GENERAL ABSTRACT NONSENSE

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1 GENERAL ABSTRACT NONSENSE MARCELLO DELGADO Abstract. In this paper, we seek to understand limits, a uniying notion that brings together the ideas o pullbacks, products, and equalizers. To do this, we will build up the basic ramework o category theory, starting rom the deinition o a category. With this done, we will deine pullbacks, products, and equalizers, and we will close this paper by showing two results: irst, that having products and equalizers is equivalent to having pullbacks and a terminal object, and second, that having all inite limits is equivalent to having products and equalizers o all cardinalities. Contents 1. The Basics Deinitions Constructions Duality Abstract Structures 6 2. Studying Objects in a Category Elements and Arrows Projective Objects Subobjects The Three Faces o Limits Products Equalizers Pullbacks Equivalences Limits Basic Deinitions Plateau Climax Resolution 22 Acknowledgments 22 Reerences The Basics 1.1. Deinitions. Category theory grew out o a generalization o abstract algebra. Thus, most o the deinitions we will see are analogous to ones in algebra. We start by deining a category. Date: DEADLINE AUGUST 22,

2 2 MARCELLO DELGADO Deinition 1.1 (Category). A category C consists o the ollowing data A collection o objects A, B, C, denoted C 0. A collection o arrows, g, h, denoted C 1 For every arrow C 1, there exist objects dom(), cod() C 0, and we write : dom() cod(). Given arrows : A B, g : B C, i.e. cod() = dom(g), there exists an arrow g C 1 such that g : A C. For all objects A, there exists an identity arrow Id A : A A These data are required to satisy the ollowing two properties: Associativity: For all arrows : A B, g : B C, and h : C D, h (g ) = (h g), i.e. the ollowing diagram commutes: A B h g g g C h D Unit: For all arrows : A B, Id A = = Id B, i.e. the ollowing commutes: A Id A A B IdB B Remark 1.2. We denote the collection o all arrows rom an object A to an object B in a category C as C(A, B). Remark 1.3. Assuming that an object A has two identity arrows, by the unit property we have that Id A = Id A Id A = Id A, which shows that the identity arrow must be unique. Deinition 1.4 (Isomorphism). An arrow : A B is an iso(morphism) i there exists an arrow g : B A such that g = Id A and g = Id B. In algebra, we have structure-preserving maps, like group homomorphisms or groups and linear transormations or vector spaces. Similarly, we have structurepreserving maps or categories. Deinition 1.5 (Functor). A unctor is a map F : C D between categories C and D that sends objects to objects (i.e. F 0 : C 0 D 0 ) and arrows to arrows (i.e. F 1 : C 1 D 1 ) such that F ( : A B) = F : F A F B F (g ) = F g F F (Id A ) = Id F A It s not hard to see that or each category there is a canonical identity unctor, Id C which takes every arrow and object to itsel and thus trivially satisies the structure-preserving properties o a unctor. It s helpul to seem some examples o categories and unctors at this point.

3 GENERAL ABSTRACT NONSENSE 3 Example 1.6. (1) The category Sets is the category which has as its objects all sets and Sets(X, Y ) = Y X. Composition o arrows is simply the composition o unctions, and the identity arrow or any set is simply its identity unction. (2) The category Grp is the category which has as its objects all groups and Grp(G, H) is the collection o all group homomorphisms rom G to H ora any two groups. (3) Two ubiquitous unctors in category theory are the identity unctor, describes above, and the orgetul unctor, U. Generally, the orgetul unctor takes a category o structured sets and their unctions (e.g homomorphisms) to Sets and sends each object in that category to its underlying set and each homomorphism to its underlying unction o sets. (4) The category Cat is the category where the objects are all locally small categories and the arrows are all the unctors between them. (5) Consider a monoid M. Then M is an single object category where the elements are arrows rom the object to itsel and the composition o arrows is the product o the corresponding elements, i.e. mn = m n, and the unit element u M is the identity arrow Id. Further, any monoid homomorphism h : M N is a unctor between their respective categories. (6) Similarly, since a group is a monoid with inverse elements, then a group is equivalent to a single element category where every arrow is an isomorphism. (7) A poset can also be considered a be considered a category where the elements o the poset are the objects o the category and there is a unique arrow : a b i a b. This should not be conused with Pos, the category o posets where the objects are posets and the arrows are monotonic unctions. (8) Let an ordered pair (A, a) be called a pointed set, where A is a set and a A. Then we can consider a category o pointed sets, where the objects are pointed sets and an arrow : (A, a) (B, b) is a unction : A B such that (a) = b. We denote this category Sets (9) For a collection o sets, Rel 0, deine an arrow : A B as a relation rom A to B, i.e. such that A B. The identity relation on a set A is Id A = { a, a A A a A}. Composition or relations A B and g B C is deined as g = { a, c A C b B such that a, b and b, c g}. I we denote the collection o relations among sets in Rel 0 as Rel 1, then Rel 0 and Rel 1 orm a category, Rel. The notion o a category is rather lexible and can come in a wide variety o lavors. Despite the very abstract eel o most categories, it can be shown that any category is isomorphic to one in which the objects are sets and the arrows are unctions. Theorem 1.7. Every category C is isomorphic to one in which the objects are sets and the arrows are unctions The proo ollows rom something akin to the Cayley representation theorem o groups. For a given category C, deine a collection o sets C = { arrows C 1 cod() = C} or every C C 0. Then or all arrows : C D, deine a unction

4 4 MARCELLO DELGADO : C D such that (h) = h. One can check that this collection o sets and unctions is a category, where Id C = Id C Constructions. Having established what categories are and seen some examples o them in nature, we can discuss some categories we can construct rom ones we already have. Deinition 1.8 (Dual Category). For a given category C, deine the dual category o C, denoted C op, such that C 0 = C op 0 and ( : A B) C op 1 i and only i ( : B A) C 1. Intuitively, the dual category is exactly the same as C, except all the arrows have been ormally turned around. What this means is that or arrows : X Y and g : Y Z in C, g = g in C op. A B g C g A B Deinition 1.9 (Product Category). For categories C and D, we deine C D, the product category o C and D, as the category with the ollowing data: (C D) 0 = {(C, D) C C 0, D D 0 } (C D) 1 = {(, g) : (C, D) (C, D ) C(C, D), g D(C, D )} composition is deined component-wise: (, g) (, g ) = (, g g ) Id (C,D) = (Id C, Id D ) g There are two canonical projection unctors here: π 1 C C D π 2 D deined such that π 1 (C, D) = C and π 1 (, g) =, with π 2 deined analogously. Deinition 1.10 (Arrow Cateogory). Given a category C, deine C such that: C 0 = C 1 An arrow g = g 1, g 2 : ( : A B) ( : A B ) in C is a pair o arrows g 1 : A A and g 2 : B B such that g 2 = g 1. Diagrammatically: A g1 A B g2 B Id = Id A, Id B or : A B. For arrows h = h 1, h 2 and g = g 1, g 2 g h = g 1 h 1, g 2 h 2, which Diagrammatically looks like: A h1 A g 1 A B h2 B g 2 B C g

5 GENERAL ABSTRACT NONSENSE 5 This construction has two canonical unctors as well: dom and cod. Deinition 1.11 (Slice Category). The slice category o a category C over an object C in C, denoted C/C, is given by the ollowing data: (C/C) 0 is the collection o arrows in C with cod() = C. An arrow in C/C, : (x : X C) (x : X C), is an arrow : X X in C such that x = x, that is, the ollowing commutes: X X x C The identity arrow or an object x : X C is simply the arrow Id X, which makes the above diagram commute trivially: xid X = x. Let x, x, and x be objects in C/C, and let C/C(x, x ) and g C/C(x, x ). Then we deine composition so that the ollowing commutes: X X x C g x x x X Thus, g C/C(x, x ) is the composite arrow g : X X in C such that x (g) = x Duality. The deinition includes objects and arrows and our operations (dom, cod, Id A, and g ) that satisy the ollowing: dom(id A ) = cod(id A ) = A Id dom() = = Id cod() dom(g ) = dom(), cod(g ) = cod(g) h (g ) = (h g) Note that the operation g is only deined where dom(g) = cod(). So or any statement in category theory Σ, we can orm the dual statement Σ by replacing g with g, cod or dom, and dom or cod (i.e. inverting all the arrows and orders o composition. Σ will also be a well-ormed sentence. Thus, supposing that a well-ormed sentence Σ entails another well-ormed sentence, by inversion o arrows we ind that Σ entails. Knowing this, we can also see that the axioms o category theory are sel-dual, i.e. CT = CT. Combining these two ideas, we arrive at what is known as the Duality Principle o category theory. Proposition 1.12 (Duality). For any statement Σ in category theory that ollows by the axioms o category theory, the dual o that statement Σ, also ollows rom the axioms.

6 6 MARCELLO DELGADO Proo. Suppose CT entails a statement Σ. Then as demonstrated above, CT entails Σ. But since CT = CT, then CT entails Σ. Along these lines, since C op is C with inverted arrows, then any true statement Σ in C necessarily implies the truth o Σ in C op. Combining this with the observation that (C op ) op = C, we arrive at a slightly more ormal version o the duality principle. Theorem 1.13 (Duality 2.0). I Σ is a well-ormed sentence that holds in all categories C, then so is/does Σ. Proo. Suppose Σ is a statement in the language o category theory that holds in all categories. Given a category C, Σ necessarily holds in C op since it holds in all categories. This implies that Σ holds in (C op ) op. Since (C op ) op = C, then Σ holds in C. So Σ holds in all categories. Many times we will come across pairs o notions which are dual to each other, and it will suice to show something about just one o them because the dual statement ollows rom the duality principle. As an example o this, we can add to our list o constructed categories the coslice category o a category C under an object C, denoted C/C. By reversing the arrows in the slice category C/C, we see that the objects in C/C are the arrows in C whose domain is C. The rest ollows starting rom this and keeping the duality principle in mind. Remark It is common practice to use the preix co- or the dual o an existing notion, e.g. coslice is the dual o slice. Along these lines, we will encounter coproducts, coequalizers, colimits, and pushouts Abstract Structures. There are several types o structure we can talk about in the context o a category. Deinition 1.15 (Monomorphism and Epimorphism). An arrow : A B is a mono(morphism), or is said to be monic, i or any set o parallel arrows g, h : X A such that g = h, i.e. X g A B commutes, we have that g = h. h An arrow : A B is an epi(morphism), or is said to be epic, i or any set o parallel arrows i, j : B Z such that i = j, i.e. A B i Z commutes, we have that i = j. Note that the notion o a monomorphism is dual to the notion o epimorphism. Despite their abstract nature, these structures do have some very concrete examples in nature. Proposition A unction : A B (arrow in Sets) is monic i and only i it s injective. Proo. Suppose is monic. Let a and a be distinct points o A. And let { } be any singleton set. Then we have two unctions a, a : { } A such that a( ) = a and a ( ) = a. Since is monic and since a a then a a. Since (a( )) = (a) and (a ( )) = (a ), then (a) (a ). Suppose is 1-to-1 and g, h : C A are parallel arrows into A such that g h. Then there exists c C such that g(c) h(c). Since is 1-to-1, then (g(c)) (h(c)), which implies that g h. By contrapositive, g = h implies g = h. j

7 GENERAL ABSTRACT NONSENSE 7 Proposition A unction : A B is epic i and only i it s onto. Proo. Consider a surjective map : X Y, and let g, h : Y Z be parallel arrows such that g = h. Consider y Y. Since is onto, there exists x X such that (x) = y. By construction, g((x)) = h((x)), which means that g(y) = h(y). Since this holds or any y Y, g = h. Let : X Y be epic. Suppose is not onto. Then Range() Y. Let g : Y {0.1} be deined such that g(y) = 1 i y Range() and g(y) = 0 otherwise. Further, deine h : Y {0, 1} such that or all y, h(y) = 1. Then g = h. However, h g since or all y / Range() g(y) = 0, and we know there is at least one such y. This contradicts being epic. Thus, is onto. Proposition In a ixed poset category P, every arrow p q is both monic and epic. Proo. By construction, a poset category has at most one arrow between any two objects. Thus, any parallel arrows must be equal. This makes all arrows trivially monic and epic. Proposition Every iso is both monic and epic. Proo. Let : X Y be an iso and suppose that g : Y X is the arrow such that g = Id X and g = Id Y. Let x, y : A X be parallel arrows such that x = y and let i, j : X B be parallel arrows such that ig = jg. Diagrammatically: A x X y Y Id X g X i B j I x = y then g(x) = g(y). Since g = Id X then x = y, so is monic. Similarly, i ig = jg then (ig) = (jg). Since g = Id Y then i = j, so g is monic. Reversing the situation and applying the same logic will show that g is monic and is epic. Thus, every iso is both monic and epic. It should be noted that arrows that are both monic and epic are not necessarily iso. Let (R, ) denote R with the standard order topology, let (R, F) denote R with the ull topology, i.e. every subset o R is open, and consider both as objects in the category Top o topological spaces and continuous unctions. Then or any topological space (X, A), any unction : (R, F) (X, A) is continuous since or any open subset V X 1 (V ) R is necessarily open. Consider the unction h : (R, F) (R, ) deined such that or all x R h(x) = x. Since the underlying set unction o this map is h = Id R, it s airly easy to check that its both monic, epic, and bijective. However it is not an isomorphism. In Top, an isomorphism is a continuous unction with a continuous inverse, which is exactly a homeomorphism. The inverse arrow o the underlying set unction is again g = Id R, and since h = Id R is iso and isos are both monic and epic then g is unique. However, g : (R, ) (R, F) is not continuous since or any x R, the preimage o the open set {x} in (R, F), g 1 (x) = x, is the same single point set, which is not open under the order topology.

8 8 MARCELLO DELGADO Deinition 1.20 (Initial and Terminal Objects). In a category C, an object is an an initial object, denoted 0, i or all objects C there exists a unique arrow! C : 0 C. An object is a terminal object, denoted 1, i or all objects C there exists a unique arrow! C : C 1. In Sets, the initial object is the empty set, and any singleton set is a terminal object. The uniqueness o the initial (terminal) object is addressed by the ollowing proposition: Proposition Initial (terminal) objects are unique up to isomorphism. Proo. Suppose a category C has two initial objects 0 and 0. Then by the universal mapping property o initial objects, there exist unique arrows! : 0 0 and! : 0 0. Thus,!! : 0 0 and!! : 0 0. By the universal mapping property o initial objects,!! = Id 0 and!! = Id 0. Thus, 0 and 0 are isomorphic. Since terminal objects are the dual notion o initial objects, an analogous proo with arrows reversed shows that terminal objects are unique up to isomorphism. It is worth noting two things at this point. First, a category does not need an initial or terminal object. Take or example the poset category (Z, ) which has neither terminal nor initial object. Second, along those same duality lines, it s worth noting that the initial (terminal) object o a category C is the terminal (initial) object o C op. Knowing this, the uniqueness up to isomorphism o terminal objects in any category C ollows rom the uniqueness up to isomorphism o the initial object in C op. In some categories, the initial and terminal object are the same 2. Studying Objects in a Category 2.1. Elements and Arrows. Arrows rom terminal object provide some insight into the structure o the objects o the category. In Sets, the terminal object is any singleton set (which are clearly isomorphic to each other). Arrows a : 1 A, where a( ) = a A, are exactly the elements o the set A. In act, we have in Sets an isomorphism X = Sets(1, X) or any set X. Further, we can generalize this notion to any category with a terminal object. Deinition 2.1 (Points or Elements). In a category C with a terminal object 1, arrows a : 1 A are called the elements or points o A. In Sets, the elements o A are enough to distinguish between parallel arrows. Proposition 2.2. I, g : A B are parallel arrows in Sets, then = g i and only i or all elements a : 1 A o A, a = ga. Proo. The orward direction is trivial. For the reverse direction, suppose that g. Then there exists a A such that (a) g(a). Deine a :! A by a( ) = a. Then ga a. We ve shown that i g then there exists an element o A such that a ga, so the result ollows by contrapositive. This argument also holds in Pos, but it does not hold in general. In Grp, the terminal object, the single element group that consists o just an identity element, is also initial. Thus, or all groups G, there exists only one arrow g : 1 G, so groups have only one point. This tells us that our notion o element or point is not general enough to capture the inormation we would like to obtain. Thus, we must

9 GENERAL ABSTRACT NONSENSE 9 abstract urther to generalized elements, which can be used to distinguish between arrows in any category.. Deinition 2.3 (Generalized Elements). Arbitrary arrows x : X A (or any domain X) are regarded as generalized or variable elements o A. Think o these arrows o things in A that are shaped like an arbitrary X. In Sets, arrows rom the terminal object point out the parts o set that look like the terminal object that is, they are exactly the elements, in the usual sense, o A. Similarly, arrows rom or a two point set {1, 2} A are exactly the subsets o A that have exactly two elements. A more illustrative example comes rom looking at generalized elements in categories o structured sets. Example 2.4. (1) Consider the three element group G = {u G, g, g 1 }. Then arrows in Grp G H rom G to other groups H are exactly the subgroups o G that consist o an element, its inverse, and the identity element u H. (2) In Pos, consider the poset P = {1 2 3}. Then the arrows p : P A are exactly the subsets o A consisting o three totally ordered elements. The useulness o generalized arrows comes rom their ability to distinguish parallel arrows. Proposition 2.5. In any category C, or all parallel arrows, g : C C, = g i and only i x = gx or all generalized elements x : D C. Proo. The orward direction is easy. For the reverse direction, i or all generalized elements x : D C we have gx = x, then we have g Id C = Id C. Since Id C = and g Id C = g, then = g. Generalized elements are part o a class o tools used to elucidate the ine structure o a category. To do so, we need to loosen how strict our tools are. Isomorphic pairs o arrows are a very strong and useul tool or establishing relations among objects. However, isomorphisms require both compositions o the arrows to yield an identity. I we require only that one composition yield identity, then we have a more lexible notions: split monomorphism and split epimorphisms. Proposition 2.6. I : A B has a let inverse g : B A (equivalently, g has a right inverse ) such that g = Id A then is monic and g is epic. Proo. Since the statement that g is epic is dual to the one that is monic, we ll omit its proo and claim that it ollows rom the duality principle. Consider parallel arrows h, k : X A such that h = k. Then we have g(h) = g(hk) by composition. By associativity, g(h) = (g)h = Id A h = h. Similarly, g(k) = k. Thus h = k, so is monic. Deinition 2.7 (Split Mono(Epi)). A split mono (epi) is an arrow with a let (right) inverse. Given arrows e : X A and s : A X such that es = Id A, we say that s is a section or splitting o e and that e is a retraction o s. A is called a retract o X. In a similar way that unctors preserve isomorphisms, unctors preserve split monos and epis.

10 10 MARCELLO DELGADO Proposition 2.8. In Sets, every mono splits. Proo. As shown previously, in Sets the monic arrows are exactly the injective unctions. Thus all monos in Sets are bijective on their images. For every x Im() there s a unique y A such that (y) = x by injectivity. So or : A B that s monic, deine g : B A by g(b) = a i a is the unique element o A such that (a) = b. Fix any y A and deine g on b B \ I() by g(b) = y. Then g = Id A, which is equivalent to saying that splits. Fact 2.9. The condition that every epi splits in Sets is the categorical equivalent o the axiom o choice. To see this, consider an epi e : E X. As shown previously, s must be onto since it is epic. Thus, or all x X, there is a nonempty set e x := e 1 {x}. A splitting o e is exactly a choice unction on the amily o sets (e x ) x X ; that is, a unction s : X E such that es = Id X. Conversely, given a amily o nonempty sets e x or every x X, taking E = {(x, y) x X, y e x }, we can deine the unction e : E X such that (x, y) x. A splitting o e would thus be a choice unction on the amily (e x ) x X Projective Objects. Projective objects are another tool that let us establish relations among the objects o a category by using something akin to a mapping property. Deinition 2.10 (Projective Objects). An object P is said to be projective i or any epi e : E X and or any arrow : P X, there exists a not necessarily unique arrow : P E such that e =. Diagrammatically: We say that lits across e. P X e E Let us consider what this rather abstract deinition means in Sets. Proposition The axiom o choice implies that all sets are projective in Sets. Proo. The axiom o choice is equivalent to saying that all epis split. So or a set A, consider the arrow : A B and the epi e : E B. Since epis split, there exists an arrow s : B E such that es = Id B. It ollows that (es) = Id B =. And by composition, we have that the ollowing commutes: A B e s E Letting = s, we ve shown that A is projective. s Proposition In any category C, the retract o a projective object is also projective.

11 GENERAL ABSTRACT NONSENSE 11 Proo. Let P be a projective object and let A be a retract o P, that is, there exists a split epi e : P A with a right inverse s : A P (which has to be monic). Consider any arrow : A B and any epi i : E B. Then e : P B by composition. Since P is projective, there exists e : P E such that i e = e, and thus the ollowing commutes: e P A B s i e E By precomposition, we get that (e)s = i e s. Since (e)s = (es) = Id A =, then = i e s and the ollowing commutes showing that A is projective: e P A B s es e E 2.3. Subobjects. Our inal piece o machinery in elucidating the ine structure o objects allows us to establish a sort o partial order among the objects by establishing certain objects as subobjects o each other. Deinition 2.13 (Subobject). A subobject o an object X in a category C is a mono m : M X. Given subobjects m and m o X, a morphism : m m is an arrow such that the ollowing commutes: M M Thus, we can talk about Sub C (X), the category o subobjects o X in C. m Remark Note that such an arrow is an arrow in C/X. In reality, Sub C (X) is a subcategory o C/X, and so composition and identity are exactly the same as in C/X. Since m is monic, there is at most one rom m to m ; thus Sub C (X) is a preorder category. We can use this to deine an inclusion relation, where m m i and only i there exists : m m. Two subobjects are then equivalent, denoted m m, i and only i m m and m m. This is equivalent to both triangles commuting: X m M M m X It s worth noting that our diagram shows that m = m. Since m is monic, this implies that = Id M. By the same argument, we see that = Id M, which implies that M = M. Thus, equivalent subobjects have isomorphic domains. m i

12 12 MARCELLO DELGADO Proposition Let m and m be subobjects o X. Then an arrow : m m is also monic, so is a subobject o M. Proo. Consider subobjects m and m o X such that m m, that is, there exists : m m such that m = m. Suppose there are parallel arrows i, j : Z M such that i = j. Then by composition, m i = m j. Since m = m, then mi = mj. Finally, since m is monic, then i = j. Thus, is monic and is a subobject o M by composition. This can be read o the ollowing diagram: Z i j M M m X m For generalized elements z : Z X o X, we can deine a local membership relation, z X m, between generalized elements and subobjects m : M X by the ollowing: z X m i and only i there exists : Z M such that Z M z X commutes. Since m is monic, then must be unique. Fact It is an entirely surprising result that Sub Sets (X) = P (X). Despite the appearance o what appears to be a terrible pun, you can see the truth o this statement by seeing that any monic unction into a set X is an injective inclusion arrow. Consider any two subobjects z : Z X and z : Z X. Then z z means that there s a monic (injective) arrow : Z Z that commutes appropriately. I we have both z z and z z, then by our deinition we have z z. But this allows us to see just how clever the deinition o this relation is, because i we have z z and z z, then we have injective arrows : Z Z and : Z Z, which by the Schroeder-Bernstein Theorem gives us that Z = Z, which is equivalent to saying that the respective sets are isomorphic in Sets m 3. The Three Faces o Limits In this section, we will introduce the notions o products, pullbacks, and equalizers, which we will go on to show later as being speciic examples o a more abstract notion: limits Products. We know rom set theory that we can take the cartesian products o sets: or any sets A and B, let A B = {(a, b) a A, b B}. And rom this product, we have two obvious projection unctions, π 1 : A B A and π 2 : A B B, where (a, b) π1 a and (a, b) π2 b. Thus, or any c A B, we

13 GENERAL ABSTRACT NONSENSE 13 have c = (π 1 c, π 2 c). This is equivalent to the ollowing diagram commuting: 1 a b (a,b) A π 1 A B π 2 B This example works with elements o A and B. But i we abstract to generalized elements, we arrive at a deinition o products in any category. Deinition 3.1 (Products). In any category C, a product diagram or the objects A and B consists o an object P and arrows p 1 : P A and p 2 : P B, p 1 p 2 A P B, that satisies the ollowing universal mapping property: x 1 Given any diagram o the orm A X x 2 B, there exists a unique arrow u : X P making the ollowing diagram commute: That is, x 1 = p 1 u and x 2 = p 2 u. X x 1 x 2 u p 1 p 2 A P B Example 3.2. In Sets, the cartesian product A B o sets A and B along with the standard projection unctions π 1 and π 2 satisies the universal mapping property o products. Proo. Consider a set X and unctions 1 : X A and 2 : X B. Then deine the unction = ( 1, 2 ) : X A B such that p ( 1 (p), 2 (p)). Then clearly 1 = π 1 and 2 = π 2. Suppose there exists u = (u 1, u 2 ) : X A B that also commutes appropriately. Then 1 = π 1 u = u 1, and a similar argument shows that 2 = u 2. Thus, u = ( 1, 2 ) =. Proposition 3.3. Products are unique up to isomorphism. Proo. Suppose or objects A and B you have two product diagrams: A q 1 q 2 and A Q B. Then by universal mapping property, there exists i : P Q such that p 1 = q 1 i and p 2 = q 2 i. Analogously, there exists j : Q P such that q 1 = p 1 j and q 2 = p 2 j. Then by precomposition, q 1 i = p 1 ji and q 2 i = p 2 ji. Since p 1 = q 1 i and p 2 = q 2 i, then we have p 1 = p 1 ji and p 2 = p 2 ji. A similar precomposition argument will shows that q 1 = q 1 ij and q 2 = q 2 ij. Diagrammatically, the ollowing diagram commutes: p 1 A i q 1 P Q p 2 q2 Since p 1 = p 1 ji and p 2 = p 2 ji, it ollows rom the uniqueness o identity arrows that ji = Id Q, and analogously that ij = Id P. Thus, P = Q. j B p 1 P p 2 B

14 14 MARCELLO DELGADO Proposition 3.4. A B = B A. Proo. Follows rom previous proposition. As a general rule, given X, x 1, and x 2 as in the above diagram we write x 1, x 2 or u : X A B. We can also talk about arrows out o products, which can be written essentially as unctions o two variables, and as such cannot be reduced. For example, 1, 2 : A B X. Remark 3.5. As a general rule, products o structured sets can be constructed as products o the underlying sets with operations deined component wise. For example, the product o two groups G and H, G H, has as its underlying set the cartesian product o the underlying sets o G and H. Further, g, h g, h = gg, hh u G H = u G, u H a, b 1 = a 1, b 1 We wont go into detail again about the construction o the product category C D, though one can check that it satisies the universal mapping property required by the deinition. Deinition 3.6 (Binary Products). A category C is said to have binary products i or every pair o objects A and B there exists an object A B and a pair o projection arrows p 1 : A B A and p 2 : A B B that satisies the universal mapping property o product diagrams. Let s examine then what an arrow rom a product to a product looks like. Consider two arrows : A B and : A B in a category C with binary products. Suppose that the ollowing diagram commutes: A A A p 2 A p 1 q 1 B B B q 2 B Then there exists a map which we denote : A A B B such that we say that both squares below commute: A p 1 A A p 2 A q 1 B B B q 2 B Remark 3.7. It should be noted that a category that has binary products will necessarily have a terminal object. Imagine taking the nullary product, that is, a product o no objects. This product clearly has no projection arrows since there are no codomains to deine those arrows on. Let us denote this product P. Thus or any other object C, we trivially have no arrows to the nonexistent objects that we used to deine this product, so by the universal mapping property o products there exists a unique arrow! : C P that makes all these nonexistent arrows commute trivially. It s clear then that P must be a terminal object since every element will trivially have arrows to the nonexistent objects that we took the product o to deine P and so all objects must have a unique arrow to P.

15 GENERAL ABSTRACT NONSENSE 15 Proposition 3.8. C 1 = C. Proo. Consider the product diagram C p 1 p 2 C 1 1, where we necessarily have that p 2 =! C 1, the unique arrow rom C 1 to the terminal object. Then or the pair o arrows Id C : C C and! C : C 1, there exists a unique arrow u : C C 1 such that Id C = p 1 u and! C =! C 1 u. That is, the ollowing commutes: C Id C! C u p 1! C 1 C C 1 1 By composition, we have! C 1 up 1 : C 1 1. By uniqueness,! C 1 up 1 =! C 1. Thus, up 1 = Id C 1. Since we have up 1 = Id C 1 and p 1 u = Id C, the result ollows. A similar, but much less exciting, proo also shows us that products are associative up to isomorphism Equalizers. In Sets, given any two unctions, g : A B, we can consider the subset E = {a A (a) = g(a)}. Then deining an inclusion unction e : E A, we have made it so that e = ge by restricting and g to the subset o A on which they are equal. It is worth noting that an inclusion unction is monic and thus that E is a subobject o A, since we will want our generalization to retain these properties. Deinition 3.9 (Equalizer). In any category C, given parallel arrows, g : A B, an equalizer o and g consists o an object E and an arrow e : E A such that e = ge, that is, the ollowing commutes: E e A B g Further, they satisy the ollowing universal mapping property: Given any z : Z A such that z = gz, there exists a unique arrow u : Z E such that eu = z. Diagrammatically, the ollowing commutes: E e A B g z Z u As promised, e is monic, which is equivalent to saying that it (or E) is a subobject o A. Proposition Equalizer arrows are monic. Proo. Given the equalizer diagram E e A B, suppose we have parallel g arrows z 1, z 2 : Z E such that ez 1 = ez 2. By precomposition, ez 1 = gez 1 : Z A. Thus, by the universal mapping property o equalizers, there exists a unique arrow u : Z E such that eu = ez 1. Thus, u = z 1. But since ez 1 = ez 2, then u = z 2. Thereore, z 1 = z 2, so e is monic.

16 16 MARCELLO DELGADO 3.3. Pullbacks. Much like an equalizer, pullbacks attempt to equalize arrows through the use o generalized elements. In the case o equalizers, the two arrows share both a domain and a codomain. In the case o pullbacks, they only share a codomain. Deinition 3.11 (Pullback). Given a corner o arrows : A C and g : B C, a pullback o and g is a pair o arrows p 1 : P A and p 2 : P B such that p 1 = gp 2, and are universal in this property: Given any pair o arrows z 1 : Z A and z 2 : Z B such that z 1 = gz 2, there exists a unique arrow u : Z P such that z 1 = p 1 u and z 2 = p 2 u. Diagrammatically, Z z 2 u P B z 1 p 1 p 2 A C Remark We usually denote the pullback object using product notation, as in A C B. In a manner almost identical to all the previous examples with universal mapping properties, we can demonstrate that pullbacks are unique up to isomorphism. Thus, we will omit the proo and just state the result. Proposition Pullbacks are unique up to isomorphism. As stated earlier, pullbacks and equalizers are very similar notions in motivation. It is probably due to the cleverness o Saunders Mac Lane that the notions themselves are also very closely related. The ollowing lemma begins to establish this relationship. Lemma E p 1 p 2 e A B A In a category C with products and equalizers, given a corner o arrows : A C and g : B C, consider the diagram where e : E A B is an equalizer or π 1 and gπ 2. I we deine p 1 = π 1 e and p 2 = π 2 e, then E, p 1, p 2 is a pullback o and g. Conversely, i E, p 1, p 2 is given as a pullback o and g, then the arrow e deined as e := (p 1, p 2 ) : E A B is an equalizer or π 1 and gπ 2. We will quickly sketch both parts o the proo. For the ormer, suppose we have arrows z 1 : Z A and z 2 : Z B such that z 1 = gz 2. This deines an arrow (z 1, z 2 ) : Z A B. Since E and e orm an equalizer, there s a unique arrow π 1 π 2 g B C g

17 GENERAL ABSTRACT NONSENSE 17 u : Z E such that eu = z by the universal mapping property o equalizers. Componentwise, this gives us ep 1 = z 1 and ep 2 = z 2, which yields the universal mapping property o pullbacks. For the second part, given the pullback, deine e = (p 1, p 2 ) : E A B. Supposing that there exists z : Z A B such that z(π 1 ) = z(gπ 2 ), we see that such an arrow into a product is the same as two arrows z 1 : Z A and z 2 : Z B. Further, we get that z 1 = z 2 g since zπ i = z i by deinition. By the universal mapping propertyo pullbacks, we get a unique arrow u : Z E such that componentwise we get z i = p i u, which when put together gives us z = eu since e = (p 1, p 2 ) by construction. This satisies the universal mapping propertyo equalizers. This leads us to a very important corollary, which will be used later to collect pullbacks, equalizers, and products under the single banner o limits. Beore we do that, we must irst tie together what we have so ar. Corollary I a category C has binary products and equalizers then it has pullbacks. Proo. Consider a corner o arrows : A C and g : B C. Since C has products, then there exists an object A B with the canonical projection arrows π 1 and π 2. Since C has equalizers, let e : E A B be an equalizer o π 1 and gπ 2. By previous lemma, i we deine p 1 = π 1 e and p 2 = π 2 e, then E, p 1, p 2 is a pullback o and g Equivalences. Having deined products, pullbacks, and equalizers, we can now prove our irst big theorem showing the equivalence o these notions. Theorem A category C has binary products and equalizers i and only i it has pullbacks and a terminal object. Proo. Let C be a category with pullbacks and a terminal object. The orward direction o this proo is exactly the corollary rom the previous section. The existence o the terminal object ollows rom the deinition i having binary products. Thus, we will restrict the given proo here to the reverse direction. Consider objects A and B. Then there exist unique arrows! A : A 1 and! B : B 1. Since we have pullbacks, there exist arrows p 1 : A 1 B A and p 2 : A 1 B B such that p 1! A = p 2! A and are universal with this property. z 1 z Suppose we have the diagram A 2 Z B. By uniqueness o arrows to the terminal object,! A z 1 =! Z =! B z 2. Diagrammatically, we have that the ollowing commutes: Z z 2 z 1 A 1 B p 2 B A p 1! A 1 By the universal mapping propertyo pullbacks, there exists a unique arrow u : Z p 1 p 2 A 1 B such that z 1 = p 1 u and z 2 = p 2 u. This means that A A 1 B B! B

18 18 MARCELLO DELGADO is a product o A and B because it satisies the universal mapping propertyo products. For parallel arrows, g : A B, consider the corner o arrows, g : A B B and Id B, Id B : B B B. Since we have pullbacks, there exist arrows e : E A and h : E B such that, g e = Id B, Id B h. This implies that e, ge = hid B, hid B = h, h. Thus, e = h = ge. To show that E and e are universal with this property, suppose we have an arrows z 1 : Z A and z 2 : Z B such that, g z 1 = Id B, Id B z 2. This gives us by the same argument as beore that z 1 = z 2 = gz 1. By the universal mapping propertyo pullbacks, there exists a unique arrow u : Z E such that z 1 = eu and z 2 = hu. From this we see that this satisies the universal mapping property o equalizers: or an arrow z 1 : Z A such that z 1 = gz 1 there exists unique u : Z E such that z 1 = eu. We can see this rom the ollowing commutative diagram: Z u z 1 E A e z 2 h B,g B B 4. Limits Id B,Id B In this inal section, we will build up the ramework or limits and then prove our main theorem: that having limits is equivalent to having products and equalizers Basic Deinitions. Here, we will present the deinitions relevant to limits. Deinition 4.1 (Type-J Diagram). A type-j diagram in a category C is a unctor D : J C. Remark 4.2. Objects in the index category J will be denoted by lower case letters i, j, k,..., and their images under D will be denoted respectively as D i, D j, D k,.... Deinition 4.3 (Cone). A cone to a diagram D consists o some object C C 0 and a amily o arrows (c j : C D j ) C 1 or each j J 1 such that or all arrows (α : i j) J 1 we have that c j = Dα c i. Equivalently, the ollowing commutes or all such α: C cj D j c i Dα D i Deinition 4.4 (Morphism o Cones). A morphism o cones ϑ : (C, c j ) (C, c j ) is an an arrow ϑ : C C in C such that or all j J 0 we have that c j = c j ϑ. We can now talk about the category o cones to a diagram D, Cone(D). Checking that this is a category is somewhat tedious and rather trivial, so the proo will be omitted. Having this, we can inally give a deinition or limits.

19 GENERAL ABSTRACT NONSENSE 19 Deinition 4.5 (Limit). A limit o a diagram D : J C is a terminal object in Cone(D). We call it a inite limit i J is a inite index category. We denote the limit (L D, p j ). Remark 4.6. It is worth clariying to prevent conusion that the limit o a diagram has the ollowing universal mapping property as terminal objects o Cone(D): Given any cone (C, c j ) to D, there exists a unique morphism o cones u : (C, c j ) (L D, p j ) such that or all j J 0 we have that c j = p j u, i.e. the ollowing commutes: u c j C D j p j L D 4.2. Plateau. Since the notion o a limit is somewhat abstract, it helps to clariy it with a ew examples. Speciically, we will show how products, pullbacks, equalizers, and terminal objects are each special cases o limits. Example 4.7 (Products as Limits). Take J = {1, 2} to be the discrete category on two elements with no nonidentity arrows. A diagram D : J C is a pair o objects D 1, D 2 C 0. A cone to D is an object C C 0 with a pair o arrows: D c 1 1 C c 2 D 2. Let D p 1 1 L p 2 D D 2 be the terminal cone to D. Then or any other cone (C, c j ), there exists a unique morphism o cones u : (C, c j ) (L D, p j ) such that the ollowing commutes: D 1 c 1 p 1 C u L D c 2 p 2 D 2 Thus, the limit exactly satisies the universal mapping property o products. Example 4.8 (Equalizers as Limits). Take J such that J 0 = {1, 2} and the only nonidentity arrows are the parallel arrows α, β : 1 2. A typej diagram then looks like: Then any cone to D looks like D Dα 1 D 2 Dβ D Dα 1 D 2 c 1 C Dβ c 2 where both triangles commute. Note that this means that Dα c 1 = c 2 = Dβ c 1, beginning to look like an equalizer. Thus, when we take the limit, we get the necessary universal property. The limit then is the terminal cone (L D, p j ) with the property that or any other cone (C, c j ) there exists a unique morphism o cones

20 20 MARCELLO DELGADO u : (C, c j ) (L D, p j ) such that the ollowing commutes: Dα D 1 D 2 Dβ p 1 c 2 C c 1 u There is an arrow p 1 : L D D 1 such that Dα p 1 = Dβ p 1 since both must equal p 2. Further, or any arrow c 1 : C D 1 such that Dα c 1 = Dβ c 1 there exists a unique u : C L D such that c 1 = p 1 u. Thus the limit satisies the universal mapping propertyo equalizers. Example 4.9 (Pullbacks as Limits). Take J to be the inite category with objects J 0 = {i, j, k} and with the only nonidentity arrows being the corner o arrows α : i k and β : j k. Then the type-j diagram (irst igure) and any cone to it (second igure) look like the ollowing: D j Dβ L D C c i p 2 c j D j c k D i Dα D k D i Dα D k Note rom the igure o the cone to D that we have that Dβ c j = c k = Dα c i by commutation. So let s consider a limit (L D, p j ) o the diagram. L D p i p j D j p k D i Dα D k Then or any other cone (C, c j ), there exists a unique u : (C, c j ) (L D, p j ) such that the ollowing commutes: C u c i c k L D p i D i c j p j p k Dβ D j Thus or any pair o arrows c i : C D i and c j : C D j such that Dα c i = Dβ c j, which ollows rom both equaling c k, there exists a unique arrow u : C L D by the universal mapping property o limits such that c i = p i u and cj = p j u. Thus, the limit satisies the universal mapping propertyo pullbacks and is a pullback o α and β. Example 4.10 (Terminal Objects as Limits). Take J to be the empty category. Then a cone to a type-j diagram D consists o just a single object in C. Let L D be the limit o this diagram. Then or every cone to D, which is every object C in C, there exists a unique arrow u : C L D, which makes the limit satisy the universal mapping propertyo terminal objects. Dα D k Dβ Dβ

21 GENERAL ABSTRACT NONSENSE Climax. Here we state and prove the theorem we ve been working towards. One inal deinition is missing. Deinition 4.11 (All Finite Limits). A category C has all inite limits i every inite diagram D : J C has a limit in C. And the theorem itsel: Theorem A category has all inite limits i and only i it has all binary products and equalizers. We should note three things things at this point. As we showed earlier, having all binary products and equalizers is equivalent to having pullbacks and a terminal object, so a quick corollary o this theorem would be Corollary A category has all inite limits i and only i it has pullbacks and a terminal object. Second, courtesy o the duality principle, once this theorem is proven we will have its dual statement and the dual o the previous corollary: Corollary A category has all inite colimits i and only i it has all binary coproducts and coequalizers. Corollary A category has all inite colimits i and only i it has pushouts and an initial object. Here, a colimit is the dual notion o a limit. To sketch it out, the colimit is the initial object is the category o cocones (dual o cones) to a diagram D. Third, since we ve already shown how using inite limits we can construct products, equalizers, pullbacks, and terminal objects, we will omit the orward direction o the proo o the theorem and restrict ourselves to the reverse direction. Proo. Consider a inite diagram D : J C, and consider the products i J 0 D i (α:i j) J 1 D j Deine two arrows φ, ψ : i D i α D j by taking taking their composites with the projections π α rom the second product to be π α φ = φ α = π cod(α) π α ψ = ψ α = Dα π dom(α) or equivalently, the ollowing diagrams commute: ψ i J 0 D i (α:i j) J 1 D j ψ α π dom(α) π α D dom(α) D cod(α) Dα Then, we take the equalizer E e i J 0 D φ i ψ i J 0 D i φ (α:i j) J 1 D j φ α=π cod(α) (α:i j) J 1 D j π α D cod(α)

22 22 MARCELLO DELGADO I or all j J 0 we deine e j = π j e, we want to show that (E, e j ) is the limit o the diagram D. Letting c : C i D i, we can write that c = c i where c i = π i c or all i J 0, noting that (C, c i ) is a cone to D i and only i φc = ψc. To see this, suppose we have a cone, then or all (α : i j) J 1 we should have Dα c i = c j. Since c i = π i c and c j = π j c, and since i = cod(α) and j = cod(α), then we have Dαπ dom(α) c = π cod(α) c, which is the same as ψ α c = φ α or all α. Conversely, suppose that we have ψc = φc. Then or all any α, we have Dαπ dom(α) c = π cod(α) c. We know that c dom(α) = π dom(α) c and c cod(α) = π cod(α) c, so it ollows that or any α Dα c dom(α) = c cod(α), which satisies the deinition o cones. Since ψe = φe by construction, (E, e j ) is a cone. Suppose we have another cone (C, c j ). Then as shown above, this gives us that ψc = φc. Then by the universal mapping property o equalizers, there exists a unique arrow u : C E such that c = eu. Then by composition, we have π i c = π i eu, which gives us c i = e i u or all i J 0, which is exactly a morphism o cones. Since or all cones, we have a unique morphism o cones to (E, e j ), then it is the limit Resolution. To close, we will discuss one inal aspect o limits, speciically their preservation and creation under unctors. Deinition 4.16 (Preserving Limits and Continuous Functors). A unctor F C D is said to preserve limits o type-j i whenever we have a limit (L D, p j ) o a diagram D : J C, the cone (F L D, F p j ) is a limit or the diagram F D : J D. Symbolically, F (L D, p j ) = (L F D, F p j ). We say that such a unctor is continuous. Deinition 4.17 (Representable Functor). In a locally small category C and or any ixed object A C 0, we deine the representable unctor o A, denoted C(A, ) : C Sets, to be the unctor deined such that or any object B, B C(A, B), and or any arrow : X Y, C(A, ) : C(A, X) C(A, Y ). Proposition Representable unctors are continuous. We will omit the proo, but we will mention that it suices to show that representable unctors preserve products and equalizers. Finally, let us recall the orgetul unctor U : C Sets. We will close this paper with one inal proposition, which we will also leave unproven. Deinition 4.19 (Creating Limits). A unctor F : C D is said to create limits o type-j i or all diagrams D : J C and any type-j limit (L, p j : L F C j ) in D, there exists a unique cone (L, p j : L C j ) to D in C with F L = L and F p j = p j such that (L, p j : L C j ) is a type-j limit in C. Proposition The orget unctor U : Grp Sets creates all limits. Acknowledgments. It is a pleasure to thank my mentors, Claire Tomesch and John Lind, or their amazing help throughout the summer, without whose help this paper would be a good deal shorter. Reerences [1] Steve Awodey. Category Theory may/vigre/vigre2009/awodey.pd