Limits and colimits { Part II Category theory and its applications { ITI9200 https://compose.ioc.ee 15 March 2021 1 S = ffinite setsg C has finite products, or is cartesian 2 S = fsmall setsg C has products 3 S = fParg C has equalisers 4 S = ffinite catsg C has finite limits, or is finitely complete 5 S = fsmall catsg 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 2 S and functors F : I!C, there is a limit cone over F in C. 2 S = fsmall setsg C has products 3 S = fParg C has equalisers 4 S = ffinite catsg C has finite limits, or is finitely complete 5 S = fsmall catsg 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 2 S and functors F : I!C, there is a limit cone over F in C. 1 S = ffinite setsg C has finite products, or is cartesian 3 S = fParg C has equalisers 4 S = ffinite catsg C has finite limits, or is finitely complete 5 S = fsmall catsg 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 2 S and functors F : I!C, there is a limit cone over F in C. 1 S = ffinite setsg C has finite products, or is cartesian 2 S = fsmall setsg C has products 4 S = ffinite catsg C has finite limits, or is finitely complete 5 S = fsmall catsg 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 2 S and functors F : I!C, there is a limit cone over F in C. 1 S = ffinite setsg C has finite products, or is cartesian 2 S = fsmall setsg C has products 3 S = fParg C has equalisers 5 S = fsmall catsg 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 2 S and functors F : I!C, there is a limit cone over F in C. 1 S = ffinite setsg C has finite products, or is cartesian 2 S = fsmall setsg C has products 3 S = fParg C has equalisers 4 S = ffinite catsg 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 2 S and functors F : I!C, there is a limit cone over F in C. 1 S = ffinite setsg C has finite products, or is cartesian 2 S = fsmall setsg C has products 3 S = fParg C has equalisers 4 S = ffinite catsg C has finite limits, or is finitely complete 5 S = fsmall catsg C has (small) limits, or is complete 1 S = ffinite setsg C has finite coproducts, or is cocartesian 2 S = fsmall setsg C has coproducts 3 S = fParg C has coequalisers 4 S = ffinite catsg C has finite colimits, or is finitely cocomplete 5 S = fsmall catsg 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 2 S and functors F : I!C, there is a colimit cone under F in C. 2 S = fsmall setsg C has coproducts 3 S = fParg C has coequalisers 4 S = ffinite catsg C has finite colimits, or is finitely cocomplete 5 S = fsmall catsg 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 2 S and functors F : I!C, there is a colimit cone under F in C. 1 S = ffinite setsg C has finite coproducts, or is cocartesian 3 S = fParg C has coequalisers 4 S = ffinite catsg C has finite colimits, or is finitely cocomplete 5 S = fsmall catsg 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 2 S and functors F : I!C, there is a colimit cone under F in C. 1 S = ffinite setsg C has finite coproducts, or is cocartesian 2 S = fsmall setsg C has coproducts 4 S = ffinite catsg C has finite colimits, or is finitely cocomplete 5 S = fsmall catsg 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 2 S and functors F : I!C, there is a colimit cone under F in C. 1 S = ffinite setsg C has finite coproducts, or is cocartesian 2 S = fsmall setsg C has coproducts 3 S = fParg C has coequalisers 5 S = fsmall catsg 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 2 S and functors F : I!C, there is a colimit cone under F in C. 1 S = ffinite setsg C has finite coproducts, or is cocartesian 2 S = fsmall setsg C has coproducts 3 S = fParg C has coequalisers 4 S = ffinite catsg 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 2 S and functors F : I!C, there is a colimit cone under F in C. 1 S = ffinite setsg C has finite coproducts, or is cocartesian 2 S = fsmall setsg C has coproducts 3 S = fParg C has coequalisers 4 S = ffinite catsg C has finite colimits, or is finitely cocomplete 5 S = fsmall catsg C has (small) colimits, or is cocomplete 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 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 2 Ob(I); h 2 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 2 Ob(I); h 2 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 2 Ob(I); h 2 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 i2Ob(I) F (i). If we also have equalisers, Q we can restrict i2Ob(I) F (i) to generalised elements that satisfy equation (1).