Chapter 3. Limits and Colimits (pp. 73–114)
FC03 — Riehl, Category Theory in Context.
This chapter contains 19 definitions, 10 theorems, 12 propositions, 7 lemmas, 3 corollaries, 31 examples, 2 constructions, 6 remarks, 1 convention and 2 other statements.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition: For c∈C, the constant functor c:J→C is constant on c and 1c
For c∈C, the constant functor c:J→C is constant on c and 1_c; these assemble into the diagonal embedding Δ:C→C^J.
Definition: A cone over F:J→C with summit c is a natural transformation c⇒F with legs c→Fj
A cone over F:J→C with summit c is a natural transformation c⇒F with legs c→Fj; a cocone/cone under F with nadir c is F⇒c.
Definition: A limit of F represents Cone(-,F):C^op→Set, with universal cone lim F⇒F;…
A limit of F represents Cone(-,F):C^op→Set, with universal cone lim F⇒F; dually a colimit represents Cone(F,-), with universal cocone F⇒colim F.
Definition: Equivalently, a limit is terminal in the category of cones over F and a…
Equivalently, a limit is terminal in the category of cones over F and a colimit initial in the category of cocones under F; cone morphisms are maps of summits/nadirs commuting with legs.
Convention/notation
Write lim_J F and colim_J F when the indexing shape matters; “limit cone” or “colimit cone” includes the universal cone data.
Proposition: Any two limit cones over one diagram have a unique isomorphism between…
Any two limit cones over one diagram have a unique isomorphism between summits commuting with all legs; dually for colimit cones.
Remark: A limit diagram is F together with its limit cone and is indexed by J with a…
A limit diagram is F together with its limit cone and is indexed by J with a freely adjoined initial object; dually a colimit diagram adjoins a terminal object.
Definition: A product is a limit of a discrete diagram
A product is a limit of a discrete diagram; projections π_k:∏_jF_j→F_k satisfy C(c,∏F_j)≅∏_k C(c,F_k).
Example: Topological products are categorical products with the coarsest topology…
Topological products are categorical products with the coarsest topology making projections continuous; for infinite families the box topology generally fails the universal property.
Definition: A terminal object is the nullary product, i.e
A terminal object is the nullary product, i.e. the limit of the empty diagram.
Example: The category 1 is terminal in Cat and CAT.
The category 1 is terminal in Cat and CAT.
Definition: The equalizer of f,g:A⇉B is the universal h:E→A with fh=gh
The equalizer of f,g:A⇉B is the universal h:E→A with fh=gh; every other equalizing a:C→A factors uniquely through h.
Example: For groups, ker φ is the equalizer of φ and the trivial homomorphism
For groups, ker φ is the equalizer of φ and the trivial homomorphism; Eq(φ,ψ)={g|φg=ψg}, and for abelian H it is ker(φ-ψ).
Definition: A pullback/fiber product B×A C is the universal commutative square over a…
A pullback/fiber product B×_A C is the universal commutative square over a cospan B→A←C; pulling back a generalized element defines a fiber.
Example: For ρ:R→S¹, t↦e^{2πit}, the fiber/pullback over 1∈S¹ is the discrete…
For ρ:R→S¹, t↦e^{2πit}, the fiber/pullback over 1∈S¹ is the discrete subspace Z⊂R.
Example: In Ab, the pullback of multiplication-by-m and multiplication-by-n on Z…
In Ab, the pullback of multiplication-by-m and multiplication-by-n on Z consists of (x,y) with nx=my and is Z with legs determined by an lcm when m,n are not both zero.
Definition: An inverse limit is the limit of a tower indexed by ω^op (more generally a…
An inverse limit is the limit of a tower indexed by ω^op (more generally a limit ordinal opposite); “inverse limit” is sometimes used for arbitrary limits.
Example: The p-adic integers are the inverse limit Zp=limn Z/p^n along canonical…
The p-adic integers are the inverse limit Z_p=lim_n Z/p^n along canonical quotient maps.
Definition: Dual limit notions
Dual limit notions: coproduct (discrete colimit), initial object (empty colimit), coequalizer, pushout, and sequential/direct colimit (ω-indexed colimit), with coproduct injections and the stated universal properties.
Example: S¹∨S¹ is a pushout in Top (and coproduct in based spaces)
S¹∨S¹ is a pushout in Top (and coproduct in based spaces); attaching D² by the commutator loop via pushout yields the torus.
Example: The cokernel of a group homomorphism φ:G→H is the coequalizer of φ and the…
The cokernel of a group homomorphism φ:G→H is the coequalizer of φ and the trivial map, i.e. H modulo the normal closure of im φ.
Example: A sequential colimit of nested sets is their union
A sequential colimit of nested sets is their union; a CW complex is the colimit of its skeleta.
Remark: Maps out of a coproduct and into a product are naturally families of…
Maps out of a coproduct and into a product are naturally families of component maps; hence a map coproduct→product is a matrix of component maps. In Ab/Mod_R/Ch_R finite coproducts and products are canonically isomorphic direct sums, with composition by matrix multiplication.
Definition: A diagram is small when its indexing category is small
A diagram is small when its indexing category is small; C is complete if it has all small limits and cocomplete if it has all small colimits.
Definition: For small F:J→Set, define lim F=Cone(1,F)
For small F:J→Set, define lim F=Cone(1,F); its j-th projection sends a cone μ:1⇒F to its leg μ_j.
Remark: Smallness of J ensures Cone(1,F) is a set because Set^J is locally small.
Smallness of J ensures Cone(1,F) is a set because Set^J is locally small.
Theorem: Set is complete
Set is complete: the construction lim F=Cone(1,F) is universal for every small set-valued diagram.
Example: Products in Set are ordinary cartesian products of indexed families.
Products in Set are ordinary cartesian products of indexed families.
Example: The terminal object in Set is any singleton set.
The terminal object in Set is any singleton set.
Example: The equalizer of f,g:X⇉Y in Set is {x∈X | f(x)=g(x)}.
The equalizer of f,g:X⇉Y in Set is {x∈X | f(x)=g(x)}.
Example: The inverse limit of a tower of sets is the set of coherent sequences (xn)…
The inverse limit of a tower of sets is the set of coherent sequences (x_n) with transition equations f_{n,n-1}(x_n)=x_{n-1}.
Example: The pullback in Set is B×A C={(b,c)∈B×C | f(b)=g(c)}.
The pullback in Set is B×_A C={(b,c)∈B×C | f(b)=g(c)}.
Example: For a left G-set X:BG→Set, lim X is the fixed-point set X^G.
For a left G-set X:BG→Set, lim X is the fixed-point set X^G.
Construction/comparison
A pullback B×_A C in Set is the equalizer of the two maps B×C⇉A induced by f and g.
Theorem: Every small set-valued limit is an equalizer ∏{j∈ob J}Fj ⇉ ∏{f∈mor J}F(cod…
Every small set-valued limit is an equalizer ∏{j∈ob J}Fj ⇉ ∏{f∈mor J}F(cod f), comparing λ_cod(f) with Ff(λ_dom(f)).
Remark: The two maps in Theorem 3.2.13 are defined categorically componentwise
The two maps in Theorem 3.2.13 are defined categorically componentwise: one by codomain projections and one by domain projections followed by Ff.
Example: For idempotent e:A→A in Set, lim e=A^e=Eq(1,e)
For idempotent e:A→A in Set, lim e=A^e=Eq(1,e); the factorization e=sr splits e with rs=1, and conversely every retract yields a split idempotent.
Definition: For a specified class of diagrams, F:C→D preserves limits by sending limit…
For a specified class of diagrams, F:C→D preserves limits by sending limit cones to limit cones, reflects them by detecting limit cones, and creates them by lifting codomain limits to domain limits while reflecting the class; dual definitions apply to colimits.
Proposition: If F creates a class of limits and D has them, then C has them and F…
If F creates a class of limits and D has them, then C has them and F preserves them; dually for colimits.
Definition: A presheaf F:O(X)^op→Set is a sheaf when it carries every open-cover colimit…
A presheaf F:O(X)^op→Set is a sheaf when it carries every open-cover colimit diagram to a limit; equivalently the usual restriction diagram is an equalizer.
Lemma: Every full and faithful functor reflects all limits and colimits that exist…
Every full and faithful functor reflects all limits and colimits that exist in its codomain.
Lemma: Every equivalence of categories preserves, reflects, and creates all limits…
Every equivalence of categories preserves, reflects, and creates all limits and colimits present in either category.
Definition: F strictly creates a class of limits when every limit cone over FK has a…
F strictly creates a class of limits when every limit cone over FK has a unique lifted cone over K and that lift is limiting; dual for colimits.
Definition: An indexing category is connected when nonempty and every two objects are…
An indexing category is connected when nonempty and every two objects are linked by a finite zig-zag; a diagram is connected when its index is.
Proposition: For c∈C, Π:c/C→C strictly creates all limits admitted by C and all connected…
For c∈C, Π:c/C→C strictly creates all limits admitted by C and all connected colimits admitted by C.
Proposition: For small A, CA→C{ob A} strictly creates all limits and colimits existing…
For small A, CA→C{ob A} strictly creates all limits and colimits existing in C; they are computed objectwise, so every evaluation ev_a preserves them.
Theorem: If F:J→C has a limit in locally small C, then C(X,limJ F)≅limJ C(X,F-),…
If F:J→C has a limit in locally small C, then C(X,lim_J F)≅lim_J C(X,F-), naturally in X.
Example: The I-fold product A^I is a power/cotensor, characterized by C(X,AI)≅C(X,A)I.
The I-fold product A^I is a power/cotensor, characterized by C(X,AI)≅C(X,A)I.
Proposition: Every covariant representable C(X,-) preserves all limits that exist in…
Every covariant representable C(X,-) preserves all limits that exist in locally small C.
Theorem: Covariant representables preserve existing limits
Covariant representables preserve existing limits; the covariant Yoneda embedding C→Set{Cop} preserves and reflects limits.
Theorem: If F:J→C has a colimit, then C(colimJ F,X)≅lim{J^op} C(F-,X), naturally in X.
If F:J→C has a colimit, then C(colim_J F,X)≅lim_{J^op} C(F-,X), naturally in X.
Example: Coproducts satisfy C(X⊔Y,Z)≅C(X,Z)×C(Y,Z).
Coproducts satisfy C(X⊔Y,Z)≅C(X,Z)×C(Y,Z).
Example: The I-fold coproduct I·A is a copower/tensor, characterized by…
The I-fold coproduct I·A is a copower/tensor, characterized by C(I·A,X)≅C(A,X)^I; in Set it is I×A.
Example: Applying C(-,X) to a pushout produces a pullback of hom-sets, giving the…
Applying C(-,X) to a pushout produces a pullback of hom-sets, giving the representable universal property of pushouts.
Theorem: Contravariant representables C(-,X) send colimits to limits in Set
Contravariant representables C(-,X) send colimits to limits in Set; the contravariant Yoneda embedding Cop→SetC preserves and reflects limits in C^op, equivalently detects colimit cones in C.
Theorem: Every small colimit is a coequalizer of a pair of maps between coproducts
Every small colimit is a coequalizer of a pair of maps between coproducts; hence coproducts+coequalizers imply cocompleteness. Dually, products+equalizers imply completeness.
Remark: A generalized element of A is a morphism X→A. Yoneda implies morphisms are…
A generalized element of A is a morphism X→A. Yoneda implies morphisms are equal iff they agree on all generalized elements, and a cone is limiting iff all represented hom-set cones are limits in Set.
Lemma: In any category with terminal object 1, the pullback of A→1←B is the product…
In any category with terminal object 1, the pullback of A→1←B is the product A×B.
Lemma: In a category with binary products, Eq(f,g) is obtained as the pullback of…
In a category with binary products, Eq(f,g) is obtained as the pullback of (f,g):A→B×B along the diagonal (1_B,1_B):B→B×B.
Definition: A diagram is finite when its indexing category has finitely many morphisms.
A diagram is finite when its indexing category has finitely many morphisms.
Theorem: Pullbacks plus a terminal object imply all finite limits
Pullbacks plus a terminal object imply all finite limits; dually pushouts plus an initial object imply all finite colimits.
Proposition: Set is cocomplete
Set is cocomplete: coproducts are disjoint unions and coequalizers are quotients by the equivalence relation generated by f(a)~g(a).
Proposition: Top is complete and cocomplete
Top is complete and cocomplete; limits/colimits have the set-theoretic carrier with respectively initial/coarsest or final/finest topology making cone legs continuous, and U:Top→Set preserves them.
Example: Topological (co)limits are formed on underlying-set (co)limits with the…
Topological (co)limits are formed on underlying-set (co)limits with the coarsest/finest leg-continuity topology; the inverse limit of p-th power maps S¹→S¹ is the p-adic solenoid.
Example: PConfn(X)⊂X^n is the ordered configuration space
PConf_n(X)⊂X^n is the ordered configuration space; Conf_n(X) is the colimit/orbit space of the Σ_n-action on PConf_n(X).
Proposition: If C is complete and cocomplete, then every slice c/C and C/c is complete…
If C is complete and cocomplete, then every slice c/C and C/c is complete and cocomplete.
Proposition: Cat and CAT are complete
Cat and CAT are complete; products are objectwise products of categories and equalizers are subcategories on objects/arrows where parallel functors agree.
Example: For F:C→Set, the category of elements ∫F is the pullback in CAT of C→Set←Set.
For F:C→Set, the category of elements ∫F is the pullback in CAT of C→Set←Set_*.
Example: In Cat, the pushout identifying source and target of the walking arrow gives…
In Cat, the pushout identifying source and target of the walking arrow gives the free one-object category on one endomorphism, i.e. monoid N; replacing the arrow by the walking isomorphism gives group Z.
Comparison
Completeness and cocompleteness are invariant under equivalence of categories.
Proposition: A poset is complete and cocomplete as a category iff it is a complete lattice
A poset is complete and cocomplete as a category iff it is a complete lattice; limits are infima and colimits suprema.
Example: The kernel pair of f:X→Y is its pullback along itself
The kernel pair of f:X→Y is its pullback along itself. It defines an internal equivalence relation R↪X×X with reflexivity, symmetry, and transitivity maps; its coequalizer is the quotient object when it exists.
Proposition: If C has all J-shaped limits, choosing one limit cone for each diagram…
If C has all J-shaped limits, choosing one limit cone for each diagram defines a functor lim_J:C^J→C; dually chosen colimits define colim_J.
Example: The path-components functor π₀:Top→Set is the composite sending X to…
The path-components functor π₀:Top→Set is the composite sending X to Path(X)⇉Point(X) by endpoint evaluations and then taking the coequalizer.
Corollary: A natural isomorphism between diagrams induces a naturally defined…
A natural isomorphism between diagrams induces a naturally defined isomorphism between their limits or colimits whenever those exist.
Example: Objectwise-isomorphic G-sets need not be naturally/equivariantly isomorphic…
Objectwise-isomorphic G-sets need not be naturally/equivariantly isomorphic and can have nonisomorphic fixed-point limits and orbit colimits.
Example: Sym(X) and Ord(X) define species Finiso→Fin with equal objectwise…
Sym(X) and Ord(X) define species Fin_iso→Fin with equal objectwise cardinalities but no natural isomorphism; their orbit colimits (unlabeled structures) differ.
Comparison
Chosen limits are functorial but generally noncanonical: even canonical pullbacks cannot be chosen to make both horizontal and vertical pullback pasting strictly canonical.
Lemma: Binary products are naturally associative
Binary products are naturally associative: there is a unique natural isomorphism X×(Y×Z)≅(X×Y)×Z commuting with projections.
Example: Isbell example
Isbell example: in a skeletal category of sets, product objects can be literally equal while the canonical associativity isomorphism is not the identity.
Lemma: The identity functor 1C:C→C has a limit iff C has an initial object.
The identity functor 1_C:C→C has a limit iff C has an initial object.
Definition: The cardinality of a small category is the cardinality of its morphism set
The cardinality of a small category is the cardinality of its morphism set; κ-small means cardinality <κ, and a κ-small diagram has κ-small index.
Proposition: Freyd
Freyd: any κ-small category admitting all κ-small limits, or all κ-small colimits, is a preorder.
Theorem: For F:I×J→C, whenever both iterated limits exist they are canonically…
For F:I×J→C, whenever both iterated limits exist they are canonically isomorphic to each other and to lim_{I×J}F; dually, iterated colimits commute.
Corollary/comparison
For any chosen limit/colimit functors, the commuting-of-iterated-(co)limits isomorphisms are natural.
Lemma: For F:I×J→C, whenever displayed objects exist there is a canonical…
For F:I×J→C, whenever displayed objects exist there is a canonical comparison κ:colim_i lim_j F(i,j)→lim_j colim_i F(i,j).
Corollary: For f:X×Y→R, whenever the extrema exist, supx infy f(x,y) ≤ infy supx f(x,y).
For f:X×Y→R, whenever the extrema exist, sup_x inf_y f(x,y) ≤ inf_y sup_x f(x,y).
Remark: The usual minimax inequality proof is exactly the universal-property…
The usual minimax inequality proof is exactly the universal-property construction of the comparison in Lemma 3.8.3, using inf_y f(x,y)≤f(x,y)≤sup_x f(x,y).
Example: The extended real line is complete/cocomplete as a poset, with limits=infima…
The extended real line is complete/cocomplete as a poset, with limits=infima and colimits=suprema; liminf=colim_n lim_m x_{n+m}, limsup=lim_n colim_m x_{n+m}, hence liminf≤limsup.
Definition: A category J is filtered when every finite diagram in J admits a cone under it.
A category J is filtered when every finite diagram in J admits a cone under it.
Example: Every ordinal category, and the one-object category with one nonidentity…
Every ordinal category, and the one-object category with one nonidentity idempotent, is filtered; the parallel-pair category, a span category, and nontrivial discrete categories are not.
Theorem: Filtered colimits commute with finite limits in Set.
Filtered colimits commute with finite limits in Set.
Construction: For small filtered J and F:J→Set, colim F is the coproduct modulo eventual…
For small filtered J and F:J→Set, colim F is the coproduct modulo eventual equality: x∈Fj and y∈Fk agree iff they map to a common element in some Ft; finite compatible families can be represented simultaneously at one stage.