LIMITS AND COLIMITS. m : M X. in a category G of structured sets of some sort call them gadgets the image subset

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1 5 LIMITS ND COLIMITS In this chapter we irst briely discuss some topics namely subobjects and pullbacks relating to the deinitions that we already have. This is partly in order to see how these are used, but also because we will need this material soon. Then we approach things more systematically, deining the general notion o a limit, which subsumes many o the particular abstract characterizations we have met so ar. O course, there is a dual notion o colimit, which also has many interesting applications. ter a brie look at one more elementary notion in the next chapter, we shall go on to what may be called higher category theory. 5.1 Subobjects We have seen that every subset U X o a set X occurs as an equalizer and that equalizers are always monomorphisms. So it is natural to regard monos as generalized subsets. That is, a mono in Groups can be regarded as a subgroup, a mono in Top as a subspace, and so on. The rough idea is this: given a monomorphism, m : M X in a category G o structured sets o some sort call them gadgets the image subset {m(y) y M} X which may be written m(m), is oten a sub-gadget o X to which M is isomorphic via m. m : M m(m) X More generally, we can think o the mono m : M X itsel as determining a part o X, even in categories that do not have underlying unctions to take images o. Deinition 5.1. subobject o an object X in a category C is a monomorphism, m : M X.

2 78 LIMITS ND COLIMITS Given subobjects m and m o X, a morphism : m m is an arrow in C/X, as in: M M Thus we have a category, m Sub C (X) m X o subobjects o X in C. In this deinition, since m is monic, there is at most one as in the diagram above, so that Sub C (X) is a preorder category. We deine the relation o inclusion o subobjects by: m m i there exists some : m m Finally, we say that m and m are equivalent, written m m, i and only i they are isomorphic as subobjects, that is, m m and m m. This holds just i there are and making both triangles below commute. M m M m X Observe that, in the above diagram, m = m = m, and since m is monic, = 1 M and similarly = 1 M. So M = M via. Thus we see that equivalent subobjects have isomorphic domains. We sometimes abuse notation and language by calling M the subobject when the mono m : M X is clear. Remark 5.2. It is oten convenient to pass rom the preorder Sub C (X) to the poset given by actoring out the equivalence relation. Then a subobject is an equivalence class o monos under mutual inclusion. In Sets, under this notion o subobject, one then has an isomorphism, Sub Sets (X) = P (X) that is, every subobject is represented by a unique subset. We shall use both notions o subobject, making clear when monos are intended, and when equivalence classes thereo are intended.

3 PULLBCKS 79 Note that i M M then the arrow which makes this so in M M X is also monic, so also M is a subobject o M. Thus we have a unctor Sub(M ) Sub(X) deined by composition with (since the composite o monos is monic). In terms o generalized elements o an object X, z : Z X one can deine a local membership relation, z X M between such elements and subobjects m : M X by z X M i there exists : Z M such that z = m Since m is monic, i z actors through it then it does so uniquely. Example 5.3. n equalizer is a subobject o with the property E z E g i (z) = g(z) B Thus, we can regard E as the subobject o generalized elements z : Z such that (z) = g(z), suggestively: E = {z Z (z) = g(z)} In categorical logic, one develops a way o making this intuition even more precise by giving a calculus o such subobjects. 5.2 Pullbacks The notion o a pullback, like that o a product, is one that comes up very oten in mathematics and logic. It is a generalization o both intersection and inverse image.

4 80 LIMITS ND COLIMITS We begin with the deinition. Deinition 5.4. In any category C, given arrows, g with cod() = cod(g), B g C the pullback o and g consists o arrows P p 2 B p 1 such that p 1 = gp 2 and universal with this property. That is, given any z 1 : Z and z 2 : Z B with z 1 = gz 2, there exists a unique u : Z P with z 1 = p 1 u and z 2 = p 2 u. The situation is indicated in the ollowing diagram. Z...u... z 2 z 1 P p 2 B p 1 g C

5 PULLBCKS 81 One sometimes uses product-style notation or pullbacks. Z z 1, z 2 z 2 z 1 C B p2 B p 1 g C Pullbacks are clearly unique up to isomorphism since they are given by a UMP (universal mapping property). Here, this means that given two pullbacks o a given pair o arrows, the uniquely determined maps between the pullbacks are mutually inverse. In terms o generalized elements, any z C B, can be written uniquely as z = z 1, z 2 with z 1 = gz 2. This makes C B = { z 1, z 2 B z 1 = gz 2 } look like a subobject o B, determined as an equalizer o π 1 and g π 2. In act, this is so. Proposition 5.5. In a category with products and equalizers, given a corner o arrows B g C

6 82 LIMITS ND COLIMITS Consider the diagram E e p 2 p 1 B π 1 B π 2 g C in which e is an equalizer o π 1 and gπ 2 and p 1 = π 1 e, p 2 = π 2 e. Then E, p 1, p 2 is a pullback o and g. Conversely, i E, p 1, p 2 are given as such a pullback, then the arrow is an equalizer o π 1 and gπ 2. Proo. Take e = p 1, p 2 : E B Z z 2 B z 1 with z 1 = gz 2. We have z 1, z 2 : Z B so π 1 z 1, z 2 = gπ 2 z 1, z 2. Thus, there is a u : Z E to the equalizer with eu = z 1, z 2. Then and p 1 u = π 1 eu = π 1 z 1, z 2 = z 1 p 2 u = π 2 eu = π 2 z 1, z 2 = z 2. I also u : Z E has p i u = z i, i = 1, 2, then π i eu = z i so eu = z 1, z 2 = eu whence u = u since e in monic. The converse is similar. Corollary 5.6. I a category C has binary products and equalizers, then it has pullbacks.

7 PULLBCKS 83 The oregoing gives an explicit construction o a pullback in Sets as a subset o the product: { a, b a = gb} = C B B Example 5.7. In Sets, take a unction : B and a subset V B. Let, as usual, and consider 1 (V ) = {a (a) V } 1 (V ) V j i B where i and j are the canonical inclusions and is the evident actorization o the restriction o to 1 (V ) (since a 1 (V ) (a) V ). This diagram is a pullback (observe that z 1 (V ) z V or all z : Z ). Thus, the inverse image 1 (V ) is determined uniquely up to isomorphism as a pullback. s suggested by the previous example, we can use pullbacks to deine inverse images in categories other than Sets. Indeed, given a pullback in any category: B M M m m B i m is monic, then m is monic. (Exercise!) Thus we see that, or ixed : B, taking pullbacks induces a map 1 : Sub(B) Sub() m m We will show that 1 also respects equivalence o subobjects, M N 1 (M) 1 (N) by showing that 1 is a unctor; that is our next goal.

8 84 LIMITS ND COLIMITS 5.3 Properties o pullbacks We start with the ollowing simple lemma, which seems to come up all the time. Lemma 5.8. (Two-pullbacks) Consider the commutative diagram below in a category with pullbacks: F E g D h h h B g C 1. I the two squares are pullbacks, so is the outer rectangle. Thus, B (B C D) = C D 2. I the right square and the outer rectangle are pullbacks, so is the let square. Proo. Diagram chase. Corollary 5.9. The pullback o a commutative triangle is a commutative triangle. Speciically, given a commutative triangle as on the right end o the ollowing prism diagram h α α... γ α γ B h β B C β h C or any h : C C, i one can orm the pullbacks α and β as on the let end, then there exists a unique γ as indicated, making the let end a commutative triangle, and the upper ace a commutative rectangle, and indeed a pullback. Proo. pply the two-pullbacks lemma. β

9 PROPERTIES OF PULLBCKS 85 Proposition Pullback is a unctor. That is, or ixed h : C C in a category C with pullbacks, there is a unctor deined by h : C/C C/C ( α C) (C C α C ) where α is the pullback o α along h, and the eect on an arrow γ : α β is given by the oregoing corollary. Proo. One must check that and h (1 X ) = 1 h X h (g ) = h (g) h () These can easily be veriied by repeated applications o the two-pullbacks lemma. For example, or the irst condition, consider h 1 h 1 α C h α C I the lower square is a pullback, then plainly so is the outer rectangle, whence the upper square is, too, and we have h 1 X = 1 X = 1 h X. Corollary Let C be a category with pullbacks. For any arrow : B in C we have the ollowing diagram o categories and unctors: Sub() 1 Sub(B) C/ C/B

10 86 LIMITS ND COLIMITS This commutes simply because 1 is deined to be the restriction o to the subcategory Sub(B). Thus, in particular, 1 is unctorial: M N 1 (M) 1 (N) It ollows that M N implies 1 (M) 1 (N), so that 1 is also deined on equivalence classes. 1 / : Sub(B)/ Sub()/ Example Consider a pullback in Sets: E B We saw that g can be constructed as an equalizer g C E = { a, b (a) = g(b)} E, g B π 1 gπ 2 C Now let B = 1, C = 2 = {, }, and g = : 1 2. Then the equalizer E 1 π 1 π 2 2 is how we already described the extension o the propositional unction : 2. Thereore we can rephrase the correspondence between subsets U and their characteristic unctions χ U : 2 in terms o pullbacks: U! 1 Precisely, the isomorphism, 2 χ U 2 = P ()

11 PROPERTIES OF PULLBCKS 87 given by taking a unction ϕ : 2 to its extension can be described as a pullback. Now suppose we have any unction V ϕ = {x ϕ(x) = } V ϕ = {x ϕ(x) = } = ϕ 1 ( ) : B and consider the induced inverse image operation 1 : P () P (B) given by pullback, as in example 5.9 above. Taking the extension V ϕ, consider the two-pullback diagram 1 (V ϕ ) V ϕ 1 B We thereore have (by the two-pullbacks lemma) ϕ 2 1 (V ϕ ) = 1 (ϕ 1 ( )) = (ϕ) 1 ( ) = V ϕ which rom a logical point o view expresses the act that the substitution o a term or the variable x in the propositional unction ϕ is modeled by taking the pullback along o the corresponding extension 1 ({x ϕ(x) = }) = {y B ϕ((y)) = }. Note that we have shown that or any unction : B the ollowing square commutes 2 = P () 2 2 B 1 P (B) = where 2 : 2 2 B is precomposition 2 (g) = g. In a situation like this, one says that the isomorphism 2 = P ()

12 88 LIMITS ND COLIMITS is natural in, which is obviously a much stronger condition than just having isomorphisms at each object. We will consider such naturality systematically later. It was in act one o the phenomena that originally gave rise to category theory. Example Let I be an index set, and consider an I-indexed amily o sets: ( i ) i I Given any unction α : J I, there is a J-indexed amily ( α(j) ) j J, obtained by reindexing along α. This reindexing can also be described as a pullback. Speciically, or each set i take the constant, i-valued unction p i : i I and consider the induced map on the coproduct p = [p i ] : i I i I The reindexed amily ( α(j) ) j J can be obtained by taking a pullback along α, as indicated in the ollowing diagram: j J α(j) i I q J α i p I where q is the indexing projection or ( α(j) ) j J analogous to p. In other words, we have J I ( i ) = i I j J α(j) The reader should work out the details as an instructive exercise. 5.4 Limits We have already seen that the notions o product, equalizer, and pullback are not independent; the precise relation between them is this. Proposition category has inite products and equalizers i it has pullbacks and a terminal object. Proo. The only i direction has already been done. For the other direction, suppose C has pullbacks and a terminal object 1.

13 LIMITS 89 For any objects, B we clearly have B = 1 B, as indicated in the ollowing: B B 1 For any arrows, g : B, the equalizer e : E is constructed as the ollowing pullback: E h B e In terms o generalized elements,, g B B E = {(a, b), g (a) = b} where, g (a) = a, ga and (b) = b, b. So, E = { a, b (a) = b = g(a)} = {a (a) = g(a)} = 1 B, 1 B which is just what we want. n easy diagram chase shows that is indeed an equalizer. E e g B Product, terminal object, pullback, and equalizer, are all special cases o the general notion o a limit, which we will consider now. First, we need some preliminary deinitions. Deinition Let J and C be categories. diagram o type J in C is a unctor. D : J C. We will write the objects in the index category J lower case, i, j,... and the values o the unctor D : J C in the orm D i, D j, etc. cone to a diagram D consists o an object C in C and a amily o arrows in C, c j : C D j

14 90 LIMITS ND COLIMITS one or each object j J, such that or each arrow α : i j in J, the ollowing triangle commutes. C c j Dj c i D α morphism o cones D i ϑ : (C, c j ) (C, c j) is an arrow ϑ in C making each triangle, C ϑ C c j D j c j commute. That is, such that c j = c j ϑ or all j J. Thus, we have an evident category o cones to D. Cone(D) We are here thinking o the diagram D as a picture o J in C. cone to such a diagram D is then imagined as a many-sided pyramid over the base D and a morphism o cones is an arrow between the apexes o such pyramids. (The reader should draw some pictures at this point!) Deinition limit or a diagram D : J C is a terminal object in Cone(D). inite limit is a limit or a diagram on a inite index category J. We oten denote a limit in the orm p i : lim j D j D i. Spelling out the deinition, the limit o a diagram D has the ollowing UMP: given any cone (C, c j ) to D, there is a unique arrow u : C lim D j such that j or all j, p j u = c j.

15 LIMITS 91 Thus the limiting cone (lim j D j, p j ) can be thought o as the closest cone to the diagram D, and indeed any other cone (C, c j ) comes rom it just by composing with an arrow at the vertex, namely u : C lim j D j. u C... lim j D j c i p j D i D α D j Example Take J = {1, 2} the discrete category with two objects and no nonidentity arrows. diagram D : J C is a pair o objects D 1, D 2 C. cone on D is an object o C equipped with arrows D 1 c 1 C c 2 D2. nd a limit o D is a terminal such cone, that is, a product in C o D 1 and D 2, Thus, in this case, D 1 p 1 D 1 D 2 p 2 D2. lim D j = D1 D 2. j Example Take J to be the ollowing category: α β diagram o type J looks like and a cone is a pair o arrows D 1 D 1 c 1 D α D β D 2 D α D 2 D β c 2 C such that D α c 1 = c 2 and D β c 1 = c 2 ; thus, D α c 1 = D β c 1. limit or D is thereore an equalizer or D α, D β.

16 92 LIMITS ND COLIMITS Example I J is empty, there is just one diagram D : J C, and a limit or it is thus a terminal object in C, lim j 0 D j = 1. Example I J is the inite category we see that a limit or a diagram o the orm B g is just a pullback o and g, C lim D j = C B. j Thus, we have shown hal o the ollowing: Proposition category has all inite limits i it has inite products and equalizers (resp. pullbacks and a terminal object by the last proposition). Here a category C is said to have all inite limits i every inite diagram D : J C has a limit in C. Proo. We need to show that any inite limit can be constructed rom inite products and equalizers. Take a inite diagram D : J C. s a irst approximation, the product i J 0 D i (5.1) over the set J 0 o objects at least has projections p j : i J 0 D i D j o the right sort. But these can t be expected to commute with the arrows D α : D i D j in the diagram D, as they must. So, as in making a pullback rom a product and an

17 LIMITS 93 equalizer, we consider also the product (α:i j) J 1 D j over all the arrows (the set J 1 ), and two special maps, i D i φ ψ α:i j which record the eect o the arrows in the diagram on the product o the objects. Speciically, we deine φ and ψ by taking their composites with the projections π α rom the second product to be, respectively: π α φ = φ α = π cod(α) D j π α ψ = ψ α = D α π dom(α) where π cod(α) and π dom(α) are projections rom the irst product. Now, in order to get the subobject o the product 5.1 on which the arows in the diagram D commute, we take the equalizer: E e i D i φ ψ α:i j We will show that (E, e i ) is a limit or D, where e i = π i e. To that end, take any arrow c : C i D i, and write c = c i or c i = π i c. Observe that the amily o arrows (c i : C D i ) is a cone to D i and only i φc = ψc. Indeed, i or all α, But, and φ c i = ψ c i π α φ c i = π α ψ c i. D j π α φ c i = φ α c i = π cod(α) c i = c j π α ψ c i = ψ α c i = D α π dom(α) c i = D α c i. Whence φc = ψc i or all α : i j we have c j = D α c i thus, i (c i : C D i ) is a cone, as claimed. It ollows that (E, e i ) is a cone, and that any cone (c i : C D i ) gives an arrow c i : C i D i with φ c i = ψ c i, thus there is a unique actorization u : C E o c i through E, which is clearly a morphism o cones. Since we made no real use o the initeness o the index category apart rom the existence o certain products, essentially the same proo yields the ollowing: Corollary category has all limits o some cardinality i it has all equalizers and products o that cardinality, where C is said to have limits (resp.

18 94 LIMITS ND COLIMITS products) o cardinality κ i C has a limit or every diagram D : J C where card(j 1 ) κ (resp. C has all products o κ many objects). The notions o cones and limits o course dualize to give those o cocones and colimits. One then has the ollowing dual theorem. Theorem category C has inite colimits i it has inite coproducts and coequalizers (resp. i it has pushouts and an initial object). C has all colimits o size κ i it has coequalizers and coproducts o size κ. 5.5 Preservation o limits Here is an application o the construction o limits by products and equalizers. Deinition unctor F : C D is said to preserve limits o type J i, whenever p j : L D j is a limit or a diagram D : J C; the cone F p j : F L F D j is then a limit or the diagram F D : J D. Briely, F (lim D j ) = lim F (D j ). unctor that preserves all limits is said to be continuous. For example, let C be a locally small category with all small limits, such as posets or monoids. Recall the representable unctor Hom(C, ) : C Sets or any object C C, taking : X Y to where (g : C X) = g. : Hom(C, X) Hom(C, Y ) Proposition The representable unctors Hom(C, ) preserve all limits. Since limits in C can be constructed rom products and equalizers, it suices to show that Hom(C, ) preserves products and equalizers. (ctually, even i C does not have all limits, the representable unctors will preserve those limits that do exist; we leave that as an exercise.) Proo. C has a terminal object 1, or which, Hom(C, 1) = {! C } = 1. Consider a binary product X Y in C. Then we already know that, Hom(C, X Y ) = Hom(C, X) Hom(C, Y ) by composing any : C X Y with the two product projections p 1 : X Y X, and p 2 : X Y Y.

19 PRESERVTION OF LIMITS 95 For arbitrary products i I X i one has analogously: Hom(C, X i ) = Hom(C, X i ) i Given an equalizer in C, E consider the resulting diagram, e X i g Y Hom(C, E) e Hom(C, X) g Hom(C, Y ). To show this is an equalizer in Sets, let h : C X Hom(C, X) with h = g h. Then h = gh, so there is a unique u : C E such that eu = h. Thus, we have a unique u Hom(C, E) with e u = eu = h. So e : Hom(C, E) Hom(C, X) is indeed the equalizer o and g. Deinition unctor o the orm F : C op D is called a contravariant unctor on C. Explicitly, such a unctor takes : B to F () : F (B) F () and F (g ) = F () F (g). typical example o a contravariant unctor is a representable unctor o the orm, Hom C (, C) : C op Sets or any C C (where C is any locally small category). Such a contravariant representable unctor takes : X Y to : Hom(Y, C) Hom(X, C) by (g : X C) = g. The dual version o the oregoing proposition is then this: Corollary Contravariant representable unctors map all colimits to limits. For example, given a coproduct X + Y in any locally small category C, there is a canonical isomorphism, Hom(X + Y, C) = Hom(X, C) Hom(Y, C) (5.2) given by precomposing with the two coproduct inclusions. From an example in Section 2.3 we can thereore conclude that the ultrailters in a coproduct + B o Boolean algebras correspond exactly to pairs o

20 96 LIMITS ND COLIMITS ultrailters (U, V ), with U in and V in B. This ollows because we showed there that the ultrailter unctor Ult : B op Sets is representable: Ult(B) = Hom B (B, 2). nother case o the above iso (5.2) is the amiliar law o exponents or sets: C X+Y = C X C Y The arithmetical law o exponents k m+n = k n k m is actually a special case o this! 5.6 Colimits Let us briely discuss some special colimits, since we did not really say much about them in the oregoing section. First, we consider pushouts in Sets. Suppose we have two unctions g C B We can construct the pushout o and g like this. Start with the coproduct (disjoint sum): B B + C C Now identiy those elements b B and c C such that, or some a, (a) = b and g(a) = c That is, we take the equivalence relation on B +C generated by the conditions (a) g(a) or all a. Finally, we take the quotient by to get the pushout (B + C)/ = B + C, which can be imagined as B placed next to C, with the respective parts that are images o pasted together or overlapping. This construction ollows simply by dualizing the one or pullbacks by products and equalizers. Example Pushouts in Top are similarly ormed rom coproducts and coequalizers, which can be made irst in Sets and then topologized as sum and quotient spaces. Pushouts are used e.g. to construct spheres rom disks. Indeed, let D 2 be the (two-dimensional) disk and S 1 the one-dimensional sphere (i.e. the

21 COLIMITS 97 circle), with its inclusion i : S 1 D 2 as the boundary o the disk. Then the 2-sphere S 2 is the pushout, S 1 i D 2 i D 2 S 2. Can you see the analogous construction o S 1 at the next lower dimension? In general, a colimit or a diagram D : J C is o course an initial object in the category o cocones. Explicitly, a cocone rom the base D consists o an object C (the vertex) and arrows c j : D j C or each j J, such that or all α : i j in J, c j D(α) = c i morphism o cocones : (C, (c j )) (C, (c j )) is an arrow : C C in C such that c j = c j or all j J. n initial cocone is the expected thing: one that maps uniquely to any other cocone rom D. We write such a colimit in the orm: lim D j j J Now let us consider some examples o a particular kind o colimit that comes up quite oten, namely over a linearly ordered index category. Our irst example is what is sometimes called a direct limit o a sequence o algebraic objects, say groups. similar construction will work or any sort o algebras (but nonequational conditions are not always preserved by direct limits). Example Direct limit o groups. Suppose we are given a sequence, G 0 g0 G 1 g1 G 2 g2 o groups and homomorphisms, and we want a colimiting group G with homomorphisms u n : G n G satisying u n+1 g n = u n. Moreover, G should be universal with this property. I think you can see the colimit setup here: the index category is the ordinal number ω = (N, ), regarded as a poset category, the sequence G 0 g0 G 1 g1 G 2 g2

22 98 LIMITS ND COLIMITS is a diagram o type ω in the category Groups, the colimiting group is the colimit o the sequence: G = lim G n n ω This group always exists, and can be constructed as ollows. Begin with the coproduct (disjoint sum) o sets G n. n ω Then make identiications x n y m, where x n G n and y m G m, to ensure in particular that x n g n (x n ) or all x n G n and g n : G n G n+1. This means, speciically, that the elements o G are equivalence classes o the orm [x n ], x n G n or any n, and [x n ] = [y m ] i or some k m, n, where, generally, i i j, we deine g n,k (x n ) = g m,k (y m ) g i,j : G i G j by composing consecutive g s as in g i,j = g j 1... g i. The reader can easily check that this is indeed the equivalence relation generated by all the conditions x n g n (x n ). The operations on G are now deined by [x] [y] = [x y ] where x x, y y, and x, y G n or n suiciently large. The unit is just [u 0 ], and we take, [x] 1 = [x 1 ]. One can easily check that these operations are well deined, and determine a group structure on G, which moreover makes all the evident unctions u n : G n G, u n (x) = [x] into homomorphisms. The universality o G and the u n results rom the act that the construction is essentially a colimit in Sets, equipped with an induced group structure. Indeed, given any group H and homomorphisms h n : G n H with h n+1 g n = h n deine

23 COLIMITS 99 h : G H by h ([x n ]) = h n (x n ). This is easily seen to be well deined and indeed a homomorphism. Moreover, it is the unique unction that commutes with all the u n. The act that the ω-colimit G o groups can be constructed as the colimit o the underlying sets is a case o a general phenomenon, expressed by saying that the orgetul unctor U : Groups Sets creates ω-colimits. Deinition unctor F : C D is said to create limits o type J i or every diagram C : J C and limit p j : L F C j in D there is a unique cone p j : L C j in C with F (L) = L and F (p j ) = p j, which, urthermore, is a limit or C. Briely, every limit in D is the image o a unique cone in C, which is a limit there. The notion o creating colimits is deined analogously. In these terms, then, we have the ollowing proposition, the remaining details o which have in eect already been shown. Proposition The orgetul unctor U : Groups Sets creates ω- colimits. It also creates all limits. The same act holds quite generally or other categories o algebraic objects, that is, sets equipped with operations satisying some equations. Observe that not all colimits are created in this way. For instance, we have already seen (in example ) that the coproduct o two abelian groups has their product as underlying set. Example Cumulative hierarchy. nother example o an ω-colimit is the cumulative hierarchy construction encountered in set theory. Let us set V 0 = V 1 = P( ) V n+1 = P(V n ) Then there is a sequence o subset inclusions, = V 0 V 1 V 2 since, generally, B implies P() P(B) or any sets and B. The colimit o the sequence. V ω = lim V n n is called the cumulative hierarchy o rank ω. One can o course continue this construction through higher ordinals ω + 1, ω + 2,.... More generally, let us start with some set (o atoms ), and let V 0 () =

24 100 LIMITS ND COLIMITS and then put V n+1 () = + P(V n ()), i.e. the set o all elements and subsets o. There is a sequence V 0 () V 1 () V 2 ()... as ollows. Let v 0 : V 0 () = + P() = V 1 () be the let coproduct inclusion. Given v n 1 : V n 1 () V n (), let v n : V n () V n+1 () be deined by v n = 1 + P! (v n 1 ) : + P(V n 1 ()) + P(V n ()) where P! denotes the covariant powerset unctor, taking a unction : X Y to the image under operation P! () : P(X) P(Y ), deined by taking U X to P! ()(U) = {(u) u U} Y. The idea behind the sequence is that we start with, then add all the subsets o, then add all the new subsets that can be ormed rom all o those elements, and so on. The colimit o the sequence V ω () = lim V n () n is called the cumulative hierarchy (o rank ω) over. O course, V ω = V ω ( ). Now suppose we have some unction Then there is a map : B. V ω () : V ω () V ω (B), determined by the colimit description o V ω, as indicated in the ollowing diagram. V 0 () V1 () V2 ()... Vω () 0 V 0 (B) V 1 (B) V 2 (B)... V ω (B) ω Here the n are deined by 0 = : B, 1 = + P! () : + P() B + P(B),. n+1 = + P! ( n ) : + P(V n ()) B + P(V n (B)).

25 COLIMITS 101 Since all the squares clearly commute, we have a cocone on the diagram o V n () s with vertex V ω (B), and there is thus a unique ω : V ω () V ω (B) that completes the diagram. Thus we see that the cumulative hierarchy is unctorial. Example ωcpos. n ωcpo is a poset that is ω-cocomplete, meaning it has all colimits o type ω = (N, ). Speciically, a poset D is an ωcpo i or every diagram d : ω D, i.e. every chain o elements o D, d 0 d 1 d 2 we have a colimit d ω = lim d n. This is an element o D such that: 1. d n d ω or all n ω; 2. or all x D, i d n x or all n ω, then also d ω x. monotone map o ωcpos h : D E is called continuous i it preserves colimits o type ω, that is, h(lim d n ) = lim h(d n ). n application o these notions is the ollowing: Proposition I D is an ωcpo with initial element 0 and h : D D is continuous, then h has a ixed point h(x) = x which, moreover, is least among all ixed points. Proo. We use Newton s method, which can be used, or example, to ind ixed points o monotone, continuous unctions : [0, 1] [0, 1]. Consider the sequence d : ω D, deined by d 0 = 0 d n+1 = h(d n )

26 102 LIMITS ND COLIMITS Since 0 d 0, repeated application o h gives d n d n+1. Now take the colimit d ω = lim n ω d n. Then h(d ω ) = h(lim n ω = lim n ω = lim n ω = d ω. d n ) h(d n ) d n+1 The last step ollows because the irst term d 0 = 0 o the sequence is trivial. Moreover, i x is also a ixed point, h(x) = x, then we have d 0 = 0 x So also d ω x, since d ω is the colimit. d 1 = h(0) h(x) = x. d n+1 = h(d n ) h(x) = x. Finally, here is an example o how (co)limits depend on the ambient category. We consider colimits o posets and ωcpos, rather than in them. Let us deine the inite ωcpos ω n = {k n k ω} then we have continuous inclusion maps: ω 0 ω 1 ω 2 In Pos, the colimit exists, and is ω, as can be easily checked. But ω itsel is not ω-complete. Indeed, the sequence has no colimit. So the colimit o the ω n in the category o ωcpos, i it exists, must be something else. In act it is ω ω For then any bounded sequence has a colimit in the bounded part, and any unbounded one has ω as colimit. The moral is that even ω-colimits are not always created in Sets, and indeed the colimit is sensitive to the ambient category in which it is taken.

27 EXERCISES Exercises 1. Show that a pullback o arrows X B p2 B p 1 g X in a category C is the same thing as their product in the slice category C/X. 2. Let C be a category with pullbacks. (a) Show that an arrow m : M X in C is monic i and only i the diagram below is a pullback. M 1M M 1 M M m m X Thus as an object in C/X, m is monic i m m = m. (b) Show that the pullback along an arrow : Y X o a pullback square over X, X B B X is again a pullback square over Y. (Hint: draw a cube and use the 2-pullbacks Lemma). Conclude that the pullback unctor preserves products.

28 104 LIMITS ND COLIMITS (c) Conclude rom the oregoing that in a pullback square M M m i m is monic, then so is m. m 3. For any object in a category C and any subobjects M, N Sub C (), show M N i or every generalized element z : Z (arbitrary arrow with codomain ): z M implies z N. 4. Show that in any category, given a pullback square M M m i m is monic, then so is m. m 5. For any object in a category C and any subobjects M, N Sub C (), show M N i or every generalized element z : Z (arbitrary arrow with codomain ): z M implies z N. 6. (Equalizers by pullbacks and products) Show that a category with pullbacks and products has equalizers as ollows: given arrows, g : B, take the pullback indicated below, where = 1 B, 1 B : E B e, g B B Show that e : E is the equalizer o and g. 7. Let C be a locally small category, and D : J C any diagram or which a limit exists in C. Show that or any object C C, the representable unctor Hom C (C, ) : C Sets

29 EXERCISES 105 preserves the limit o D. 8. (Partial maps) For any category C with pullbacks, deine the category Par(C) o partial maps in C as ollows: the objects are the same as those o C, but an arrow : B is a pair (, U ) where U is a subobject and : U B is a suitable equivalence class o arrows, as indicated in the diagram: U B Composition o (, U ) : B and ( g, U g ) : B C is given by taking a pullback and then composing to get ( g, (U g )), as suggested by the ollow diagram. (U g ) Ug g C U B Veriy that this really does deine a category, and show that there is a unctor, which is the identity on objects. C Par(C) 9. Suppose the category C has limits o type J, or some index category J. For diagrams F and G o type J in C, a morphism o diagrams θ : F G consists o arrows θ i : F i Gi or each i J such that or each α : i j in J, one has θ j F (α) = G(α)θ i (a commutative square). This makes Diagrams(J, C) into a category (check this). Show that taking the vertex-objects o limiting cones determines a unctor: lim J : Diagrams(J, C) C

30 106 LIMITS ND COLIMITS Iner that or any set I, there is a product unctor, : Sets I Sets i I or I-indexed amilies o sets ( i ) i I. 10. (Pushouts) (a) Dualize the deinition o a pullback to deine the copullback (usually called the pushout ) o two arrows with common domain. (b) Indicate how to construct pushouts using coproducts and coequalizers (proo by duality ). 11. Let R X X be an equivalence relation on a set X, with quotient q : X Q. Show that the ollowing is an equalizer, PQ Pq PX Pr 1 Pr 2 PR, where r 1, r 2 : R X are the two projections o R X, and P is the (contravariant) powerset unctor. (Hint: PX = 2 X.) 12. Consider the sequence o posets [0] [1] [2]..., where [n] = {0 n}, and the arrows [n] [n + 1] are the evident inclusions. Determine the limit and colimit posets o this sequence. 13. Consider sequences o monoids, M 0 M 1 M 2... N 0 N 1 N 2... and the ollowing limits and colimits, constructed in the category o monoids: lim M n, lim M n, lim N n, lim N n. n n n n (a) Suppose all M n and N n are abelian groups. Determine whether each o the our (co)limits lim n M n etc. is also an abelian group. (b) Suppose all M n and N n are inite groups. Determine whether each o the our (co)limits lim n M n etc. has the ollowing property: or every element x there is a number k such that x k = 1 (the least such k is called the order o x).