts, or is cocartesian 2 S = {small sets} C has coproducts 3 S = {Par} C has coequalisers 5 S = {small cats} C has (small) colimits, or is cocomplete

Categories having (co)limits

Category with S -limits Let C be a category. We say that C has S -colimits if, for all categories I ∈ S and functors F : I → C, there is a colimit cone under F in C.

1 S = {finite sets} C has finite coproducts, or is cocartesian 2 S = {small sets} C has coproducts 3 S = {Par} C has coequalisers 4 S = {finite cats} C has finite colimits, or is finitely cocomplete Categories having (co)limits

Category with S -limits Let C be a category. We say that C has S -colimits if, for all categories I ∈ S and functors F : I → C, there is a colimit cone under F in C.

Limits = products + equalisers

Theorem Let C be a category. Suppose that C 1 has (small) products, and 2 has equalisers. The theorem is relative to a choice of what “small” means (i.e. a choice of an inﬁnite cardinal); you can also replace “small” with “ﬁnite”! There is a dual version: small coproducts + coequalisers = small colimits

Limits = products + equalisers

Theorem Let C be a category. Suppose that C 1 has (small) products, and 2 has equalisers. Then C has all (small) limits. you can also replace “small” with “ﬁnite”! There is a dual version: small coproducts + coequalisers = small colimits

Limits = products + equalisers

Theorem Let C be a category. Suppose that C 1 has (small) products, and 2 has equalisers. Then C has all (small) limits.

The theorem is relative to a choice of what “small” means (i.e. a choice of an inﬁnite cardinal); There is a dual version: small coproducts + coequalisers = small colimits

Limits = products + equalisers

Theorem Let C be a category. Suppose that C 1 has (small) products, and 2 has equalisers. Then C has all (small) limits.

The theorem is relative to a choice of what “small” means (i.e. a choice of an inﬁnite cardinal); you can also replace “small” with “ﬁnite”! Limits = products + equalisers

Theorem Let C be a category. Suppose that C 1 has (small) products, and 2 has equalisers. Then C has all (small) limits.

The theorem is relative to a choice of what “small” means (i.e. a choice of an inﬁnite cardinal); you can also replace “small” with “ﬁnite”! There is a dual version: small coproducts + coequalisers = small colimits Morphisms C → lim F (i.e. “generalised elements” of lim F ) correspond to cones b over F with tip(b) = C.

Such a cone is a family of morphisms bi : C → F (i) satisfying the equations

F (h) ◦ bi = bj , i, j ∈ Ob(I), h ∈ HomI (i, j).

Limits = products + equalisers

The idea of the proof: Such a cone is a family of morphisms bi : C → F (i) satisfying the equations

F (h) ◦ bi = bj , i, j ∈ Ob(I), h ∈ HomI (i, j).

Limits = products + equalisers

The idea of the proof: Morphisms C → lim F (i.e. “generalised elements” of lim F ) correspond to cones b over F with tip(b) = C. Limits = products + equalisers

The idea of the proof: Morphisms C → lim F (i.e. “generalised elements” of lim F ) correspond to cones b over F with tip(b) = C.

Such a cone is a family of morphisms bi : C → F (i) satisfying the equations

F (h) ◦ bi = bj , i, j ∈ Ob(I), h ∈ HomI (i, j). If we have small products, and I is small, a family of morphisms with codomain F (i) Q corresponds to a generalised element of i∈Ob(I) F (i). If we also have equalisers, Q we can restrict i∈Ob(I) F (i) to generalised elements that satisfy equation (1).

Limits = products + equalisers

The idea of the proof (cont.d):

Such a cone is a family of morphisms bi : C → F (i) satisfying the equations

F (h) ◦ bi = bj , i, j ∈ Ob(I), h ∈ HomI (i, j). (1) If we also have equalisers, Q we can restrict i∈Ob(I) F (i) to generalised elements that satisfy equation (1).

Limits = products + equalisers

The idea of the proof (cont.d):

Such a cone is a family of morphisms bi : C → F (i) satisfying the equations

F (h) ◦ bi = bj , i, j ∈ Ob(I), h ∈ HomI (i, j). (1)

If we have small products, and I is small, a family of morphisms with codomain F (i) Q corresponds to a generalised element of i∈Ob(I) F (i). Limits = products + equalisers

The idea of the proof (cont.d):

Such a cone is a family of morphisms bi : C → F (i) satisfying the equations

F (h) ◦ bi = bj , i, j ∈ Ob(I), h ∈ HomI (i, j). (1)

If we have small products, and I is small, a family of morphisms with codomain F (i) Q corresponds to a generalised element of i∈Ob(I) F (i). If we also have equalisers, Q we can restrict i∈Ob(I) F (i) to generalised elements that satisfy equation (1). Since C has small products, we can form the products Y Y F (i), F (j) i∈Ob(I) i,j∈Ob(I), h∈HomC (i,j)

the ﬁrst indexed by all objects, the second by all morphisms of I, 0 with families of projections πi and πh, respectively. For each morphism h: i → j, we have a pair of morphisms Y F (h) ◦ πi , πj : F (i) → F (j); i∈Ob(I)

this corresponds to the equation F (h) ◦ bi = bj .

Limits = products + equalisers

Proof Let F : I → C be a functor from a small category I. For each morphism h: i → j, we have a pair of morphisms Y F (h) ◦ πi , πj : F (i) → F (j); i∈Ob(I)

this corresponds to the equation F (h) ◦ bi = bj .

Limits = products + equalisers

Proof Let F : I → C be a functor from a small category I. Since C has small products, we can form the products Y Y F (i), F (j) i∈Ob(I) i,j∈Ob(I), h∈HomC (i,j)

the ﬁrst indexed by all objects, the second by all morphisms of I, 0 with families of projections πi and πh, respectively. this corresponds to the equation F (h) ◦ bi = bj .

Limits = products + equalisers

Proof Let F : I → C be a functor from a small category I. Since C has small products, we can form the products Y Y F (i), F (j) i∈Ob(I) i,j∈Ob(I), h∈HomC (i,j)

this corresponds to the equation F (h) ◦ bi = bj . this can be seen as packaging all the equations into a single parallel pair. Q Let e : lim F → i∈Ob(I) F (i) be the equaliser of this parallel pair.

Limits = products + equalisers

Proof, cont.d The families

(F (h) ◦ πi ) i,j∈Ob(I), (πj ) i,j∈Ob(I), h∈HomI (i,j) h∈HomI (i,j) produce a parallel pair of morphisms Y Y hF (h) ◦ πi i, hπj i: F (i) → F (j); i∈Ob(I) i,j∈Ob(I), h∈HomI (i,j) Q Let e : lim F → i∈Ob(I) F (i) be the equaliser of this parallel pair.

Limits = products + equalisers

Proof, cont.d The families

(F (h) ◦ πi ) i,j∈Ob(I), (πj ) i,j∈Ob(I), h∈HomI (i,j) h∈HomI (i,j) produce a parallel pair of morphisms Y Y hF (h) ◦ πi i, hπj i: F (i) → F (j); i∈Ob(I) i,j∈Ob(I), h∈HomI (i,j)

this can be seen as packaging all the equations into a single parallel pair. Limits = products + equalisers

Proof, cont.d The families

this can be seen as packaging all the equations into a single parallel pair. Q Let e : lim F → i∈Ob(I) F (i) be the equaliser of this parallel pair. 1 It is a cone over F :

0 0 F (h) ◦ πi ◦ e = πh ◦ hF (h) ◦ πi i ◦ e = πh ◦ hπj i ◦ e = πj ◦ e

2 Given another cone b over F , the family (bi )i∈Ob(I) induces Y hbi i: tip(b) → F (i) i∈Ob(I)

We need to show that

hF (h) ◦ πi i ◦ hbi i = hπj i ◦ hbi i

Limits = products + equalisers

Proof, cont.d We claim that lim F together with π ◦ e := (πi ◦ e)i∈Ob(I) is a limit cone over F . 2 Given another cone b over F , the family (bi )i∈Ob(I) induces Y hbi i: tip(b) → F (i) i∈Ob(I)

We need to show that

hF (h) ◦ πi i ◦ hbi i = hπj i ◦ hbi i

Limits = products + equalisers

Proof, cont.d We claim that lim F together with π ◦ e := (πi ◦ e)i∈Ob(I) is a limit cone over F .

1 It is a cone over F :

0 0 F (h) ◦ πi ◦ e = πh ◦ hF (h) ◦ πi i ◦ e = πh ◦ hπj i ◦ e = πj ◦ e We need to show that

hF (h) ◦ πi i ◦ hbi i = hπj i ◦ hbi i

Limits = products + equalisers

Proof, cont.d We claim that lim F together with π ◦ e := (πi ◦ e)i∈Ob(I) is a limit cone over F .

1 It is a cone over F :

0 0 F (h) ◦ πi ◦ e = πh ◦ hF (h) ◦ πi i ◦ e = πh ◦ hπj i ◦ e = πj ◦ e

2 Given another cone b over F , the family (bi )i∈Ob(I) induces Y hbi i: tip(b) → F (i) i∈Ob(I) Limits = products + equalisers

Proof, cont.d We claim that lim F together with π ◦ e := (πi ◦ e)i∈Ob(I) is a limit cone over F .

1 It is a cone over F :

0 0 F (h) ◦ πi ◦ e = πh ◦ hF (h) ◦ πi i ◦ e = πh ◦ hπj i ◦ e = πj ◦ e

2 Given another cone b over F , the family (bi )i∈Ob(I) induces Y hbi i: tip(b) → F (i) i∈Ob(I)

We need to show that

hF (h) ◦ πi i ◦ hbi i = hπj i ◦ hbi i Because e is the equaliser of hF (h) ◦ πi i and hπj i, we obtain a unique morphism

k : tip(b) → lim F

such that e ◦ k = hbi i, so

(πi ◦ e) ◦ k = bi , i ∈ Ob(I).

Limits = products + equalisers

Proof, cont.d Because b is a cone over F , for all morphisms h: i → j, we have

0 πh ◦ hF (h) ◦ πi i ◦ hbi i = F (h) ◦ πi ◦ hbi i = F (h) ◦ bi = bj 0 = πj ◦ hbi i = πh ◦ hπj i ◦ hbi i

which implies hF (h) ◦ πi i ◦ hbi i = hπj i ◦ hbi i. so

(πi ◦ e) ◦ k = bi , i ∈ Ob(I).

Limits = products + equalisers

Proof, cont.d Because b is a cone over F , for all morphisms h: i → j, we have

0 πh ◦ hF (h) ◦ πi i ◦ hbi i = F (h) ◦ πi ◦ hbi i = F (h) ◦ bi = bj 0 = πj ◦ hbi i = πh ◦ hπj i ◦ hbi i

which implies hF (h) ◦ πi i ◦ hbi i = hπj i ◦ hbi i.

Because e is the equaliser of hF (h) ◦ πi i and hπj i, we obtain a unique morphism

k : tip(b) → lim F

such that e ◦ k = hbi i, Limits = products + equalisers

Proof, cont.d Because b is a cone over F , for all morphisms h: i → j, we have

0 πh ◦ hF (h) ◦ πi i ◦ hbi i = F (h) ◦ πi ◦ hbi i = F (h) ◦ bi = bj 0 = πj ◦ hbi i = πh ◦ hπj i ◦ hbi i

which implies hF (h) ◦ πi i ◦ hbi i = hπj i ◦ hbi i.

Because e is the equaliser of hF (h) ◦ πi i and hπj i, we obtain a unique morphism

k : tip(b) → lim F

such that e ◦ k = hbi i, so

(πi ◦ e) ◦ k = bi , i ∈ Ob(I). This is a solution to the system

(πi ◦ e) ◦ x = bi , i ∈ Ob(I)

and we can prove that it is unique from the uniqueness properties of products and equalisers.

This proves that π ◦ e is a limit cone over F .

Limits = products + equalisers

Proof, cont.d We have constructed k : tip(b) → lim F such that

(πi ◦ e) ◦ k = bi , i ∈ Ob(I). This proves that π ◦ e is a limit cone over F .

Limits = products + equalisers

Proof, cont.d We have constructed k : tip(b) → lim F such that

(πi ◦ e) ◦ k = bi , i ∈ Ob(I).

This is a solution to the system

(πi ◦ e) ◦ x = bi , i ∈ Ob(I)

and we can prove that it is unique from the uniqueness properties of products and equalisers. Limits = products + equalisers

Proof, cont.d We have constructed k : tip(b) → lim F such that

(πi ◦ e) ◦ k = bi , i ∈ Ob(I).

This is a solution to the system

(πi ◦ e) ◦ x = bi , i ∈ Ob(I)

and we can prove that it is unique from the uniqueness properties of products and equalisers.

This proves that π ◦ e is a limit cone over F . When C is small, the functor category [C, Set] has all small limits and colimits. A poset P has all limits (resp. colimits) if and only if it has greatest lower bounds (resp. least upper bounds) of arbitrary collections of elements.

Complete and cocomplete categories

Set has all small limits and colimits. A poset P has all limits (resp. colimits) if and only if it has greatest lower bounds (resp. least upper bounds) of arbitrary collections of elements.

Complete and cocomplete categories

Set has all small limits and colimits. When C is small, the functor category [C, Set] has all small limits and colimits. Complete and cocomplete categories

Set has all small limits and colimits. When C is small, the functor category [C, Set] has all small limits and colimits. A poset P has all limits (resp. colimits) if and only if it has greatest lower bounds (resp. least upper bounds) of arbitrary collections of elements. Let b be a cone over F with components

bi : tip(b) → F (i), i ∈ Ob(I).

Then Gb with components

Gbi : G(tip(b)) → GF (i), i ∈ Ob(I),

is a cone over GF .

Functors and limit cones

Let F : I → C and G : C → D be functors. Then Gb with components

Gbi : G(tip(b)) → GF (i), i ∈ Ob(I),

is a cone over GF .

Functors and limit cones

Let F : I → C and G : C → D be functors.

Let b be a cone over F with components

bi : tip(b) → F (i), i ∈ Ob(I). Functors and limit cones

Let F : I → C and G : C → D be functors.

Let b be a cone over F with components

bi : tip(b) → F (i), i ∈ Ob(I).

Then Gb with components

Gbi : G(tip(b)) → GF (i), i ∈ Ob(I),

is a cone over GF . Preservation of limits The functor G preserves limits of F if, whenever f is a limit cone over F , Gf is a limit cone over GF . The functor G preserves S -limits if, for all I ∈ S and functors F : I → C, G preserves limits of F .

“if it is a limit, then its image is a limit”

Functors and limit cones

Let F : I → C and G : C → D be functors. Let S be a class of categories. The functor G preserves S -limits if, for all I ∈ S and functors F : I → C, G preserves limits of F .

“if it is a limit, then its image is a limit”

Functors and limit cones

Let F : I → C and G : C → D be functors. Let S be a class of categories. Preservation of limits The functor G preserves limits of F if, whenever f is a limit cone over F , Gf is a limit cone over GF . “if it is a limit, then its image is a limit”

Functors and limit cones

“if it is a limit, then its image is a limit” Reflection of limits The functor G reflects limits of F if, for all cones f over F , if Gf is a limit cone over GF , then f is a limit cone. The functor G reflects S -limits if, for all I ∈ S and functors F : I → C, G reflects limits of F .

“if its image is a limit, then it is a limit”

Functors and limit cones

Let F : I → C and G : C → D be functors. Let S be a class of categories. The functor G reﬂects S -limits if, for all I ∈ S and functors F : I → C, G reﬂects limits of F .

“if its image is a limit, then it is a limit”

Functors and limit cones

Let F : I → C and G : C → D be functors. Let S be a class of categories. Reﬂection of limits The functor G reﬂects limits of F if, for all cones f over F , if Gf is a limit cone over GF , then f is a limit cone. “if its image is a limit, then it is a limit”

Functors and limit cones

“if its image is a limit, then it is a limit” Creation of limits The functor G creates limits of F if 1 G preserves and reﬂects limits of F , and 2 if GF has a limit, so does F . The functor G creates S -limits if, for all I ∈ S and functors F : I → C, G creates limits of F .

“from limits of GF , we get limits of F ”

Functors and limit cones

Let F : I → C and G : C → D be functors. Let S be a class of categories. The functor G creates S -limits if, for all I ∈ S and functors F : I → C, G creates limits of F .

“from limits of GF , we get limits of F ”

Functors and limit cones

Let F : I → C and G : C → D be functors. Let S be a class of categories. Creation of limits The functor G creates limits of F if 1 G preserves and reﬂects limits of F , and 2 if GF has a limit, so does F . “from limits of GF , we get limits of F ”

Functors and limit cones

“from limits of GF , we get limits of F ” There are results in CT of the form

“all functors with certain properties create S -(co)limits” So if we have a functor G : C → D that creates S -(co)limits, know that D has S -(co)limits (for example, D = Set), then we automatically know that C has S -(co)limits!

Preservation of (co)limits is also very important in the theory of adjunctions (next lecture!)

Functors and limit cones

Why are these deﬁnitions useful? So if we have a functor G : C → D that creates S -(co)limits, know that D has S -(co)limits (for example, D = Set), then we automatically know that C has S -(co)limits!

Preservation of (co)limits is also very important in the theory of adjunctions (next lecture!)

Functors and limit cones

Why are these deﬁnitions useful? There are results in CT of the form

“all functors with certain properties create S -(co)limits” know that D has S -(co)limits (for example, D = Set), then we automatically know that C has S -(co)limits!

Preservation of (co)limits is also very important in the theory of adjunctions (next lecture!)

Functors and limit cones

Why are these deﬁnitions useful? There are results in CT of the form

“all functors with certain properties create S -(co)limits” So if we have a functor G : C → D that creates S -(co)limits, then we automatically know that C has S -(co)limits!

Preservation of (co)limits is also very important in the theory of adjunctions (next lecture!)

Functors and limit cones

Why are these deﬁnitions useful? There are results in CT of the form

“all functors with certain properties create S -(co)limits” So if we have a functor G : C → D that creates S -(co)limits, know that D has S -(co)limits (for example, D = Set), Preservation of (co)limits is also very important in the theory of adjunctions (next lecture!)

Functors and limit cones

Why are these deﬁnitions useful? There are results in CT of the form

Preservation of (co)limits is also very important in the theory of adjunctions (next lecture!) For each i ∈ Ob(I), we have a functor F (i, −): J → C. Proposition Suppose that, for each i ∈ I, F (i, −) has a (chosen) limit with tip limJ F (i, −). Then i 7→ limJ F (i, −) extends to a functor

limJ F : I → C.

Doubly indexed limits

Let F : I × J → C be a functor. Proposition Suppose that, for each i ∈ I, F (i, −) has a (chosen) limit with tip limJ F (i, −). Then i 7→ limJ F (i, −) extends to a functor

limJ F : I → C.

Doubly indexed limits

Let F : I × J → C be a functor.

For each i ∈ Ob(I), we have a functor F (i, −): J → C. Then i 7→ limJ F (i, −) extends to a functor

limJ F : I → C.

Doubly indexed limits

Let F : I × J → C be a functor.

For each i ∈ Ob(I), we have a functor F (i, −): J → C. Proposition Suppose that, for each i ∈ I, F (i, −) has a (chosen) limit with tip limJ F (i, −). Doubly indexed limits

Let F : I × J → C be a functor.

For each i ∈ Ob(I), we have a functor F (i, −): J → C. Proposition Suppose that, for each i ∈ I, F (i, −) has a (chosen) limit with tip limJ F (i, −). Then i 7→ limJ F (i, −) extends to a functor

limJ F : I → C. For each j ∈ Ob(J ), we have morphisms

fj : limJ F (i, −) → F (i, j), F (h, j): F (i, j) → F (i 0, j),

whose composites (F (h, j) ◦ fj )j∈Ob(J ) form 0 a cone over F (i , −) with tip limJ F (i, −).

Doubly indexed limits

Sketch of proof Let h: i → i 0 be a morphism in I, let f be the chosen limit cone over F (i, −), and let f 0 be the chosen limit cone over F (i 0, −). whose composites (F (h, j) ◦ fj )j∈Ob(J ) form 0 a cone over F (i , −) with tip limJ F (i, −).

Doubly indexed limits

Sketch of proof Let h: i → i 0 be a morphism in I, let f be the chosen limit cone over F (i, −), and let f 0 be the chosen limit cone over F (i 0, −).

For each j ∈ Ob(J ), we have morphisms

fj : limJ F (i, −) → F (i, j), F (h, j): F (i, j) → F (i 0, j), Doubly indexed limits

Sketch of proof Let h: i → i 0 be a morphism in I, let f be the chosen limit cone over F (i, −), and let f 0 be the chosen limit cone over F (i 0, −).

For each j ∈ Ob(J ), we have morphisms

fj : limJ F (i, −) → F (i, j), F (h, j): F (i, j) → F (i 0, j),

whose composites (F (h, j) ◦ fj )j∈Ob(J ) form 0 a cone over F (i , −) with tip limJ F (i, −). We can show that this assignment deﬁnes a functor (i.e. is compatible with identities and composition).

Doubly indexed limits

Sketch of proof, cont.d By the universal property of the cone f 0, 0 whose tip is limJ F (i , −), we obtain a unique morphism

0 limJ F (i, −) → limJ F (i , −),

and we deﬁne limJ F (h, −) to be this morphism. Doubly indexed limits

Sketch of proof, cont.d By the universal property of the cone f 0, 0 whose tip is limJ F (i , −), we obtain a unique morphism

0 limJ F (i, −) → limJ F (i , −),

and we deﬁne limJ F (h, −) to be this morphism.

We can show that this assignment deﬁnes a functor (i.e. is compatible with identities and composition). Commutativity of (co)limits

If we are in the conditions of the previous proposition, we can ask if limJ F : I → C has a limit. Theorem (commutativity of limits) Suppose that

limJ F : I → C, limI F : J → C

are deﬁned and have limit cones with tips

limI limJ F , limJ limI F .

Then limI limJ F and limJ limI F are canonically isomorphic, and are tips of limit cones over F . In this sense,

limits commute with limits, colimits commute with colimits

However limits seldom commute with colimits...

Commutativity of (co)limits

Dually, we have isomorphisms

colimI colimJ F ' colimJ colimI F

whenever both sides are deﬁned. However limits seldom commute with colimits...

Commutativity of (co)limits

Dually, we have isomorphisms

colimI colimJ F ' colimJ colimI F

whenever both sides are deﬁned. In this sense,

limits commute with limits, colimits commute with colimits Commutativity of (co)limits

Dually, we have isomorphisms

colimI colimJ F ' colimJ colimI F

whenever both sides are deﬁned. In this sense,

limits commute with limits, colimits commute with colimits

However limits seldom commute with colimits... Proposition There is a canonical morphism

colimI limJ F → limJ colimI F .

...but it is not, in general, an isomorphism.

Non-commutativity of limits with colimits

Let F : I × J → C be a functor, and suppose

colimI limJ F , limJ colimI F

are both deﬁned. ...but it is not, in general, an isomorphism.

Non-commutativity of limits with colimits

Let F : I × J → C be a functor, and suppose

colimI limJ F , limJ colimI F

are both deﬁned. Proposition There is a canonical morphism

colimI limJ F → limJ colimI F . Non-commutativity of limits with colimits

Let F : I × J → C be a functor, and suppose

colimI limJ F , limJ colimI F

are both deﬁned. Proposition There is a canonical morphism

colimI limJ F → limJ colimI F .

...but it is not, in general, an isomorphism. Then

colimI limJ F = (11,1 × 11,2) + (12,1 × 12,2) ' 2,

limJ colimI F = (11,1 + 12,1) × (11,2 + 12,2) ' 4.

Non-commutativity of limits with colimits

A simple counterexample: Let F : 2 × 2 → Set be deﬁned by

F (i, j) := 1 ≡ 1i,j

for all i, j ∈ 2. Non-commutativity of limits with colimits

A simple counterexample: Let F : 2 × 2 → Set be deﬁned by

F (i, j) := 1 ≡ 1i,j

for all i, j ∈ 2. Then

colimI limJ F = (11,1 × 11,2) + (12,1 × 12,2) ' 2,

limJ colimI F = (11,1 + 12,1) × (11,2 + 12,2) ' 4. A category with finite colimits is filtered. A poset is filtered when every finite collection of elements has an upper bound.

Filtered colimits

Filtered category A category C is filtered if, for all finite categories I and functors F : I → C, there exists a cone under F . A poset is filtered when every finite collection of elements has an upper bound.

Filtered colimits

Filtered category A category C is ﬁltered if, for all ﬁnite categories I and functors F : I → C, there exists a cone under F .

A category with ﬁnite colimits is ﬁltered. Filtered colimits

Filtered category A category C is ﬁltered if, for all ﬁnite categories I and functors F : I → C, there exists a cone under F .

A category with finite colimits is filtered. A poset is filtered when every finite collection of elements has an upper bound. Theorem (Filtered colimits commute with finite limits) Let I be a filtered category, J a finite category, and F : I × J → C a functor. Then the canonical morphism

colimI limJ F → limJ colimI F

is an isomorphism.

Filtered colimits

A ﬁltered colimit is a colimit of a functor whose domain is a ﬁltered category. Then the canonical morphism

colimI limJ F → limJ colimI F

is an isomorphism.

Filtered colimits

A ﬁltered colimit is a colimit of a functor whose domain is a ﬁltered category.

Theorem (Filtered colimits commute with finite limits) Let I be a filtered category, J a finite category, and F : I × J → C a functor. Filtered colimits

A ﬁltered colimit is a colimit of a functor whose domain is a ﬁltered category.

Theorem (Filtered colimits commute with finite limits) Let I be a filtered category, J a finite category, and F : I × J → C a functor. Then the canonical morphism

colimI limJ F → limJ colimI F

is an isomorphism. A functor F : N → C is a sequence of objects and morphisms

F (0) → F (1) → ... → F (n) → ...

A colimit of such a functor is called a sequential colimit. For example, an increasing sequence of sets

X0 ⊆ X1 ⊆ ... ⊆ Xn ⊆ ...

deﬁnes a functor N → Set, whose colimit is [ Xn n∈N

Filtered colimits: example

Let (N, ≤) be the poset of natural numbers with the usual order. A colimit of such a functor is called a sequential colimit. For example, an increasing sequence of sets

X0 ⊆ X1 ⊆ ... ⊆ Xn ⊆ ...

deﬁnes a functor N → Set, whose colimit is [ Xn n∈N

Filtered colimits: example

Let (N, ≤) be the poset of natural numbers with the usual order. A functor F : N → C is a sequence of objects and morphisms

F (0) → F (1) → ... → F (n) → ... For example, an increasing sequence of sets

X0 ⊆ X1 ⊆ ... ⊆ Xn ⊆ ...

deﬁnes a functor N → Set, whose colimit is [ Xn n∈N

Filtered colimits: example

Let (N, ≤) be the poset of natural numbers with the usual order. A functor F : N → C is a sequence of objects and morphisms

F (0) → F (1) → ... → F (n) → ...

A colimit of such a functor is called a sequential colimit. Filtered colimits: example

Let (N, ≤) be the poset of natural numbers with the usual order. A functor F : N → C is a sequence of objects and morphisms

F (0) → F (1) → ... → F (n) → ...

A colimit of such a functor is called a sequential colimit. For example, an increasing sequence of sets

X0 ⊆ X1 ⊆ ... ⊆ Xn ⊆ ...

deﬁnes a functor N → Set, whose colimit is [ Xn n∈N It follows that sequential colimits commute with ﬁnite limits. We recover, for example, the fact that [ [ Xn × Y ' (Xn × Y ) n∈N n∈N in Set.

Filtered colimits: example

N has least upper bounds of all finite collection of numbers, so it has finite colimits. In particular, it is a filtered category. We recover, for example, the fact that [ [ Xn × Y ' (Xn × Y ) n∈N n∈N in Set.

Filtered colimits: example

N has least upper bounds of all finite collection of numbers, so it has finite colimits. In particular, it is a filtered category.

It follows that sequential colimits commute with ﬁnite limits. Filtered colimits: example

N has least upper bounds of all finite collection of numbers, so it has finite colimits. In particular, it is a filtered category.

It follows that sequential colimits commute with ﬁnite limits. We recover, for example, the fact that [ [ Xn × Y ' (Xn × Y ) n∈N n∈N in Set. For all objects C in C, we also have a functor

op HomC(−, C): C → Set

Theorem The following are equivalent: 1 F has a limit; 2 there exist an object lim F of C and a natural isomorphism

limI HomC(−, F −) ' HomC(−, lim F ).

Representable property of limits

When F : I → C is a functor from a small category, we have

op limI HomC(−, F −): C → Set Theorem The following are equivalent: 1 F has a limit; 2 there exist an object lim F of C and a natural isomorphism

limI HomC(−, F −) ' HomC(−, lim F ).

Representable property of limits

When F : I → C is a functor from a small category, we have

op limI HomC(−, F −): C → Set

For all objects C in C, we also have a functor

op HomC(−, C): C → Set Representable property of limits

When F : I → C is a functor from a small category, we have

op limI HomC(−, F −): C → Set

For all objects C in C, we also have a functor

op HomC(−, C): C → Set

Theorem The following are equivalent: 1 F has a limit; 2 there exist an object lim F of C and a natural isomorphism

limI HomC(−, F −) ' HomC(−, lim F ). Interpretation: generalised elements of the limits, i.e. x ∈ HomC(C, lim F ), are the same as elements x ∈ lim HomC(C, F −) of the limit of sets of elements

Representable property of limits

The following are equivalent: 1 F has a limit; 2 there exist an object lim F of C and a natural isomorphism

limI HomC(−, F −) ' HomC(−, lim F ). Representable property of limits

The following are equivalent: 1 F has a limit; 2 there exist an object lim F of C and a natural isomorphism

limI HomC(−, F −) ' HomC(−, lim F ).

Interpretation: generalised elements of the limits, i.e. x ∈ HomC(C, lim F ), are the same as elements x ∈ lim HomC(C, F −) of the limit of sets of elements But because [Cop, Set] is complete, the limit of the diagram

HomC(−, F −)

always exists in [Cop, Set], even when the limit of F does not exist in C... So [Cop, Set] is a kind of “complete extension” of C.

Representable property of limits

A little teaser of future lectures: the Yoneda lemma will allow us to identify

op C ∈ Ob(C) and HomC(−, C): C → Set

i.e. an object of C with its presheaf of generalised elements. So [Cop, Set] is a kind of “complete extension” of C.

Representable property of limits

A little teaser of future lectures: the Yoneda lemma will allow us to identify

op C ∈ Ob(C) and HomC(−, C): C → Set

i.e. an object of C with its presheaf of generalised elements.

But because [Cop, Set] is complete, the limit of the diagram

HomC(−, F −)

always exists in [Cop, Set], even when the limit of F does not exist in C... Representable property of limits

A little teaser of future lectures: the Yoneda lemma will allow us to identify

op C ∈ Ob(C) and HomC(−, C): C → Set

i.e. an object of C with its presheaf of generalised elements.

But because [Cop, Set] is complete, the limit of the diagram

HomC(−, F −)

always exists in [Cop, Set], even when the limit of F does not exist in C... So [Cop, Set] is a kind of “complete extension” of C.