Joseph Muscat 2015 1 Categories joseph.muscat@um.edu.mt 1 December 2012 1 Objects and Morphisms category is a class o objects with morphisms : (a way o comparing/substituting/mapping/processing to ) such that, (i) given morphisms :, g : C, g : C is also a morphism, (ii) or compatible morphisms, h(g) = (hg), and (iii) each object has a morphism a : satisying a =, ga = g. (Note: in a sense, an object is the morphism a; so we can even do away with objects.) Sets can be considered as 0-categories (only objects or elements), or as discrete categories with each object having one morphism a. The class o morphisms rom to is denoted Hom(, ); thus Hom(, ) is a monoid. Even at this abstract level there are at least three important categories: 1. logic (with statements as objects and as morphisms), 2. sets (with unctions as morphisms), 3. computing (with data types and algorithms). 1.1 Morphisms monomorphism : satisies C, x, y Hom(C, ), x = y x = y. x n epimorphism : satisies C y C, x, y Hom(, C), x = y x = y. x C 1. In particular, or a monomorphism, g = g = ι ; or an epimorphism g = g = ι. y
Joseph Muscat 2015 2 2. The composition o monomorphisms is a monomorphism, and o epimorphisms an epimorphism. 3. Conversely, i g is a monomorphism then so is g, and i it is an epimorphism then so is. monomorphism : is also called a sub-object o. Monomorphisms with the same codomain have a pre-order: let g or : C, g : C when = gh or some (mono)morphism h : ; h It can be made into a poset by using the equivalence relation = g when g. n isomorphism is an invertible morphism, i.e., has an inverse g such that g = ι, g = ι. In this case, and are called isomorphic (an equivalence relation); i g. n isomorphism : is called an automorphism; or example, any ι ; the automorphisms o orm a group. I g = ι then is called a split monomorphism or section (has a let-inverse), and g a split epimorphism or retraction (has a right-inverse). morphism with let and right inverses is an isomorphism (since then, g 1 = g 1 g 2 = g 2 ). n extremal monomorphism is a monomorphism such that the only way = ge with e an epimorphism is that e is an isomorphism (and g a monomorphism). n extremal epimorphism is an epimorphism such that = eg with e a monomorphism e is an isomorphism (and g an epimorphism). Thus a monomorphism which is an extremal epimorphism, or an epimorphism which is an extremal monomorphism, is an isomorphism. Let g mean gx = y u x = u, y = gu. strong monomorphism is one such that Epi. g C Isomorphisms SplitMono StrongMono ExtremalMono Monomorphisms Isomorphisms SplitEpi StrongEpi ExtremalEpi Epimorphisms Proo. I is a split monomorphism with g = ι, then is a monomorphism and g an epimorphism. I = hk with k an epimorphism, then ghk = ι and kghk = k, so kgh = ι; thus k has the inverse gh. I = ge is a strong monomorphism and e epi, then e, so ι = ge u ι = ue, g = u. So e is split and an epi, hence an isomorphism. morphism : is called idempotent when 2 = ; or example, the split idempotents = gh where hg = ι. n object is called inite, when every monomorphism : is an automorphism. In particular, i = then =. Example: For Sets, a monomorphism is a 1-1 unction; an epimorphism is an onto unction; such unctions are automatically split; an isomorphism is thus a
Joseph Muscat 2015 3 bijective unction; isomorphic sets are those with the same number o elements; a set is inite in the category sense when it is inite in the set sense. Functors (or actions) are maps between categories that preserve the morphisms (and so the objects), They preserve isomorphisms. 1.2 Constructions F : F F, F ι = ι F, F (g) = F F g Subcategory: a subset o the objects and morphisms; a ull subcategory is a subset o the objects, with all the corresponding morphisms. Dual category: C has the same objects but with reversed morphisms :, and g := (g) ; so C = C. Every concept in a theorem has a co-concept in its dual (eg monomorphisms correspond to epimorphisms); every theorem in a category has a dual theorem in the dual category. unctor between dual categories is called a dual unctor; a unctor rom a dual category to a category is called contra-variant, F (g) = F (g)f (). dagger category is one or which there is a unctor : C C, where (g) = g, =. (Set cannot be made into a dagger category because there is a morphism 1 but not vice-versa). Product o Categories: C D the objects are pairs (X, Y ) with X C and Y D, and the morphisms are (, g), where ( 1, g 1 )( 2, g 2 ) := ( 1 2, g 1 g 2 ), ι (X,Y ) = (ι X, ι Y ). The projection unctors are C D C, (, g), and C D D, (, g) g. (C D) C D (The product is the categorical product in Category) Quotient Category: given a category and an equivalence relation on morphisms (o same objects), then C/ is that category with the same objects and with equivalence classes o morphisms. The map C C/ deined by F :, [], is a unctor. rrow Category: C consists o the morphisms o C (as objects), with the morphisms g being pairs o morphisms (h, k), such that k = gh, h g k and composition (h 1, k 1 )(h 2, k 2 ) := (h 1 h 2, k 1 k 2 ), and identities (ι, ι ). Monomorphisms are those pairs (h, k) where h and k are monomorphisms. For example, the arrow category o sets is the category o unctions.
Joseph Muscat 2015 4 Slice Category (or comma category): C is the subcategory where the morphisms have the same codomain and k = ι; the morphisms simpliy to h where = gh; similarly or the morphisms with the same domain. n object is called projective when every morphism : actors through any epimorphism g : C, = gh. Dually, is called injective when : actors through any monomorphism = hg. 1.3 Functors Functors can be thought o as higher-morphisms acting on objects and morphisms; or as a model o C in D. (Examples: the constant unctor, mapping objects to a single one, and morphisms to its identity; the mapping rom a subcategory to the parent category; orgetul unctor (when structure is lost) and inclusion unctor (when structure is added, minimally); the mapping which sends to the set Hom(, ) and a morphism to the unction g g is a unctor rom any category to the category o sets; similarly or Hom(, ) and (g g ) (contra-variant).) unctor is called aithul when it is 1-1 on morphisms (and hence objects) It is ull when it is onto all morphisms in Hom(F, F ); it is called dense, when it is onto all objects up to isomorphism. It is an isomorphism on categories when it is bijective on the morphisms Hom(F, F ). dense isomorphism is called an equivalence, and the two categories are said to be equivalent. (let) adjoint o a unctor is F : D C with natural isomorphisms e, i such that e : F F 1, i : 1 F F and Hom(F, ) Hom(, F ); hence (F G) = G F. (For example, a orgetul unctor and inclusion unctor are adjoints, with i being the embedding) 2-Categories: Categories with unctors as morphisms orm a Category; the identity unctor is the one which leaves objects and morphisms untouched; (there is an initial object namely, and a terminal object, {. } It has the additional structure o a 2-unctor, called a natural transormation (or homotopy ), between unctors on the same categories, τ : F G; two such unctors map an object C to two objects F and G in D, and a natural transormation determines a morphism τ : F G between the two, such that :, (G)τ = τ (F ) (so F G). natural isomorphism is a natural transormation or which τ are isomorphisms. With these notions, two categories are equivalent when there are unctors F and F such that F F 1, F F 1 (or equivalently when F and F are isomorphisms with F F 1). The auto-equivalences o a category orm a symmetric monoidal category. More generally, a 2-category is a set o objects, with morphisms :, and 2-morphisms τ : 1 2 (or some 1, 2 Hom(, )); 2-morphisms can be combined either vertically by composition τ 2 τ 1, (and must be associative,
Joseph Muscat 2015 5 with an identity), or horizontally σ τ: g σ(g)τ(), such that g τ 1 σ 1 τ 2 σ 2 τ 2 τ 1 σ 2 σ 1 = (σ 2 τ 2 )(σ 1 τ 1 ). 2-category with 1 object gives rise to a monoidal category (o the morphisms and 2-morphisms o the object); a 2-category with 1 object and 1 morphism gives a commutative monoid (o 2-morphisms). The unctors themselves orm a category D C where morphisms are the natural transormations. C 1 C; C 2 is the category o arrows on C. 2 Limits When a category maps under a unctor F : C D to another category, the image o an object may have morphisms that were not present in C; an object D may sometimes determine a unique (up to isomorphism) object (called a universal) U in C, which makes F (U ) closest to in the sense that there is a unique morphism φ : F (U ), such that : F (),!g : U, = φ F (g). U g F (U ) φ F (g) F () co-universal is similarly an object U C with a morphism φ : F (U ) such that : F (),!g : U, = F (g)φ. In particular, sub-categories C may have universal properties: Terminal object 1:,! : 1 (or the empty sub-category). Initial object 0:,! : 0. 0 1 (0, 0) is an initial object in C D. For example, { 0 } and are the terminal and initial objects o sets; True and False are the ones or logic. Isomorphism The closest objects or an object with its identity morphism (the category 1), are its isomorphic copies. For example, sets with the same cardinality are isomorphic, while statements are so in logic. Products: For the subcategory 2 (with only the identity morphisms), the closest object o and is, with morphisms π :, π :
Joseph Muscat 2015 6 such that any other morphisms p : C, p : C actor out through a unique morphism g : C, p = π g, p = π g. p π C g p π 1 = ; = ; ( ) C = ( C). For example, the usual product, and the statement and are the products or sets and logic respectively. More generally, starting with a discrete category, the closest object o i is i i, with π i : i i i i.e., i p i : X i are morphisms then there is a morphism h : X i i with p i = π i h. repeated product gives C (starting with a constant unctor rom a discrete category). relation on objects, is a monomorphism R : ρ. Sums (or Co-products): i i is the dual o the product in the dual category i.e., it is the closest object with morphisms π i : i i i. For example, + (disjoint union) and or. Equalizer: starting rom the category with two objects,, and morphisms i :, their equalizer is the closest object E with (extremal mono-)morphism eq : E, i, j, i eq = j eq. eq 1 E For example, or Sets, { x : 1 (x) = 2 (x) }. Equalizers are monomorphisms: let e = eq, i xe = ye then xe = xeg, so!u, xe = ue, x = u; similarly y = u = x. Co-equalizer: similarly an (extremal epi-)morphism coeq : Y E, i, j, coeq i = coeq j. For example, the co-equalizer o a relation on a set X is the partition on it (or an equivalence relation, this partition is compatible with the relation). Pullback (ibre product): starting rom the category with objects X i and morphisms i : X i Z, then the pullback is the (unique... ) closest object Z X i with morphisms π i : Z 2 X i X i, i π i = π Z.
Joseph Muscat 2015 7 π Z π The equalizer is a special case when the morphisms start rom the same object. I Z is the terminal object, then Z X i = i X i. For example, the pullback on sets is X Z Y = { (x, y) : (x) = g(y) }; in particular when g is the identity, X Z Y = 1 Y. Pullback lemma: pullbacks orm squares (X Z Y, X, Z, Y ); i two adjacent squares orm pullbacks, then so does the outer rectangle; i the outer rectangle and the right (or bottom) square are pullbacks, then so is the let (or upper) square. Pullbacks preserve monomorphisms: I u = vg with mono, and gx = gy, then ux = vgx = vgy = uy, so ux = uy and x = y by uniqueness o pullbacks. Push-out is that closest object Z X i with g Z π i : X i Z X i, π i i = π Z. For example, or sets, the push-out X Z Y is the set X Y with the elements (z) X and g(z) Y identiied. Inverse Limit: starting rom the subcategory o a chain o objects i with morphisms j,i (such that k,i = k,j j,i ), the inverse limit is the closest object lim i with morphisms π i : lim i i, π j = j,i π i. π i 23 12 lim i 3 2 1 (More generally, can start with a topology o objects rather than a chain.) The pullback is a special case. For example, the inverse limit o sets X i is the set o sequences x i X i such that x j = j,i (x i ). Co-limit (Direct Limit) is similar with lim i and morphisms π i : i lim i, π i = π j j,i. More generally, or any subcategory, or any unctor, F : C D there may be a limit object lim F in D with (unique) morphisms π : lim F ( C) such that or any :,, C, π = π π lim F π
Joseph Muscat 2015 8 and it is the closest such object in the sense that or any other C D with p = p then!u : C lim F, π u = p. limit, i it exists, is unique up to isomorphism. co-limit is similar with : F () colimf such that :, F () =. In general, any unctor rom a category with an initial object to C has a limit; and any unctor rom a category with a terminal object has a co-limit. complete category is one in which every subcategory (or unctor) has a limit. For example, the category o sets is complete and co-complete. So, every unctor has an adjoint F : D C mapping U and g; so that F F 1, and similarly F F 1. unctor is said to be continuous when it preserves limits (e.g. rightadjoints) i.e., G, lim(f G) = F (lim G). It is co-continuous (e.g. let-adjoints) when it preserves co-limits. The existence o products and equalizers implies the existence o all inite limits. The Hom(,.) unctor is continuous, so it represents these limits by sets (and Hom(., ) takes colimits to limits). amily o zero morphisms 0 are such that, g, 0 = g0 or example, when 0 = 1 (called a zero object), 0 : 0 are zero morphisms. 0 0 g 0 In this case, the kernel o a morphism is the equalizer o and 0 i.e., the closest (mono)morphism k : K such that k = 0. k K The co-kernel is the co-equalizer ie the closest (epi)morphism k : K such that k = 0. pre-shea is a contra-variant unctor rom a pre-order (or topology) to a category F : O C (the F (x) are called sections o F over x) such that x y there is a restriction morphism F (x) F (y) with res x,x = ι F (x) and x y z res y,x res z,y = res z,x. shea is a continuous pre-shea (preserves limits). On a topological space X, the stalk at x X is the direct limit o the open neighborhoods o x. So there is a morphism F (U) F x or x U open (i the morphism is a unction x, where x is called the germ at x). The etale space E is the space o stalks, with the continuous map E X, F x x. (the set o sheaves orm 0
Joseph Muscat 2015 9 a topos, with Ω = the disjoint union o all open sets) The space E is locally homeomorphic to X (i.e., there are isomorphic open sets in E and X that cover F x and x). For example, a shea o sets is a bundle, i.e., a collection o disjoint sets i with a map π : i i I, π 1 (i) = i ; the category o bundles over I is the same as the comma category. 2.1 Monoidal Categories Objects have an associative unctor tensor product and an object I (called unit) such that I = = I ( ) C = ( C) ( I) = = (I ) (the isomorphisms in the irst two lines are called the two unitor and one associator natural isomorphisms; more generally, any product o n objects are isomorphic to each other). Product o morphisms g : C D. The tensor product is like treating two objects in parallel; so a morphism :... C... D takes n objects and maps them to m objects, and looks like a Feynman diagram. The unit object is null, so : I creates one object. The tensor product is dierent rom the categorical product in that there need not be projections. The morphisms Hom(I, I) now have two operations: ( g)(h k) = (h) (gk); but rom universal algebras, this implies that g = g and is commutative. Set with is monoidal (in act cartesian-closed); Set with disjoint union is also monoidal. The (right) dual o an object is another object (unique up to isomorphism), such that there are annihilation/creation morphisms I, I, called the co-unit o and the unit o, respectively, satisying the zig-zag equations, i.e., creating then annihilating and leaves nothing I; ( can be represented as a line in the opposite direction o ; is called the let dual o ). 2.1.1 raided Monoidal categories monoidal category in which there is a natural isomorphism that switches objects around, =,
C Joseph Muscat 2015 10 such that all permutations o products become isomorphic, e.g. ( ) C = C ( ), i.e., C C It need not be its own inverse! Its inverse is: The Yang-axter equation states C = C C Let duals are duals. braided monoidal category is called symmetric when the switching isomorphism is its own inverse. 2.2 Closed Monoidal Categories monoidal category is closed when every set o morphisms Hom(, ) has an associated object, with Hom(, C) = Hom(, C ) (or alternatively Hom(, C) = Hom(, C )) (via currying natural isomorphisms). That is, every morphism can be treated as an object (without inputs). In particular : is associated to I. For example, in sets, the powerset axiom asserts that Hom(, ) is a set ; in logic the distinction is between the morphism and the object. monoidal category is compact (or autonomous) when every object has a dual and a let dual. In this case it is closed, with :=, i.e., = Hom(, I); in particular the unit I corresponds to a unit inside. The reverse o currying, changing an object into a morphism, is an evaluation morphism eval :, eval( ι ) =.
Joseph Muscat 2015 11 (So morphisms o two variables become morphisms o one variable.) For example, in sets (and unctional programming languages), eval(, x) = (x); in logic, it is modus ponens, and gives. 2.3 Cartesian-closed categories Finite products exist and are closed, i.e., every unctor has a right-adjoint, called exponentiation, Hom(, C) = Hom(, C ) This means that every morphism : i i C can be represented by an ordered set o morphisms i : i C. It is thus symmetric braided monoidal, with being and the unit being the terminal object 1; but has more properties in that it can duplicate objects via : ; and delete objects by mapping to 1, i.e.,! : 1; every morphism : 1 is o the type (1, 1) : 1, 1. (e.g. the adjoint o X (X, X) is (X, Y ) X Y.) g : C D can be deined as that unique morphism induced by π, gπ. In particular, (1 a, 1 b ) = 1 a b. Similarly, can deine the sum + g. 2.3.1 Evaluation eval :, eval( ι ) =. n element or point o is a morphism x : 1 ; so eval(, x) = x. In particular a morphism : corresponds to an element 1 (called the name o ). In such categories, dual concepts lose their symmetry: There are no morphisms 0 unless = 0, in particular i 0 = 1, then all objects are isomorphic; 0 is monic. 0 = 0, 1 =, 0 = 1, 1 = 1 (proos: there is only one morphism 0, so only one morphism 0 so 0 = 0, and 0 = 0 orces them to be isomorphisms; eval : 1 is an isomorphism; 1 0 corresponds to 0 = 1 0 which is unique, so 1 0 and 0 1 are inverses; 1 1 must be ι and 1 1 corresponds to 1 also unique; any map 0 is a unique isomorphism so g = h g = h) X + = X X, ( ) C = C C,
Joseph Muscat 2015 12 (C ) = C, X ( + ) = X + X (Proos: the inclusions, + give X + X X ; conversely, X X X + correspond to + X X X i.e., to two inclusion maps, and hence the projections X X X, X ; The projections, give rise to a map ( ) C C C ; its inverse is C C ( ) C which corresponds to C C C i.e., to C C C,, i.e., the projections C C C, C ; C (C ) corresponds to C C, i.e., the evaluation map C X, similarly (C ) C corresponds to the double evaluation (C ) C.; The maps + (X + X ) X correspond to the inclusions X, X X + X ) There is a unctor mapping morphisms : X 1 X 2 to F : X1 Y X2 Y deined by (F )g = g or g : Y X 1. There is another contra-variant unctor (restriction?) mapping morphisms : Y 1 Y 2 to F : X Y2 X Y1, deined by (F )g = g. 2.4 Topos category with inite limits, exponentials (i.e., cartesian-closed), and a subobject classiier. sub-object classiier is an object Ω (unique up to isomorphism) and a morphism True : 1 Ω such that monomorphisms : ( sub-objects ) correspond to unique morphisms χ : Ω, χ = 1 Ω 1 χ Ω In particular True corresponds to χ True = ι Ω, and the unique monomorphism 0 Ω corresponds to a morphism : Ω Ω; hence False := T rue : 1 Ω. For example, or sets Ω = 2; sub-objects : I X correspond to subsets X; subsets are maps 2 and correspond to the characteristic maps χ : 1 2 ; a singleton is a map 2. Other logical connectives are deined in terms o their characteristic maps: True and : Ω Ω Ω (True, True) : 1 Ω Ω or : Ω Ω Ω (True Ω, ι Ω ), (ι Ω, True Ω ) : Ω + Ω Ω Ω : Ω Ω Ω 2 Ω Ω(where 2 is the category 0 1) complement o χ intersections g χ g := χ and χ g unions g χ g := χ or χ g.
Joseph Muscat 2015 13 ut there may be several truth values, i.e., Ω may have several elements 1 Ω, not just True and False. Ω is injective, i.e., or any monomorphism : and any morphism g : Ω, there is a morphism ḡ : Ω such that g = ḡ. Ω can be thought o as a dual o ; the Fourier map ˆ : Ω Ω deined by ˆx() = x; = g χ = χ g ; the sub-objects o orm a bounded lattice, Sub() = Hom(, Ω). morphism is an isomorphism it is both mono and epi (called a bi-morphism) (since an epi monomorphism : is the equalizer o χ and Trueι ). Every morphism actors as = gh where h is epi and g is mono (via the object obtained by the pushout o with itsel). The pull-back o an epimorphism is also epi. Coproducts preserve pullbacks. (implies inite co-limits also exist) Every category can be extended to a topos. The product o topoi is a topos. comma category C/ o a topos is also a topos; its elements are bundles o elements (i.e., sections) o. Every topos has power objects P () := Ω, meaning objects P () and ɛ and a monomorphism : ɛ P () such that every relation (i.e., monomorphism) r : R has an associated unique morphism r : P () such that R P () = R ɛ P (). ɛ R r P () Ω = P (1). Conversely every category with inite limits and power objects is a topos. 2.4.1 Well-pointed topos topos that satisies the extensionality axiom, elements are epi: x : 1, x = gx = g. morphism is mono it is 1-1, i.e., x = y x = y or all x, y : 1. morphism is epi it is onto, i.e., y : 1, x : 1, x = y. The only non-empty object (i.e., without any elements 1 ) is the initial object (since χ 1 χ 0 ). The only elements o Ω are T rue and F alse (bivalent), and Ω = 1 + 1 (oolean). In act a topos is well-pointed the only non-empty object is the initial one, and Ω = 1 + 1. The arrow category Set is neither oolean nor bivalent; Set 2 is oolean but not bivalent; the category o actions o a monoid (that is not a group) is bivalent but not oolean. 2.4.2 With xiom o Choice category is called balanced when is an isomorphism it is a monomorphism and an epimorphism.
Joseph Muscat 2015 14 category satisies an xiom o Choice when every epimorphism is rightinvertible (splits). So balanced. For example, in sets, every monomorphism has a let-inverse, except or 0 ; the axiom o choice says that every epimorphism has a right-inverse. Strong xiom o Choice:, g, = g. topos with the axiom o choice has the localic property: i : C 1 monomorphism and g 1 g 2 : C, g 1 g 2. lso every object has a complement X = +. 2.5 Pre-additive Categories When Hom(, ) is an abelian group, distributive over composition o morphisms ie (g + h) = g + h, ( + g)h = h + gh. (then Hom(, ) is a ring) Can be extended to an belian category. 2.5.1 dditive Categories pre-additive category with inite products and sums; 2.5.2 belian Categories an additive category in which every morphism has a kernel and a co-kernel (so there is a zero object), and every monomorphism is a kernel and every epimorphism is a co-kernel. 2.6 Concrete category one in which the objects are sets and the morphisms are unctions; ie a category which has a aithul unctor C Sets (called the orgetul unctor). 2.6.1 Category o Sets One can even consider set theory rom the categorical point o view with the ollowing axioms: 1. Sets and unctions orm a category; 2. Sets have inite limits and co-limits; 3. Sets allow exponentiation; 4. Sets have a sub-object classiier (so orm a topos); this is a orm o comprehension axiom; 5. With a morphism T : 1 2; 6. Sets are oolean in the sense that the truth-value object 2 is given by 1 + 1;
Joseph Muscat 2015 15 7. 2 has two elements (up to isomorphism); 8. xiom o Choice (every epimorphism has a right-inverse); 9. There is an ininite (inductive) set. It then ollows that or every 0, 1 epimorphism and x : 1 morphisms (since 1 is unique, which gives 1 where is an epimorphism; but 0 0, so = 1; the axiom o choice gives a morphism x : 1 ); every monomorphism induces a complement monomorphism (the pullback o Ω along F : 1 Ω). 3 Research Questions Most grand questions in pure mathematics are o the ollowing type: 1. Syntax: given a set o mathematical structures/examples, to ind a minimal set o axioms common to all. 2. Semantics: given a set o axioms, to discover all mathematical examples satisying them; classiy all possible spaces X in a category i.e., give a concrete description o the spaces, up to isomorphism. This problem may be too hard or even impossible to answer, so the irst attempt is to restrict X to the smaller ones, or else ask an easier question 2a. Find a way o distinguishing spaces: given any two spaces X, Y is there a way o showing whether they are isomorphic or not? 2b. Can one show whether X is isomorphic to a known space? 2c. In particular is X isomorphic to the trivial space?