How Category Theory Works: The Elements & Distinctions Analysis of the Morphisms, Duality, and Universal Constructions in Sets David Ellerman University of Ljubljana, Slovenia Abstract The purpose of this paper is to show that the dual notions of elements & distinctions are the basic analytical concepts needed to unpack and analyze morphisms, duality, and universal constructions in the Sets, the category of sets and functions. The analysis extends directly to other concrete categories (groups, rings, vector spaces, etc.) where the objects are sets with a certain type of structure and the morphisms are functions that preserve that structure. Then the elements & distinctions-based denitions can be abstracted in purely arrow-theoretic way for abstract category theory. In short, the language of elements & distinctions is the conceptual language in which the category of sets is written, and abstract category theory gives the abstract arrows version of those denitions. Contents 1 Elements & Distinctions Analysis 2 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 The elements & distinctions analysis of canonical maps . . . . . . . . . . . . . . . . 2 1.3 Quantitative measures of elements & distinctions . . . . . . . . . . . . . . . . . . . . 3 2 The Elements & Distinctions Characterization of the Morphisms in Sets 3 2.1 Set functions transmit elements and reect distinctions . . . . . . . . . . . . . . . . . 3 2.2 Abstracting to arrow-theoretic denitions . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Duality interchanges elements & distinctions . . . . . . . . . . . . . . . . . . . . . . 5 3 The Elements & Distinctions Analysis of Products and Coproducts 5 3.1 The coproduct in Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 The product in Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4 The Elements & Distinctions Analysis of Equalizers and Coequalizers 6 4.1 The coequalizer in Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4.2 The equalizer in Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5 The Elements & Distinctions Analysis of Cartesian and Co-Cartesian Squares 7 5.1 The pushout or co-Cartesian square in Sets . . . . . . . . . . . . . . . . . . . . . . . 7 5.2 The pullback or Cartesian square in Sets . . . . . . . . . . . . . . . . . . . . . . . . 8 6 Speculative Concluding Remarks 8 1 1 Elements & Distinctions Analysis 1.1 Introduction The purpose of this paper is to show that the dual notions of "elements & distinctions" are the basic analytical concepts needed to unpack and analyze morphisms, duality, and universal constructions in the Sets, the category of sets and functions. The analysis extends directly to other concrete categories (groups, rings, vector spaces, etc.) where the objects are sets with a certain type of structure and the morphisms are functions that preserve or reect that structure. Then the elements & distinctionsbased denitions can be abstracted in purely arrow-theoretic way for abstract category theory. One way to approach the concepts of "elements" and "distinctions" is to start with the categorytheoretic duality between subsets and quotient sets (= partitions = equivalence relations): "The dual notion (obtained by reversing the arrows) of part[subobject] is the notion of partition." [8, p. 85]. That leads to the two dual forms of mathematical logic: the Boolean logic of subsets and the logic of partitions ([3]; [4]). If partitions are dual to subsets, then what is the dual concept that corresponds to the notion of elements of a subset? The notion dual to the elements of a subset is the notion of the distinctions of a partition (pairs of elements in distinct blocks of the partition). 1.2 The elements & distinctions analysis of canonical maps The canonical maps and the unique factor morphisms in the universal mapping properties in Sets are always constructed in the two ways that maps are canonically constructed with subsets and partitions. In the Boolean algebra of subsets } (U) of U , the partial order is the inclusion relation S  T for S; T  U , which induces the canonical injection S ! T . Thats the way canonical injective maps are dened using subsets. In the dual algebra of partitions (U) on U , the partial order is the renement relation between partitions and it induces a canonical map using renement. A partition  = fB;B0; :::g on U is a set of subsets of U (called blocks, B;B0; :::) that are mutually exclusive (i.e., disjoint) and jointly exhaustive (i.e., whose union is U). Given another partition  = fC;C 0; :::g on U , a partition  is said to rene  (or  is rened by ), written  - , if for every block B 2 , there is a block C 2  (necessarily unique) such that B  C. If we denote the set of distinctions or dits of a partition (ordered pairs of elements in di¤erent blocks) by dit (), the ditset of , then just as the partial order in } (U) is the inclusion of elements, so the renement partial order on (U) is the inclusion of distinctions, i.e.,  -  i¤ (if and only if) dit ()  dit (). And just as the inclusion ordering on subsets induces a canonical map between subsets, so the renement ordering on partitions induces a canonical surjection between partitions, namely  !  where B 2  is taken to the unique C 2  where B  C. Thats the way canonical surjective maps are dened using partitions. The claim that all "canonical" maps in Sets arise in these two ways (or by compositions of them) cannot be proven since "canonical" is an intuitive notion. But we will show that all the canonical maps and unique factor maps in the universal constructions (limits and colimits) in Sets arise in this way from the partial orders of the dual lattices (or algebras) of subsets and partitions. The top of the partition algebra (U) is the discrete partition 1 = ffuggu2U of all singletons and the bottom is the indiscrete partition 0 = fUg with only one block U . Since every partition  is rened by 1, i.e.,  1, there is the canonical surjection 1 !  that takes the singleton fug to the unique block B such that u 2 B. And similarly the top of the Boolean algebra } (U) is U , where each subset S  U induces the canonical injection S ! U . 2 Dualities Subset logic Partition logic Elements Elements u of S Dits (u; u0) of  Partial order Inclusion S  T  - : dit ()  dit () Canonical map S ! T  !  Top of partial order U all elements dit(1) = U2  , all dits Bottom of partial order ; no elements dit(0) = ;, no dits Table 1: Elements and distinctions in the dual logics 1.3 Quantitative measures of elements & distinctions The quantitative (normalized) counting measure of the elements in a subset gives the classical Laplacian notion of logicalprobability. The quantitative (normalized) counting measure of the distinctions in a partition gives the notion of logical entropy that underlies the Shannon notion of entropy (which is not a measure in the sense of measure theory) ([2]; [5]; [6]). That realizes the idea expressed in Gian-Carlo Rotas Fubini Lectures [9] (and in his lectures at MIT), where he noted that in view of duality between partitions and subsets, the lattice of partitions plays for information the role that the Boolean algebra of subsets plays for size or probability [7, p. 30] or symbolically: information partitions  probability subsets. Since Probability is a measure on the Boolean algebra of events that gives quantitatively the intuitive idea of the size of a set, we may ask by analogyfor some measure to capture a property for a partition like what size is to a set.The answer is the number of distinctions. The logical entropy h() = jdit()jjUU j of a partition  on a nite set U is that measure on the lattice of partitions on U , i.e., the normalized counting measure on the isomorphic lattice of partition relations (= ditsets), the binary relations that are the complements of the equivalence relations on U U . Since the logical entropy h () is also a normalized measure, it has a probability interpretation, i.e., h () is the probability that in two independent draws from U , one will get a distinction of , i.e.,  distinguishes, just as Pr (S) is interpreted as the probability that in one draw from U , one will get an element of S, i.e., S occurs. Duality in quant. measures Subset logic Partition logic Outcomes Elements u of U Distinctions (u; u0) 2 U  U Events Subsets S of U Partitions  of U Event occurs u 2 S (u; u0) 2 dit () Logical measure Pr (S) = jSjjU j h () = jdit()j jUU j Interpretation Prob. event S occurs Prob. partition  distinguishes Table 2: Logical measures on elements and distinctions 2 The Elements & Distinctions Characterization of the Morphisms in Sets 2.1 Set functions transmit elements and reect distinctions The duality between elements ("its") of a subset and distinctions ("dits") of a partition already appears in the very notion of a function between sets. The concepts of elements and distinctions provide the natural notions to specify the binary relations, i.e., subsets R  X  Y , that dene functions f : X ! Y . 3 A binary relation R  X  Y transmits elements if for each element x 2 X, there is an ordered pair (x; y) 2 R for some y 2 Y . A binary relation R  X  Y reects elements if for each element y 2 Y , there is an ordered pair (x; y) 2 R for some x 2 X. A binary relation R  X  Y transmits distinctions if for any pairs (x; y) and (x0; y0) in R, if x 6= x0, then y 6= y0. A binary relation R  X  Y reects distinctions if for any pairs (x; y) and (x0; y0) in R, if y 6= y0, then x 6= x0. The dual role of elements and distinctions can be seen if we translate the usual characterization of the binary relations that dene functions into the elements-and-distinctions language. In the usual treatment, a binary relation R  X  Y denes a function X ! Y if it is dened everywhere on X and is single-valued. But "being dened everywhere" is the same as transmitting (or "preserving" elements, and being single-valued is the same as reecting distinctions so the more natural denition is: a binary relation R is a function if it transmits elements and reects distinctions. What about the other two special types of relations, i.e., those which transmit (or preserve) distinctions or reect elements? The two important special types of functions are the injections and surjections, and they are dened by the other two notions: a function is injective if it transmits distinctions, and a function is surjective if it reects elements. Given a set function f : X ! Y with domain X and codomain Y , a subset of the codomain is determined as the image f(X) of an injective function f , and a partition on the domain is determined as the coimage (or inverse-image) ff 1(y)gy2Y of a surjective function f . 2.2 Abstracting to arrow-theoretic denitions One of our themes is that the concepts of elements and distinctions unpack and analyze the basic category theoretic concepts in the basic category Sets, and they are abstracted into purely arrowtheoretic denitions in abstract category theory. For instance, the elements & distinctions denitions of injections and surjections yield "arrow-theoretic" characterizations which can then be applied in any category to provide the usual category-theoretic dual denitions of monomorphisms (injections for set functions) and epimorphisms (surjections for set functions). Two set functions f; g : X  Y are di¤erent, i.e., f 6= g, if there is an element x of X such that their values f (x) and g (x) are a distinction of Y , i.e., f (x) 6= g (x). Hence if f and g are followed by a function h : Y ! Z, then the compositions hf; hg : X ! Y ! Z must be di¤erent if h preserves distinctions (so that the distinction f(x) 6= g (x) is preserved as hf (x) 6= hg (x)), i.e., if h is injective. Thus in the category of sets, h being injective is characterized by: for any f; g : X  Y , "f 6= g implies hf 6= hg" or equivalently, "hf = hg implies f = g" which is the general category-theoretic denition of a monomorphism. In a similar manner, if we have functions f; g : X ! Y where f 6= g, i.e., where there is an element x of X such that their values f (x) and g (x) are a distinction of Y , then suppose the functions are preceded by a function h : W ! X. Then the compositions fh; gh : W ! X ! Y must be di¤erent if h reects elements (so that the element x where f and g di¤er is sure to be in the image of h), i.e., if h is surjective. Thus in the category of sets, h being surjective is characterized by: for any f; g : X  Y , "f 6= g implies fh 6= fg" or "fh = gh implies f = g" which is the general category-theoretic denition of an epimorphism. Hence the dual interplay of the notions of elements & distinctions can be seen as yielding the arrow-theoretic characterizations of injections and surjections which are lifted into the general categorical dual denitions of monomorphisms and epimorphisms. 4 2.3 Duality interchanges elements & distinctions The reverse-the-arrows duality of category theory is the abstraction from the reversing of the roles of elements & distinctions in dualizing Sets to Setsop. That is, a concrete morphism in Setsop is a binary relation, which might be called a cofunction, that preserves distinctions and reects elementsinstead of preserving elements and reecting distinctions. Thus with every binary relation f  XY that is a function f : X ! Y , there is a binary relation fop  Y X that is a cofunction fop : Y ! X. The interchange of elements and distinctions means that the coimage of a function becomes the image of a cofunction and the image of a function becomes the coimage of a cofunction. For the universal constructions in Sets, the interchange in the roles of elements and distinctions interchanges each construction and its dual: products and coproducts, equalizers and coequalizers, and in general limits and colimits. That is then abstracted to make the reverse-the-arrows duality in abstract category theory. This begins to illustrate our theme that the language of elements & distinctions is the conceptual language in which the category of sets and functions is written, and abstract category theory gives the abstract-arrows version of those denitions. Hence we turn to universal constructions for further analysis. 3 The Elements & Distinctions Analysis of Products and Coproducts 3.1 The coproduct in Sets Given two sets X and Y in Sets, the idea of the coproduct is to create the set with the maximum number of elements starting with X and Y . Since X and Y may overlap, we must make two copies of the elements in the intersection. Hence the relevant operation is not the union of sets X [Y but the disjoint union XtY . To take the disjoint union of a set X with itself, a copy X = fx : x 2 Xg of X is made so that X tX can be constructed as X [X. In a similar manner, if X and Y overlap, then X t Y = X [ Y . Then the inclusions X;Y  X t Y , give the canonical injections iX : X ! X t Y and iY : Y ! X t Y . The universal mapping property for the coproduct in Sets is that given any other maps f : X ! Z and g : Y ! Z, there is a unique map f t g : X t Y ! Z such that X iX! X t Y ftg! Z = X f! Z and Y iY! X t Y ftg! Z = Y g! Z. X iX ! X t Y iY   Y &f 9! #ftg g . Z Coproduct diagram For the universal constructions with the prex "co-" (the "co-constructions") as in coproduct or coequalizer, the unique factor maps in Sets are generated by partition renements. From the data f : X ! Z and g : Y ! Z, we need to construct the unique factor map X t Y ! Z. The map f contributes the coimage (or inverse-image) f 1 partition on X and g contributes g 1 on Y so the blocks of f 1 together with the blocks of g 1 make up a partition f 1 t g 1 on X t Y:To dene the unique factor map f t g : X t Y ! Z, we use the renement relation f 1 t g 1 1XtY . For each element singleton in the discrete partition 1XtY on X t Y , there is a unique block f 1 (z) or g 1 (z) in f 1 t g 1 containing it, so the element in the singleton gets mapped to the corresponding z, and the factor mapping property automatically holds. 5 3.2 The product in Sets Given two sets X and Y in Sets, the idea of the product is to create the set with the maximum number of distinctions starting with X and Y . The product in Sets is usually constructed as the set of ordered pairs in the Cartesian product X Y . But to emphasize the point about distinctions, we might employ the same trick of markingthe elements of Y , particularly when Y = X, with an asterisk. Then an alternative construction of the product in Sets is the set of unordered pairsXY = ffx; yg : x 2 X; y 2 Y g which in the case of Y = X would beXX = ffx; xg : x 2 X;x 2 Xg. This alternative construction of the product (isomorphic to the Cartesian product) emphasizes the distinctions formed from X and Y so the ordering in the ordered pairs of the usual construction X  Y is only a way to make the same distinctions. The set X denes a partition X on XY whose blocks are Bx = f(x; y) : y 2 Y g = fxgY for each x 2 X, and Y denes a partition Y whose blocks are By = f(x; y) : x 2 Xg = Xfyg for each y 2 Y . Since X ; Y 1XY , the induced surjections are the canonical projections pX : XY ! X and pY : X  Y ! Y . The universal mapping property for the product in Sets is that given any other maps f : Z ! X and g : Z ! Y , there is a unique map hf; gi : Z ! X  Y such that Z hf;gi! X  Y pX! X = Z f! X and Z hf;gi! X  Y pY! Y = Z g! Y . Z .f 9! #hf;gi g & X pX   X  Y pY ! Y Product diagram For the universal constructions without the prex "co-" (the "non-co-constructions") as in product or equalizer, the unique factor maps in Sets are generated by set inclusions. From the data f : Z ! X and g : Z ! Y , we need to construct the unique factor map Z ! X  Y . The map f contributes the image f (U) subset of X and g contributes the image g (U) subset of Y so we have the subset f (U)  g (U) of the product X  Y where for each z 2 Z, gives the pair (f (z) ; g (z)) 2 f (U)  g (U). To dene the unique factor map hf; gi : Z ! X  Y , we use the inclusion relation f (U)  g (U)  X  Y . Each element z 2 Z determines the element (f (z) ; g (z)) 2 f (U)  g (U) so by the inclusion, we have the map hf; gi : Z ! X  Y where z 7 ! (f (z) ; g (z)) 2 X  Y , and the factor mapping property automatically holds. 4 The Elements & Distinctions Analysis of Equalizers and Coequalizers 4.1 The coequalizer in Sets For the equalizer and coequalizer, the data is not just two sets but two parallel maps f; g : X  Y . Then each element x 2 X, gives us a pair f (x) and g (x) so we take the equivalence relation  dened on Y that is generated by f (x)  g (x) for any x 2 X. Then the coequalizer is the quotient set C = Y=  . When  is represented as a partition on Y , then it is rened by the discrete partition 1Y on Y , and that renement denes the canonical map can: : Y ! Y= . For the universality property, let h : Y ! Z be such that hf = hg. Then we need to show there is a unique renement-dened map h : Y= ! Z such that hcan: = h. X f  g Y can:! Y=  &h 9! #h Z 6 Coequalizer diagram We already have one partition  on Y which was generated by f (x)  g (x). Since hf = hg, we know that hf (x) = hg (x) so the coimage or inverse-image h 1 has to at least identify f (x) and g (x) (and perhaps identify other elements) which means that h 1 -. Hence for each element of Y= , i.e., each block b in the partition , there is a unique block h 1 (z) containing that block, so induced map is h (b) = z. Commutativity requires h 1 - so the map h thereby dened Y= ! Z is thereby unique to make the triangle commute. 4.2 The equalizer in Sets The data for the equalizer construction is the same two parallel maps f; g : X  Y . The equalizer is the E = fx 2 X : f (x) = g (x)g  X so the map induced by that inclusion is the canonical map can: : E ! X. The universal property is that for any other map h : Z ! X such that fh = gh;then 9!h : Z ! E such that hcan: = h. E can:! X f  g Y 9! "h h% Z Equalizer diagram The image of h(Z) must satisfy fh (z) = gh (z) for all z 2 Z, so all the elements h (z) 2 X where f and g agree give h (Z)  E; and thus the factor map h is induced by that inclusion. That inclusion is implied by the fh = gh condition so the factor map is unique. 5 The Elements & Distinctions Analysis of Cartesian and Co-Cartesian Squares 5.1 The pushout or co-Cartesian square in Sets It is a standard theorem of category theory that if a category has products and equalizers, then it has all limits, and if it has coproducts and coequalizers, then it has all colimits. Since we have presented the elements & distinctions analysis of products and coproducts, and of equalizers and coequalizers, the analysis extends to all limits and colimits. However, the theme would be better illustrated by considering some more complicated limits and colimits such as Cartesian and co-Cartesian squares, i.e., pullbacks and pushouts. For the pushout or co-Cartesian square, the data are two maps f : Z ! X and g : Z ! Y so we have the two parallel maps Z f! X iX :! X t Y and Z g! Y iY :! X t Y and then we can take their coequalizer C formed by the equivalence relation  on the common codomain X t Y which is the equivalence relation generated by x  y if there is a z 2 Z such that f (z) = x and g (z) = y. The canonical maps X ! X t Y=  and Y ! X t Y=  are just the canonical injections into the disjoint union followed by the canonical map of the coequalizer construction analyzed above. As the composition of a canonical injection with a canonical surjection, those canonical maps need not be either injective or surjective. Z f! X = X g # & #can:: Y can:! C = X t Y=  #h q 9! &h Y h0! U 7 Pushout or co-Cartesian square diagram For the universal mapping property, consider any h : X ! U and h0 : Y ! U such that hf = h0g. Then h 1 is a partition on X and h0 1 is a partition on Y so let h 1 t h0 1 be the partition on X t Y with the blocks h 1 (u) t h0 1 (u) for some u 2 U . The condition that for any z 2 Z, hf (z) = h0g (z) = u for some u 2 U means that h 1 t h0 1 must make at least the identications of the coequalizer (and perhaps more) so that h 1 t h0 1 is rened by  as partitions on X t Y . Each block in h 1 t h0 1 has the form h 1 (u) t h0 1 (u) for some u and h 1 t h0 1 - so each block b in  is contained in a block of the form h 1 (u) t h0 1 (u) of h 1 t h0 1. Hence that block b of  is mapped by h to u, and the commutativity properties follow automatically. 5.2 The pullback or Cartesian square in Sets For the Cartesian square or pullback, the data are two maps f : X ! Z and g : Y ! Z. We then have two parallel maps X  Y  Z (the projections followed by f or g) so we take the pullback as their equalizer E. The canonical maps E ! X and E ! Y are the compositions of the canonical injective map E ! X  Y followed by the canonical projections pX : X  Y ! X and pY : X  Y ! Y . As the composition of a canonical injection with a canonical surjection, those canonical maps need not be either injective or surjective. U h! X 9! &h q #h0 E  X  Y can:! X #can: #f Y = Y g! Z Pullback or Cartesian square diagram For the universality property, consider any other maps h : U ! X and h0 : U ! Y such that fh = gh0. We know that h0 (u) and h (u) are elements such that (h (u) ; h0 (u)) 2 E so for the images, there is the inclusion h (U)  h0 (U)  E. Hence that inclusion gives the uniquely dened map h : U ! h (U)  h0 (U)  E where u 7 ! (h (u) ; h0 (u)) 2 E so that the commutativity property automatically holds. 6 Speculative Concluding Remarks We shown how the dual concepts of elements & distinctions can be used to account for the notion of morphism, duality, and for the universal constructions in Setswhich are then abstracted in abstract category theory. This suggests that the notions have some broader signicance. One possibility is they are respectively mathematical versions of the old metaphysical concepts of matter (or substance) and form (as in in-form-ation). The matter versus form idea [1] can be illustrated by comparing the two lattices of subsets and partitions on a setthe two lattices that we saw dened the canonical morphisms and unique factor maps in the universal constructions of Sets. For U = fa; b; cg, start at the bottom and move towards the top of each lattice. 8 Figure 1: Moving up the subset and partition lattices. At the bottom of the Boolean subset lattice is the empty set ; which represents no substance (no its). As one moves up the lattice, new elements of substance are created that are always fully formed until nally one reaches the top, the universe U . Thus new substance is created in moving up the lattice but each element is fully formed and thus distinguished from the other elements. At the bottom of the partition lattice is the indiscrete partition or "blob" 0 = fUg (where the universe set U makes one block) which represents all the substance or matter but with no distinctions to in-form the substance (no dits). As one moves up the lattice, no new substance is created but distinctions are created that in-form the indistinct elements as they become more and more distinct. Finally one reaches the top, the discrete partition 1, where all the elements of U have been fully formed. A partition combines indeniteness (within blocks) and deniteness (between blocks). At the top of the partition lattice, the discrete partition 1 = ffug : fug  Ug is the result making all the distinctions to eliminate any indeniteness. Thus one ends up at the "same place" (universe U of fully formed entities) either way, but by two totally di¤erent but dual creation stories:  creating elements (as in creating fully-formed matter out of nothing) versus  creating distinctions (as in starting with a totally undi¤erentiated matter and then in a big bangstart making distinctions, e.g., breaking symmetries, to give form to the matter). Moreover, we have seen that:  the quantitative increase in substance (normalized number of elements) moving up in the subset lattice is measured by logical probability, and  the quantitative increase in form (normalized number of distinctions) moving up in the partition lattice is measured by logical information ([2]; [5]). 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