Category Theory

Category theoryFrom Wikipedia, the free encyclopedia

Contents

1 Abstraction 11.1 Origins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Themes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2.1 Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Instantiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.3 Material process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.4 Ontological status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.5 Physicality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.6 Referencing and referring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.7 Simplication and ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.8 Thought processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 As used in dierent disciplines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.1 In art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.2 In computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.3 In linguistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.4 In mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.5 In music . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.6 In neurology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.7 In philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.8 In psychology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.9 In social theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Category theory 92.1 An abstraction of other mathematical concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Categories, objects, and morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.3 Natural transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Categories, objects, and morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

i

ii CONTENTS

2.3.1 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.2 Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Natural transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6 Other concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.6.1 Universal constructions, limits, and colimits . . . . . . . . . . . . . . . . . . . . . . . . . 132.6.2 Equivalent categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.6.3 Further concepts and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.6.4 Higher-dimensional categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.7 Historical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.11 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Mathematical structure 193.1 Example: the real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Chapter 1

Abstraction

For specic types of abstraction and other uses of the term, see Abstraction (disambiguation).

Abstraction in its main sense is a conceptual process by which general rules and concepts are derived from theusage and classication of specic examples, literal (real or concrete) signiers, rst principles, or other methods.An abstraction is the product of this processa concept that acts as a super-categorical noun for all subordinateconcepts, and connects any related concepts as a group, eld, or category.[1]

Conceptual abstractions may be formed by ltering the information content of a concept or an observable phenomenon,selecting only the aspects which are relevant for a particular purpose. For example, abstracting a leather soccer ballto the more general idea of a ball selects only the information on general ball attributes and behavior, eliminating theother characteristics of that particular ball.[1] In a typetoken distinction, a type (e.g., a 'ball') is more abstract thanits tokens (e.g., 'that leather soccer ball').Abstraction in its secondary use is a material process,[2] discussed in the themes below.

1.1 Origins

Thinking in abstractions is considered by anthropologists, archaeologists, and sociologists to be one of the key traitsin modern human behaviour, which is believed to have developed between 50,000 and 100,000 years ago. Its devel-opment is likely to have been closely connected with the development of human language, which (whether spoken orwritten) appears to both involve and facilitate abstract thinking.Abstraction involves induction of ideas or the synthesis of particular facts into one general theory about something. Itis the opposite of specication, which is the analysis or breaking-down of a general idea or abstraction into concretefacts. Abstraction can be illustrated with Francis Bacon's Novum Organum (1620), a book of modern scienticphilosophy written in the late Elizabethan era[3] of England to encourage modern thinkers to collect specic factsbefore making any generalizations. Bacon used and promoted induction as an abstraction tool, and it countered theancient deductive-thinking approach that had dominated the intellectual world since the times of Greek philosopherslike Thales, Anaximander, and Aristotle. Thales (c. 624 BC c. 546 BCE) believed that everything in the universecomes from one main substance, water. He deduced or specied from a general idea, everything is water, to thespecic forms of water such as ice, snow, fog, and rivers. Modern scientists can also use the opposite approach ofabstraction, or going from particular facts collected into one general idea, such as the motion of the planets (Newton(1642-1727)). When determining that the sun is the center of our solar system (Copernicus (1473-1543)), scientistshad to utilize thousands of measurements to nally conclude that Mars moves in an elliptical orbit about the sun(Kepler (1571-1630)), or to assemble multiple specic facts into the law of falling bodies (Galileo (1564-1642)).

1.2 Themes

1

2 CHAPTER 1. ABSTRACTION

1.2.1 CompressionAn abstraction can be seen as a compression process,[4] mapping multiple dierent pieces of constituent data to asingle piece of abstract data;[5] based on similarities in the constituent data, for example, many dierent physical catsmap to the abstraction CAT. This conceptual scheme emphasizes the inherent equality of both constituent andabstract data, thus avoiding problems arising from the distinction between abstract and "concrete". In this sense theprocess of abstraction entails the identication of similarities between objects, and the process of associating theseobjects with an abstraction (which is itself an object).

For example, picture 1 below illustrates the concrete relationship Cat sits on Mat.

Chains of abstractions can be construed,[6] moving from neural impulses arising from sensory perception to basicabstractions such as color or shape, to experiential abstractions such as a specic cat, to semantic abstractions suchas the idea of a CAT, to classes of objects such as mammals and even categories such as "object" as opposed toaction.

For example, graph 1 below expresses the abstraction agent sits on location. This conceptual schemeentails no specic hierarchical taxonomy (such as the one mentioned involving cats and mammals), onlya progressive exclusion of detail.

1.2.2 InstantiationThings that do not exist at any particular place and time are often considered abstract. By contrast, instances, ormembers, of such an abstract thing might exist in many dierent places and times. Those abstract things are then saidto be multiply instantiated, in the sense of picture 1, picture 2, etc., shown below. It is not sucient, however, to deneabstract ideas as those that can be instantiated and to dene abstraction as the movement in the opposite directionto instantiation. Doing so would make the concepts cat and telephone abstract ideas since despite their varyingappearances, a particular cat or a particular telephone is an instance of the concept cat or the concept telephone.Although the concepts cat and telephone are abstractions, they are not abstract in the sense of the objects in graph1 below. We might look at other graphs, in a progression from cat to mammal to animal, and see that animal is moreabstract than mammal; but on the other hand mammal is a harder idea to express, certainly in relation to marsupialor monotreme.Perhaps confusingly, some philosophies refer to tropes (instances of properties) as abstract particularse.g., theparticular redness of a particular apple is an abstract particular. This is similar to qualia and sumbebekos.

1.2.3 Material processStill retaining the primary meaning of 'abstrere' or 'to draw away from', the abstraction of money, for example, worksby drawing away from the particular value of things allowing completely incommensurate objects to be compared (seethe section on 'Physicality' below). Karl Marx's writing on the commodity abstraction recognizes a parallel process.The state (polity) as both concept and material practice exemplies the two sides of this process of abstraction.Conceptually, 'the current concept of the state is an abstraction from the much more concrete early-modern use asthe standing or status of the prince, his visible estates. At the same time, materially, the 'practice of statehood isnow constitutively and materially more abstract than at the time when princes ruled as the embodiment of extendedpower'.[7]

Further information: Power projection and Display behavior

1.2.4 Ontological statusThe way that physical objects, like rocks and trees, have being diers from the way that properties of abstract conceptsor relations have being, for example the way the concrete, particular, individuals pictured in picture 1 exist diers fromthe way the concepts illustrated in graph 1 exist. That dierence accounts for the ontological usefulness of the wordabstract. The word applies to properties and relations to mark the fact that, if they exist, they do not exist in spaceor time, but that instances of them can exist, potentially in many dierent places and times.

1.2. THEMES 3

1.2.5 PhysicalityFurther information: History of accounting Ancient history

A physical object (a possible referent of a concept or word) is considered concrete (not abstract) if it is a particularindividual that occupies a particular place and time. However, in the secondary sense of the term 'abstraction', thisphysical object can carry materially abstracting processes. For example, record-keeping aids throughout the FertileCrescent included calculi (clay spheres, cones, etc.) which represented counts of items, probably livestock or grains,sealed in containers. According to Schmandt-Besserat (1981), these clay containers contained tokens, the total ofwhich were the count of objects being transferred. The containers thus served as something of a bill of lading or anaccounts book. In order to avoid breaking open the containers for the count, marks were placed on the outside of thecontainers. These physical marks, in other words, acted as material abstractions of a materially abstract process ofaccounting, using conceptual abstractions (numbers) to communicate its meaning.[8][9]

Abstract things are sometimes dened as those things that do not exist in reality or exist only as sensory experiences,like the color red. That denition, however, suers from the diculty of deciding which things are real (i.e. whichthings exist in reality). For example, it is dicult to agree to whether concepts like God, the number three, andgoodness are real, abstract, or both.An approach to resolving such diculty is to use predicates as a general term for whether things are variously real, ab-stract, concrete, or of a particular property (e.g., good). Questions about the properties of things are then propositionsabout predicates, which propositions remain to be evaluated by the investigator. In the graph 1 below, the graphicalrelationships like the arrows joining boxes and ellipses might denote predicates.

1.2.6 Referencing and referringAbstractions sometimes have ambiguous referents; for example, "happiness" (when used as an abstraction) can referto as many things as there are people and events or states of being which make them happy. Likewise, "architecture"refers not only to the design of safe, functional buildings, but also to elements of creation and innovation which aimat elegant solutions to construction problems, to the use of space, and to the attempt to evoke an emotional responsein the builders, owners, viewers and users of the building.

1.2.7 Simplication and orderingAbstraction uses a strategy of simplication, wherein formerly concrete details are left ambiguous, vague, or unde-ned; thus eective communication about things in the abstract requires an intuitive or common experience betweenthe communicator and the communication recipient. This is true for all verbal/abstract communication.

agent

Sitting

agent

MATCAT: Elsie

Conceptual graph for A Cat sitting on the Mat (graph 1)

For example, many dierent things can be red. Likewise, many things sit on surfaces (as in picture 1, to the right). Theproperty of redness and the relation sitting-on are therefore abstractions of those objects. Specically, the conceptualdiagram graph 1 identies only three boxes, two ellipses, and four arrows (and their ve labels), whereas the picture 1shows much more pictorial detail, with the scores of implied relationships as implicit in the picture rather than withthe nine explicit details in the graph.Graph 1 details some explicit relationships between the objects of the diagram. For example the arrow between theagent and CAT:Elsie depicts an example of an is-a relationship, as does the arrow between the location and the MAT.

4 CHAPTER 1. ABSTRACTION

Cat on Mat (picture 1)

The arrows between the gerund/present participle SITTING and the nouns agent and location express the diagram'sbasic relationship; agent is SITTING on location ; Elsie is an instance of CAT.[10]

Although the description sitting-on (graph 1) is more abstract than the graphic image of a cat sitting on a mat (pic-ture 1), the delineation of abstract things from concrete things is somewhat ambiguous; this ambiguity or vaguenessis characteristic of abstraction. Thus something as simple as a newspaper might be specied to six levels, as inDouglas Hofstadter's illustration of that ambiguity, with a progression from abstract to concrete in Gdel, Escher,Bach (1979):[11]

(1) a publication(2) a newspaper

(3) The San Francisco Chronicle(4) the May 18 edition of the The San Francisco Chronicle

(5) my copy of the May 18 edition of the The San FranciscoChronicle

(6) my copy of the May 18 edition of the The San FranciscoChronicle as it was when I rst picked it up (as contrastedwith my copy as it was a few days later: in my replace,burning)

An abstraction can thus encapsulate each of these levels of detail with no loss of generality. But perhaps a detective orphilosopher/scientist/engineer might seek to learn about something, at progressively deeper levels of detail, to solvea crime or a puzzle.

1.2.8 Thought processesIn philosophical terminology, abstraction is the thought process wherein ideas[12] are distanced from objects.

1.3. AS USED IN DIFFERENT DISCIPLINES 5

1.3 As used in dierent disciplines

1.3.1 In artMain article: Abstraction (art)

Typically, abstraction is used in the arts as a synonym for abstract art in general. Strictly speaking, it refers toart unconcerned with the literal depiction of things from the visible worldit can, however, refer to an object orimage which has been distilled from the real world, or indeed, another work of art.[13] Artwork that reshapes thenatural world for expressive purposes is called abstract; that which derives from, but does not imitate a recognizablesubject is called nonobjective abstraction. In the 20th century the trend toward abstraction coincided with advances inscience, technology, and changes in urban life, eventually reecting an interest in psychoanalytic theory.[14] Later still,abstraction was manifest in more purely formal terms, such as color, freedom from objective context, and a reductionof form to basic geometric designs.[15]

1.3.2 In computer scienceMain article: Abstraction (computer science)

Computer scientists use abstraction to make models that can be used and re-used without having to re-write all theprogram code for each new application on every dierent type of computer. They communicate their solutions withthe computer by writing source code in some particular computer language which can be translated into machine codefor dierent types of computer to execute. Abstraction allows program designers to separate categories and conceptsrelated to computing problems from specic instances of implementation. This means that the program code canbe written so that it does not depend on the specic details of supporting applications, operating system software orhardware, but on an abstract concept of the solution to the problem that can then be integrated with the system withminimal additional work.

1.3.3 In linguisticsMain article: Abstraction (linguistics)

Abstraction is frequently applied in linguistics so as to allow phenomena of language to be analyzed at the desiredlevel of detail. A commonly considered abstraction is the phoneme, which abstracts speech sounds in such a wayas to neglect details that cannot serve to dierentiate meaning. Other analogous kinds of abstractions (sometimescalled "emic units") considered by linguists include morphemes, graphemes, and lexemes. Abstraction also arises inthe relation between syntax, semantics, and pragmatics. Pragmatics involves considerations that make reference tothe user of the language; semantics considers expressions and what they denote (the designata) abstracted from thelanguage user; and syntax considers only the expressions themselves, abstracted from the designata.

1.3.4 In mathematicsMain article: Abstraction (mathematics)

Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removingany dependence on real world objects with which it might originally have been connected, and generalizing it so thatit has wider applications or matching among other abstract descriptions of equivalent phenomena.The advantages of abstraction in mathematics are:

It reveals deep connections between dierent areas of mathematics Known results in one area can suggest conjectures in a related area Techniques and methods from one area can be applied to prove results in a related area

6 CHAPTER 1. ABSTRACTION

The main disadvantage of abstraction is that highly abstract concepts are more dicult to learn, and require a degreeof mathematical maturity and experience before they can be assimilated.

1.3.5 In musicIn music, the term abstraction can be used to describe improvisatory approaches to interpretation, and may sometimesindicate abandonment of tonality. Atonal music has no key signature, and is characterized by the exploration ofinternal numeric relationships.[16]

1.3.6 In neurologyFurther information: Intelligence, Mental rotation and Mental operations

A recent meta-analysis suggests that the verbal system has greater engagement for abstract concepts when the per-ceptual system is more engaged for processing of concrete concepts. This is because abstract concepts elicit greaterbrain activity in the inferior frontal gyrus and middle temporal gyrus compared to concrete concepts which elicitgreater activity in the posterior cingulate, precuneus, fusiform gyrus, and parahippocampal gyrus.[17] Other researchinto the human brain suggests that the left and right hemispheres dier in their handling of abstraction. For example,one meta-analysis reviewing human brain lesions has shown a left hemisphere bias during tool usage.[18]

1.3.7 In philosophyAbstraction in philosophy is the process (or, to some, the alleged process) in concept formation of recognizing someset of common features in individuals, and on that basis forming a concept of that feature. The notion of abstraction isimportant to understanding some philosophical controversies surrounding empiricism and the problem of universals.It has also recently become popular in formal logic under predicate abstraction. Another philosophical tool fordiscussion of abstraction is thought space.

1.3.8 In psychologyCarl Jung's denition of abstraction broadened its scope beyond the thinking process to include exactly four mutuallyexclusive, dierent complementary psychological functions: sensation, intuition, feeling, and thinking. Together theyform a structural totality of the dierentiating abstraction process. Abstraction operates in one of these functions whenit excludes the simultaneous inuence of the other functions and other irrelevancies, such as emotion. Abstractionrequires selective use of this structural split of abilities in the psyche. The opposite of abstraction is concretism.Abstraction is one of Jungs 57 denitions in Chapter XI of Psychological Types.

There is an abstract thinking, just as there is abstract feeling, sensation and intuition. Abstract thinkingsingles out the rational, logical qualities ... Abstract feeling does the same with ... its feeling-values. ... Iput abstract feelings on the same level as abstract thoughts. ... Abstract sensation would be aesthetic asopposed to sensuous sensation and abstract intuition would be symbolic as opposed to fantastic intuition.(Jung, [1921] (1971):par. 678).

1.3.9 In social theoryIn social theory, abstraction is used as both an ideational and material process. Alfred Sohn-Rethel, asked Can therebe abstraction other than by thought?"[2] He used the example of commodity abstraction to show that abstractionoccurs in practice as people create systems of abstract exchange that extend beyond the immediate physicality of theobject and yet have real and immediate consequences. This work was extended through the 'Constitutive Abstraction'approach of writers associated with the Journal Arena. Two books that have taken this theme of the abstraction ofsocial relations as an organizing process in human history are Nation Formation: Towards a Theory of Abstract Com-munity.(1996) and the second volume of Towards a Theory of Abstract Community, published in 2006: Globalism,Nationalism, Tribalism: Bringing Theory Back InVolume 2 of Towards a Theory of Abstract Community. Thesebooks argue that the nation is an abstract community bringing together strangers who will never meet as such; thus

1.4. SEE ALSO 7

constituting materially real and substantial, but abstracted and mediated relations. The books suggest that contempo-rary processes of globalization and mediatization have contributed to materially abstracting relations between people,with major consequences for how we live our lives.It can be easily argued that abstraction is an elementary methodological tool in several disciplines of social science.These disciplines have denite and dierent man concepts that highlight those aspects of man and his behaviour byidealization that are relevant for the given human science. For example, homo sociologicus is the man as sociologyabstracts and idealizes it, depicting man as a social being. Moreover, we could talk about homo cyber sapiens[19] (theman who can extend his biologically determined intelligence thanks to new technologies), or homo creativus[20] (whois simply creative).Abstraction (combined with Weberian idealization) plays a crucial role in economics. Breaking away from directlyexperienced reality was a common trend in 19th century sciences (especially physics), and this was the eort whichwas fundamentally determined the way economics tried and still tries to approach the economic aspects of social life.It is abstraction we meet in the case of both Newtons physics and the neoclassical theory, since the goal was to graspthe unchangeable and timeless essence of phenomena. For example, Newton created the concept of the materialpoint by following the abstraction method so that he abstracted from the dimension and shape of any perceptibleobject, preserving only inertia and translational motion. Material point is the ultimate and common feature of allbodies. Neoclassical economists created the indenitely abstract notion of homo oeconomicus by following the sameprocedure. Economists abstract from all individual and personal qualities in order to get to those characteristics thatembody the essence of economic activity. Eventually, it is the substance of the economic man that they try to grasp.Any characteristic beyond it only disturbs the functioning of this essential core.[21]

1.4 See also

1.5 Notes[1] Suzanne K. Langer (1953), Feeling and Form: a theory of art developed from Philosophy in a New Key p. 90: "Sculptural

form is a powerful abstraction from actual objects and the three-dimensional space which we construe ... through touchand sight.

[2] Alfred Sohn-Rethel, Intellectual and manual labour: A critique of epistemology, Humanities Press, 1977

[3] Hesse, M. B. (1964), Francis Bacons Philosophy of Science, in A Critical History of Western Philosophy, ed. D. J.O'Connor, New York, pp. 14152.

[4] Chaitin, Gregory (2006), The Limits Of Reason (PDF)

[5] Murray Gell-Mann (1995) "What is complexity? Remarks on simplicity and complexity by the Nobel Prize-winning authorof The Quark and the Jaguar" Complexity states the 'algorithmic information complexity' (AIC) of some string of bits isthe shortest length computer program which can print out that string of bits.

[6] Ross,L. (1987). The Problem of Construal in Social Inference and Social Psychology. In N. Grunberg, R.E. Nisbett, J.Singer (eds), A Distinctive Approach to psychological research: the inuence of Stanley Schacter. Hillsdale, NJ: Earlbaum.

[7] James, Paul (2006). Globalism, Nationalism, Tribalism: Bringing Theory Back InVolume 2 of Towards a Theory ofAbstract Community. London: Sage Publications., pages 318-19.

[8] Eventually (Schmandt-Besserat estimates it took 4000 years) the marks on the outside of the containers were all that wereneeded to convey the count, and the clay containers evolved into clay tablets with marks for the count.

[9] Robson, Eleanor (2008). Mathematics in Ancient Iraq. ISBN 978-0-691-09182-2.. p. 5: these calculi were in use in Iraqfor primitive accounting systems as early as 32003000 BCE, with commodity-specic counting representation systems.Balanced accounting was in use by 30002350 BCE, and a sexagesimal number system was in use 23502000 BCE.

[10] Sowa, John F. (1984). Conceptual Structures: Information Processing in Mind and Machine. Reading, MA: Addison-Wesley. ISBN 978-0-201-14472-7.

[11] Douglas Hofstadter (1979) Gdel, Escher, Bach

[12] But an idea can be symbolized. A symbol is any device whereby we are enabled to make an abstraction. -- p.xi andchapter 20 of Suzanne K. Langer (1953), Feeling and Form: a theory of art developed from Philosophy in a New Key: NewYork: Charles Scribners Sons. 431 pages, index.

8 CHAPTER 1. ABSTRACTION

[13] Encyclopdia Britannica

[14] Catherine de Zegher and Hendel Teicher (eds.), 3 X Abstraction. NY/New Haven: The Drawing Center/Yale UniversityPress. 2005. ISBN 0-300-10826-5

[15] National Gallery of Art: Abstraction.

[16] Washington State University: Glossary of Abstraction.

[17] Wang, Jing; Conder, Julie A.; Blitzer, David N.; Shinkareva, Svetlana V. (2010). Neural Representation of Abstract andConcrete Concepts: A Meta-Analysis of Neuroimaging Studies. HumanBrainMapping 31: 14591468. doi:10.1002/hbm.20950.

[18] James W. Lewis Cortical Networks Related to Human Use of Tools 12 (3): 211231 The Neuroscientist (June 1, 2006).

[19] Steels, Luc (1995). The Homo Cyber Sapiens, the Robot Homonidus Intelligens, and the Articial Life Approach to ArticialIntelligence. Brussel: Vrije Universiteit, Articial Intelligence Laboratory.

[20] Inkinen, Sam (2009). Homo Creativus Creativity and Serendipity Management in Third Generation Science and Tech-nology Parks. Science and Public Policy 36 (7): 537548.

[21] Galbcs, Peter (2015). Methodological Principles and an Epistemological Introduction. The Theory of New ClassicalMacroeconomics. A Positive Critique. Heidelberg/New York/Dordrecht/London: Springer. pp. 152. ISBN 978-3-319-17578-2.

1.6 References The American Heritage Dictionary of the English Language, 3rd edition, Houghton Miin (1992), hardcover,

2140 pages, ISBN 0-395-44895-6

James, Paul (1996). Nation Formation: Towards a Theory of Abstract Community. London: Sage Publications. James, Paul (2006). Globalism, Nationalism, Tribalism: Bringing Theory Back In - Volume 2 of Towards a

Theory of Abstract Community. London: Sage Publications. Jung, C.G. [1921] (1971). Psychological Types, Collected Works, Volume 6, Princeton, NJ: Princeton Uni-

versity Press. ISBN 0-691-01813-8. Sohn-Rethel, Alfred (1977) Intellectual and manual labour: A critique of epistemology, Humanities Press. Schmandt-Besserat, Denise (1981). Decipherment of the Earliest Tablets. Science 211 (4479): 283285.

Bibcode:1981Sci...211..283S. doi:10.1126/science.211.4479.283. PMID 17748027..

1.7 External links Abstraction at the Indiana Philosophy Ontology Project Abstraction at PhilPapers Internet Encyclopedia of Philosophy: Gottlob Frege Discussion at The Well concerning Abstraction hierarchy

Chapter 2

Category theory

Schematic representation of a category with objects X, Y, Z and morphisms f, g, g f. (The categorys three identity morphisms 1X,1Y and 1Z, if explicitly represented, would appear as three arrows, next to the letters X, Y, and Z, respectively, each having as itsshaft a circular arc measuring almost 360 degrees.)

Category theory[1] formalizes mathematical structure and its concepts in terms of a collection of objects and of

9

10 CHAPTER 2. CATEGORY THEORY

arrows (also called morphisms). A category has two basic properties: the ability to compose the arrows associativelyand the existence of an identity arrow for each object. Category theory can be used to formalize concepts of otherhigh-level abstractions such as sets, rings, and groups.Several terms used in category theory, including the term morphism, are used dierently from their uses in the restof mathematics. In category theory, a morphism obeys a set of conditions specic to category theory itself. Thus,care must be taken to understand the context in which statements are made.

2.1 An abstraction of other mathematical conceptsMany signicant areas of mathematics can be formalised by category theory as categories. Category theory is anabstraction of mathematics itself that allows many intricate and subtle mathematical results in these elds to be stated,and proved, in a much simpler way than without the use of categories.[2]

The most accessible example of a category is the category of sets, where the objects are sets and the arrows arefunctions from one set to another. However, the objects of a category need not be sets, and the arrows need not befunctions; any way of formalising a mathematical concept such that it meets the basic conditions on the behaviour ofobjects and arrows is a valid category, and all the results of category theory will apply to it.The arrows of category theory are often said to represent a process connecting two objects, or in many cases astructure-preserving transformation connecting two objects. There are however many applications where muchmore abstract concepts are represented by objects and morphisms. The most important property of the arrows is thatthey can be composed, in other words, arranged in a sequence to form a new arrow.Categories now appear in most branches of mathematics, some areas of theoretical computer science where they cancorrespond to types, and mathematical physics where they can be used to describe vector spaces. Categories wererst introduced by Samuel Eilenberg and Saunders Mac Lane in 194245, in connection with algebraic topology.Category theory has several faces known not just to specialists, but to other mathematicians. A term dating fromthe 1940s, "general abstract nonsense", refers to its high level of abstraction, compared to more classical branchesof mathematics. Homological algebra is category theory in its aspect of organising and suggesting manipulations inabstract algebra.

2.2 Utility

2.2.1 Categories, objects, and morphisms

The study of categories is an attempt to axiomatically capture what is commonly found in various classes of relatedmathematical structures by relating them to the structure-preserving functions between them. A systematic study ofcategory theory then allows us to prove general results about any of these types of mathematical structures from theaxioms of a category.Consider the following example. The class Grp of groups consists of all objects having a group structure. Onecan proceed to prove theorems about groups by making logical deductions from the set of axioms. For example, it isimmediately proven from the axioms that the identity element of a group is unique.Instead of focusing merely on the individual objects (e.g., groups) possessing a given structure, category theory em-phasizes the morphisms the structure-preserving mappings between these objects; by studying these morphisms,we are able to learn more about the structure of the objects. In the case of groups, the morphisms are the grouphomomorphisms. A group homomorphism between two groups preserves the group structure in a precise sense it is a process taking one group to another, in a way that carries along information about the structure of the rstgroup into the second group. The study of group homomorphisms then provides a tool for studying general propertiesof groups and consequences of the group axioms.A similar type of investigation occurs in many mathematical theories, such as the study of continuous maps (mor-phisms) between topological spaces in topology (the associated category is called Top), and the study of smoothfunctions (morphisms) in manifold theory.Not all categories arise as structure preserving (set) functions, however; the standard example is the category ofhomotopies between pointed topological spaces.

2.3. CATEGORIES, OBJECTS, AND MORPHISMS 11

If one axiomatizes relations instead of functions, one obtains the theory of allegories.

2.2.2 FunctorsMain article: FunctorSee also: Adjoint functors Motivation

A category is itself a type of mathematical structure, so we can look for processes which preserve this structure insome sense; such a process is called a functor.Diagram chasing is a visual method of arguing with abstract arrows joined in diagrams. Functors are representedby arrows between categories, subject to specic dening commutativity conditions. Functors can dene (construct)categorical diagrams and sequences (viz. Mitchell, 1965). A functor associates to every object of one category anobject of another category, and to every morphism in the rst category a morphism in the second.In fact, what we have done is dene a category of categories and functors the objects are categories, and the mor-phisms (between categories) are functors.By studying categories and functors, we are not just studying a class of mathematical structures and the morphismsbetween them; we are studying the relationships between various classes of mathematical structures. This is a funda-mental idea, which rst surfaced in algebraic topology. Dicult topological questions can be translated into algebraicquestions which are often easier to solve. Basic constructions, such as the fundamental group or the fundamentalgroupoid of a topological space, can be expressed as functors to the category of groupoids in this way, and theconcept is pervasive in algebra and its applications.

2.2.3 Natural transformationsMain article: Natural transformation

Abstracting yet again, some diagrammatic and/or sequential constructions are often naturally related a vaguenotion, at rst sight. This leads to the clarifying concept of natural transformation, a way to map one functor toanother. Many important constructions in mathematics can be studied in this context. Naturality is a principle, likegeneral covariance in physics, that cuts deeper than is initially apparent. An arrow between two functors is a naturaltransformation when it is subject to certain naturality or commutativity conditions.Functors and natural transformations ('naturality') are the key concepts in category theory.[3]

2.3 Categories, objects, and morphismsMain articles: Category (mathematics) and Morphism

2.3.1 CategoriesA category C consists of the following three mathematical entities:

A class ob(C), whose elements are called objects; A class hom(C), whose elements are called morphisms or maps or arrows. Each morphism f has a source

object a and target object b.The expression f : a b, would be verbally stated as "f is a morphism from a to b".The expression hom(a, b) alternatively expressed as homC(a, b), mor(a, b), or C(a, b) denotes thehom-class of all morphisms from a to b.

A binary operation , called composition of morphisms, such that for any three objects a, b, and c, we havehom(b, c) hom(a, b) hom(a, c). The composition of f : a b and g : b c is written as g f or gf,[4]governed by two axioms:

12 CHAPTER 2. CATEGORY THEORY

Associativity: If f : a b, g : b c and h : c d then h (g f) = (h g) f, and Identity: For every object x, there exists a morphism 1x : x x called the identity morphism for x, such

that for every morphism f : a b, we have 1b f = f = f 1a.

From the axioms, it can be proved that there is exactly one identity morphism for every object.Some authors deviate from the denition just given by identifying each object with its identitymorphism.

2.3.2 MorphismsRelations among morphisms (such as fg = h) are often depicted using commutative diagrams, with points (corners)representing objects and arrows representing morphisms.Morphisms can have any of the following properties. A morphism f : a b is a:

monomorphism (or monic) if f g1 = f g2 implies g1 = g2 for all morphisms g1, g2 : x a. epimorphism (or epic) if g1 f = g2 f implies g1 = g2 for all morphisms g1, g2 : b x. bimorphism if f is both epic and monic. isomorphism if there exists a morphism g : b a such that f g = 1b and g f = 1a.[5]

endomorphism if a = b. end(a) denotes the class of endomorphisms of a. automorphism if f is both an endomorphism and an isomorphism. aut(a) denotes the class of automorphisms

of a. retraction if a right inverse of f exists, i.e. if there exists a morphism g : b a with f g = 1b. section if a left inverse of f exists, i.e. if there exists a morphism g : b a with g f = 1a.

Every retraction is an epimorphism, and every section is a monomorphism. Furthermore, the following three state-ments are equivalent:

f is a monomorphism and a retraction; f is an epimorphism and a section; f is an isomorphism.

2.4 FunctorsMain article: Functor

Functors are structure-preserving maps between categories. They can be thought of as morphisms in the category ofall (small) categories.A (covariant) functor F from a category C to a category D, written F : C D, consists of:

for each object x in C, an object F(x) in D; and for each morphism f : x y in C, a morphism F(f) : F(x) F(y),

such that the following two properties hold:

For every object x in C, F(1x) = 1Fx; For all morphisms f : x y and g : y z, F(g f) = F(g) F(f).

A contravariant functor F: C D, is like a covariant functor, except that it turns morphisms around (reverses allthe arrows). More specically, every morphism f : x y in C must be assigned to a morphism F(f) : F(y) F(x)in D. In other words, a contravariant functor acts as a covariant functor from the opposite category Cop to D.

2.5. NATURAL TRANSFORMATIONS 13

2.5 Natural transformationsMain article: Natural transformation

A natural transformation is a relation between two functors. Functors often describe natural constructions andnatural transformations then describe natural homomorphisms between two such constructions. Sometimes twoquite dierent constructions yield the same result; this is expressed by a natural isomorphism between the twofunctors.If F and G are (covariant) functors between the categories C and D, then a natural transformation from F to Gassociates to every object X in C a morphism X : F(X) G(X) in D such that for every morphism f : X Y in C,we have Y F(f) = G(f) X; this means that the following diagram is commutative:

Commutative diagram dening natural transformations

The two functors F and G are called naturally isomorphic if there exists a natural transformation from F to G suchthat X is an isomorphism for every object X in C.

2.6 Other concepts

2.6.1 Universal constructions, limits, and colimits

Main articles: Universal property and Limit (category theory)

Using the language of category theory, many areas of mathematical study can be categorized. Categories includesets, groups and topologies.

14 CHAPTER 2. CATEGORY THEORY

Each category is distinguished by properties that all its objects have in common, such as the empty set or the product oftwo topologies, yet in the denition of a category, objects are considered to be atomic, i.e., we do not know whetheran object A is a set, a topology, or any other abstract concept. Hence, the challenge is to dene special objectswithout referring to the internal structure of those objects. To dene the empty set without referring to elements, orthe product topology without referring to open sets, one can characterize these objects in terms of their relations toother objects, as given by the morphisms of the respective categories. Thus, the task is to nd universal propertiesthat uniquely determine the objects of interest.Indeed, it turns out that numerous important constructions can be described in a purely categorical way. The centralconcept which is needed for this purpose is called categorical limit, and can be dualized to yield the notion of a colimit.

2.6.2 Equivalent categories

Main articles: Equivalence of categories and Isomorphism of categories

It is a natural question to ask: under which conditions can two categories be considered to be essentially the same, inthe sense that theorems about one category can readily be transformed into theorems about the other category? Themajor tool one employs to describe such a situation is called equivalence of categories, which is given by appropriatefunctors between two categories. Categorical equivalence has found numerous applications in mathematics.

2.6.3 Further concepts and results

The denitions of categories and functors provide only the very basics of categorical algebra; additional importanttopics are listed below. Although there are strong interrelations between all of these topics, the given order can beconsidered as a guideline for further reading.

The functor category DC has as objects the functors from C to D and as morphisms the natural transformationsof such functors. The Yoneda lemma is one of the most famous basic results of category theory; it describesrepresentable functors in functor categories.

Duality: Every statement, theorem, or denition in category theory has a dual which is essentially obtained byreversing all the arrows. If one statement is true in a category C then its dual will be true in the dual categoryCop. This duality, which is transparent at the level of category theory, is often obscured in applications and canlead to surprising relationships.

Adjoint functors: A functor can be left (or right) adjoint to another functor that maps in the opposite direction.Such a pair of adjoint functors typically arises from a construction dened by a universal property; this can beseen as a more abstract and powerful view on universal properties.

2.6.4 Higher-dimensional categories

Many of the above concepts, especially equivalence of categories, adjoint functor pairs, and functor categories, can besituated into the context of higher-dimensional categories. Briey, if we consider a morphism between two objects asa process taking us from one object to another, then higher-dimensional categories allow us to protably generalizethis by considering higher-dimensional processes.For example, a (strict) 2-category is a category together with morphisms between morphisms, i.e., processes whichallow us to transform one morphism into another. We can then compose these bimorphisms both horizontallyand vertically, and we require a 2-dimensional exchange law to hold, relating the two composition laws. In thiscontext, the standard example is Cat, the 2-category of all (small) categories, and in this example, bimorphisms ofmorphisms are simply natural transformations of morphisms in the usual sense. Another basic example is to considera 2-category with a single object; these are essentially monoidal categories. Bicategories are a weaker notion of 2-dimensional categories in which the composition of morphisms is not strictly associative, but only associative up toan isomorphism.This process can be extended for all natural numbers n, and these are called n-categories. There is even a notion of-category corresponding to the ordinal number .

2.7. HISTORICAL NOTES 15

Higher-dimensional categories are part of the broader mathematical eld of higher-dimensional algebra, a conceptintroduced by Ronald Brown. For a conversational introduction to these ideas, see John Baez, 'A Tale of n-categories(1996).

2.7 Historical notesIn 194245, Samuel Eilenberg and Saunders Mac Lane introduced categories, functors, and natural transformationsas part of their work in topology, especially algebraic topology. Their work was an important part of the transitionfrom intuitive and geometric homology to axiomatic homology theory. Eilenberg and Mac Lane later wrote thattheir goal was to understand natural transformations; in order to do that, functors had to be dened, which requiredcategories.Stanislaw Ulam, and some writing on his behalf, have claimed that related ideas were current in the late 1930s inPoland. Eilenberg was Polish, and studied mathematics in Poland in the 1930s. Category theory is also, in somesense, a continuation of the work of Emmy Noether (one of Mac Lanes teachers) in formalizing abstract processes;Noether realized that in order to understand a type of mathematical structure, one needs to understand the processespreserving that structure. In order to achieve this understanding, Eilenberg and Mac Lane proposed an axiomaticformalization of the relation between structures and the processes preserving them.The subsequent development of category theory was powered rst by the computational needs of homological algebra,and later by the axiomatic needs of algebraic geometry, the eld most resistant to being grounded in either axiomaticset theory or the Russell-Whitehead view of united foundations. General category theory, an extension of universalalgebra having many new features allowing for semantic exibility and higher-order logic, came later; it is now appliedthroughout mathematics.Certain categories called topoi (singular topos) can even serve as an alternative to axiomatic set theory as a foundationof mathematics. A topos can also be considered as a specic type of category with two additional topos axioms. Thesefoundational applications of category theory have been worked out in fair detail as a basis for, and justication of,constructive mathematics. Topos theory is a form of abstract sheaf theory, with geometric origins, and leads to ideassuch as pointless topology.Categorical logic is now a well-dened eld based on type theory for intuitionistic logics, with applications in functionalprogramming and domain theory, where a cartesian closed category is taken as a non-syntactic description of a lambdacalculus. At the very least, category theoretic language claries what exactly these related areas have in common (insome abstract sense).Category theory has been applied in other elds as well. For example, John Baez has shown a link between Feynmandiagrams in Physics and monoidal categories.[6] Another application of category theory, more specically: topostheory, has been made in mathematical music theory, see for example the book The Topos of Music, Geometric Logicof Concepts, Theory, and Performance by Guerino Mazzola.More recent eorts to introduce undergraduates to categories as a foundation for mathematics include those ofWilliam Lawvere and Rosebrugh (2003) and Lawvere and Stephen Schanuel (1997) and Mirroslav Yotov (2012).

2.8 See also Group theory Domain theory Enriched category theory Glossary of category theory Higher category theory Higher-dimensional algebra Important publications in category theory Outline of category theory Timeline of category theory and related mathematics

16 CHAPTER 2. CATEGORY THEORY

2.9 Notes[1] Awodey 2006

[2] Geroch, Robert (1985). Mathematical physics ([Repr.] ed.). Chicago: University of Chicago Press. p. 7. ISBN 0-226-28862-5. Retrieved 20 August 2012. Note that theorem 3 is actually easier for categories in general than it is for the specialcase of sets. This phenomenon is by no means rare.

[3] Mac Lane 1998, p. 18: As Eilenberg-Mac Lane rst observed, 'category' has been dened in order to be able to dene'functor' and 'functor' has been dened in order to be able to dene 'natural transformation' "

[4] Some authors compose in the opposite order, writing fg or f g for g f. Computer scientists using category theory verycommonly write f ; g for g f

[5] Note that a morphism that is both epic and monic is not necessarily an isomorphism! An elementary counterexample: inthe category consisting of two objects A and B, the identity morphisms, and a single morphism f from A to B, f is bothepic and monic but is not an isomorphism.

[6] Baez, J.C.; Stay, M. (2009). Physics, topology, logic and computation: A Rosetta stone (PDF). arXiv:0903.0340.

2.10 References Admek, Ji; Herrlich, Horst; Strecker, George E. (1990). Abstract and concrete categories. John Wiley &

Sons. ISBN 0-471-60922-6.

Awodey, Steve (2006). Category Theory. Oxford Logic Guides 49. Oxford University Press. ISBN 978-0-19-151382-4.

Barr, Michael; Wells, Charles (2012), Category Theory for Computing Science, Reprints in Theory and Appli-cations of Categories 22 (3rd ed.).

Barr, Michael; Wells, Charles (2005), Toposes, Triples and Theories, Reprints in Theory and Applications ofCategories 12 (revised ed.), MR 2178101.

Borceux, Francis (1994). Handbook of categorical algebra. Encyclopedia of Mathematics and its Applications50-52. Cambridge University Press.

Bucur, Ion; Deleanu, Aristide (1968). Introduction to the theory of categories and functors. Wiley. Freyd, Peter J. (1964). Abelian Categories. New York: Harper and Row. Freyd, Peter J.; Scedrov, Andre (1990). Categories, allegories. North Holland Mathematical Library 39. North

Holland. ISBN 978-0-08-088701-2.

Goldblatt, Robert (2006) [1979]. Topoi: The Categorial Analysis of Logic. Studies in logic and the foundationsof mathematics 94 (Reprint, revised ed.). Dover Publications. ISBN 978-0-486-45026-1.

Hatcher, William S. (1982). Ch. 8. The logical foundations of mathematics. Foundations & philosophy ofscience & technology (2nd ed.). Pergamon Press.

Herrlich, Horst; Strecker, George E. (2007), Category Theory (3rd ed.), Heldermann Verlag Berlin, ISBN978-3-88538-001-6.

Kashiwara, Masaki; Schapira, Pierre (2006). Categories and Sheaves. Grundlehren der Mathematischen Wis-senschaften 332. Springer. ISBN 978-3-540-27949-5.

Lawvere, F. William; Rosebrugh, Robert (2003). Sets for Mathematics. Cambridge University Press. ISBN978-0-521-01060-3.

Lawvere, F. W.; Schanuel, Stephen Hoel (2009) [1997]. Conceptual Mathematics: A First Introduction toCategories (2nd ed.). Cambridge University Press. ISBN 978-0-521-89485-2.

Leinster, Tom (2004). Higher operads, higher categories. London Math. Society Lecture Note Series 298.Cambridge University Press. ISBN 978-0-521-53215-0.

2.11. FURTHER READING 17

Leinster, Tom (2014). Basic Category Theory. Cambridge University Press. Lurie, Jacob (2009). Higher topos theory. Annals of Mathematics Studies 170. Princeton, NJ: Princeton

University Press. arXiv:math.CT/0608040. ISBN 978-0-691-14049-0. MR 2522659.

Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5(2nd ed.). Springer-Verlag. ISBN 0-387-98403-8. MR 1712872.

Mac Lane, Saunders; Birkho, Garrett (1999) [1967]. Algebra (2nd ed.). Chelsea. ISBN 0-8218-1646-2. Martini, A.; Ehrig, H.; Nunes, D. (1996). Elements of basic category theory. Technical Report (Technical

University Berlin) 96 (5).

May, Peter (1999). A Concise Course in Algebraic Topology. University of Chicago Press. ISBN 0-226-51183-9.

Guerino, Mazzola (2002). The Topos of Music, Geometric Logic of Concepts, Theory, and Performance.Birkhuser. ISBN 3-7643-5731-2.

Pedicchio, Maria Cristina; Tholen, Walter, eds. (2004). Categorical foundations. Special topics in order, topol-ogy, algebra, and sheaf theory. Encyclopedia of Mathematics and Its Applications 97. Cambridge: CambridgeUniversity Press. ISBN 0-521-83414-7. Zbl 1034.18001.

Pierce, Benjamin C. (1991). Basic Category Theory for Computer Scientists. MIT Press. ISBN 978-0-262-66071-6.

Schalk, A.; Simmons, H. (2005). An introduction to Category Theory in four easy movements (PDF). Notesfor a course oered as part of the MSc. in Mathematical Logic, Manchester University.

Simpson, Carlos. Homotopy theory of higher categories. arXiv:1001.4071., draft of a book. Taylor, Paul (1999). Practical Foundations of Mathematics. Cambridge Studies in Advanced Mathematics 59.

Cambridge University Press. ISBN 978-0-521-63107-5.

Turi, Daniele (19962001). Category Theory Lecture Notes (PDF). Retrieved 11 December 2009. Basedon Mac Lane 1998.

2.11 Further reading Jean-Pierre Marquis (2008). From a Geometrical Point of View: A Study of the History and Philosophy of

Category Theory. Springer Science & Business Media. ISBN 978-1-4020-9384-5.

2.12 External links Theory and Application of Categories, an electronic journal of category theory, full text, free, since 1995. nLab, a wiki project on mathematics, physics and philosophy with emphasis on the n-categorical point of view. Andr Joyal, CatLab, a wiki project dedicated to the exposition of categorical mathematics. Category Theory, a web page of links to lecture notes and freely available books on category theory. Hillman, Chris, A Categorical Primer, CiteSeerX: 10 .1 .1 .24 .3264, a formal introduction to category theory. Adamek, J.; Herrlich, H.; Stecker, G. Abstract and Concrete Categories-The Joy of Cats (PDF). Category Theory entry by Jean-Pierre Marquis in the Stanford Encyclopedia of Philosophy with an extensive

bibliography.

List of academic conferences on category theory Baez, John (1996). The Tale of n-categories. An informal introduction to higher order categories.

18 CHAPTER 2. CATEGORY THEORY

WildCats is a category theory package for Mathematica. Manipulation and visualization of objects, morphisms,categories, functors, natural transformations, universal properties.

The catsterss channel on YouTube, a channel about category theory. Category Theory at PlanetMath.org. Video archive of recorded talks relevant to categories, logic and the foundations of physics. Interactive Web page which generates examples of categorical constructions in the category of nite sets. Category Theory for the Sciences, an instruction on category theory as a tool throughout the sciences.

Chapter 3

Mathematical structure

For the notion of structure in mathematical logic, see Structure (mathematical logic).

In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that, insome manner, attach (or relate) to the set, endowing the collection with meaning or signicance.A partial list of possible structures are measures, algebraic structures (groups, elds, etc.), topologies, metric structures(geometries), orders, events, equivalence relations, dierential structures, and categories.Sometimes, a set is endowed with more than one structure simultaneously; this enables mathematicians to study itmore richly. For example, an order induces a topology. As another example, if a set both has a topology and is agroup, and the two structures are related in a certain way, the set becomes a topological group.Mappings between sets which preserve structures (so that structures in the domain are mapped to equivalent structuresin the codomain) are of special interest in many elds of mathematics. Examples are homomorphisms, which preservealgebraic structures; homeomorphisms, which preserve topological structures; and dieomorphisms, which preservedierential structures.N. Bourbaki suggested an explication of the concept mathematical structure in their book Theory of Sets (Chapter4. Structures) and then dened on that base, in particular, a very general concept of isomorphism.

3.1 Example: the real numbersThe set of real numbers has several standard structures:

an order: each number is either less or more than any other number. algebraic structure: there are operations of multiplication and addition that make it into a eld. a measure: intervals along the real line have a specic length, which can be extended to the Lebesgue measure

on many of its subsets.

a metric: there is a notion of distance between points. a geometry: it is equipped with a metric and is at. a topology: there is a notion of open sets.

There are interfaces among these:

Its order and, independently, its metric structure induce its topology. Its order and algebraic structure make it into an ordered eld. Its algebraic structure and topology make it into a Lie group, a type of topological group.

19

20 CHAPTER 3. MATHEMATICAL STRUCTURE

3.2 See also Abstract structure Algebraic structure Space (mathematics)

3.3 References Structure at PlanetMath.org. (provides a model theoretic denition.) D.S. Malik and M. K. Sen (2004) Discrete mathematical structures: theory and applications, ISBN 978-0-619-

21558-3 .

M. Senechal (1993) Mathematical Structures, Science 260:11703. Bernard Kolman, Robert C. Ross, and Sharon Cutler (2004) Discrete mathematical Structures, ISBN 978-0-

13-083143-9 . Stephen John Hegedes and Luis Moreno-Armella (2011)"The emergence of mathematical structures, Educational

Studies in Mathematics 77(2):36988. Journal: Mathematical structures in computer science, Cambridge University Press ISSN 0960-1295.

3.4. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 21

3.4 Text and image sources, contributors, and licenses3.4.1 Text

Abstraction Source: https://en.wikipedia.org/wiki/Abstraction?oldid=668013602 Contributors: Ed Poor, Andre Engels, Shii, Ellmist,Stevertigo, Paul Barlow, Wshun, Fred Bauder, Lexor, TakuyaMurata, Ellywa, Glenn, Palfrey, Sethmahoney, Dysprosia, Hao2lian, Max-imus Rex, Banno, Robbot, Zandperl, Altenmann, Dessimoz, Gandalf61, Kagredon, GreatWhiteNortherner, Adhib, Ancheta Wis, Giftlite,Kenny sh, Bensaccount, Siroxo, Eequor, Solipsist, Edcolins, Golbez, OldakQuill, Andycjp, Antandrus, Szajd, Brianjd, Rich Farmbrough,Cacycle, LindsayH, Sietse Snel, Cacophony, Triona, Thuresson, Nortexoid, Maurreen, Treborbassett, Melah Hashamaim, Red WingedDuck, Alansohn, Ceyockey, Ott, Uncle G, SeventyThree, Palica, Magister Mathematicae, BD2412, NeoUrfahraner, Rjwilmsi, Mayu-mashu, Dpark, MZMcBride, 25~enwiki, Haya shiloh, FlavrSavr, Margosbot~enwiki, Jrtayloriv, Cactus.man, The Rambling Man, Yurik-Bot, Wavelength, NTBot~enwiki, Rodasmith, Manop, Ziddy, Gaius Cornelius, Chaos, KSchutte, Wimt, Muntuwandi, Spike Wilbury,Deskana, Howcheng, Jpbowen, Tony1, BOT-Superzerocool, Action potential, Rwalker, Black Falcon, Tanet, RDF, Oyvind, Curpsbot-unicodify, GrinBot~enwiki, FritzSolms, TuukkaH, Sardanaphalus, Planemad, SmackBot, Artexc, Lestrade, Reedy, Xaosux, Ohnoits-jamie, Oneismany, Bluebot, Jprg1966, Go for it!, Epastore, Mladilozof, Can't sleep, clown will eat me, Chlewbot, OrphanBot, LoveMon-key, Jon Awbrey, Drunken Pirate, TenPoundHammer, Byelf2007, SashatoBot, Vildricianus, Aquilina, Grumpyyoungman01, Special-T,RichardF, Levineps, White Ash, Iridescent, Dreftymac, Courcelles, Floridi~enwiki, Hukkinen, Ternto333, Sdorrance, Gregbard, Bardak,Chrislk02, Shubh123, Letranova, Thijs!bot, Qwyrxian, Kilva, Headbomb, Missvain, Wai Wai, Transhumanist, Good Vibrations, Katewill,QuiteUnusual, Modernist, Crissidancer88, JAnDbot, The Transhumanist, Jmd2121, Albany NY, Freshacconci, VoABot II, JNW, JayGatsby, Edzhandle, DerHexer, JaGa, Gwern, MartinBot, R'n'B, Nono64, Trusilver, Rivereld, Numbo3, Maurice Carbonaro, Reedy Bot,Ncmvocalist, Samuel William~enwiki, (jarbarf), Belovedfreak, Grossdj, Sanscrit1234, CardinalDan, Fimbulfamb, Pasixxxx, Jmrowland,AlnoktaBOT, Philip Trueman, TXiKiBoT, Rei-bot, Ask123, Ocolon, The Tetrast, Ripepette, Cnilep, Cowlinator, SieBot, YonaBot,Soler97, Davidlewisbaker, Phral, Denisarona, Faithlessthewonderboy, ClueBot, GorillaWarfare, Fyyer, The Thing That Should Not Be,Rocker453, BmastonLJ, Excirial, Cenarium, Hans Adler, Warrior4321, Muro Bot, SoxBot III, BodhisattvaBot, Libcub, SilvonenBot, Ad-dbot, American Eagle, Some jerk on the Internet, Grandscribe, Jncraton, Cst17, Download, CarsracBot, Amirobot, South Bay, Auawise,AnomieBOT, Piano non troppo, ArthurBot, Gondwanabanana, Nasa-verve, GrouchoBot, Omnipaedista, SassoBot, Shadowjams, AaronKauppi, GliderMaven, LucienBOT, Paine Ellsworth, Haeinous, Winterst, Jthechemist, RedBot, TobeBot, Philocentric, Standardfact,Dinamik-bot, LilyKitty, Miracle Pen, C9cute2wall, TjBot, Me6620, Jowa fan, Slawekb, Werieth, ZroBot, Doddy Wuid, Wikiloop, Di-neshkumar Ponnusamy, G Man Unit, ClueBot NG, Shahircool, Chetanpan, Millermk, SaintGeorgeIV, Widr, Ramaksoud2000, JeraphineGryphon, Ingotian, ChristineAlicia, Wingroras, Cleopeter36579, Leon gillingham, Yousefarbash, Victor Yus, ChrisGualtieri, Pwdent,Onepebble, Kilternom, Yamaha5, Nigellwh, Heloiseabelard, KasparBot, Pgalbacs and Anonymous: 185

Category theory Source: https://en.wikipedia.org/wiki/Category_theory?oldid=666333705 Contributors: AxelBoldt, LC~enwiki, BryanDerksen, Zundark, The Anome, Toby~enwiki, Toby Bartels, Oliverkroll, Roadrunner, Youandme, Chas zzz brown, Michael Hardy,Erik Zachte, JereyYasskin, Dominus, Dcljr, Cyde, TakuyaMurata, Mpagano, Marco Krohn, Jiang, Zhaoway~enwiki, Revolver, CharlesMatthews, Reddi, Dysprosia, Wik, Phys, Bevo, Jerzy, Phil Boswell, Robbot, Jmabel, Gandalf61, Choni, Bkell, Hadal, Fuelbottle, To-bias Bergemann, Giftlite, Markus Krtzsch, Bogdanb, Inkling, Lethe, Lupin, Fropu, Dratman, Semorrison, Jason Quinn, Cambyses,Matt Crypto, Gdr, MarkSweep, CSTAR, APH, Tzanko Matev, Creidieki, Barnaby dawson, David Sneek, Smimram, Mikolt, Rich Farm-brough, Guanabot, Luqui, Mat cross, Paul August, Bender235, Rec syn, Elwikipedista~enwiki, Chalst, Szquirrel, Mike Schwartz, Lys-dexia, Phils, Msh210, Diego Moya, Campani~enwiki, Ceyockey, Oleg Alexandrov, Linas, Mpatel, Cbcarlson, Ryan Reich, Marudub-shinki, Graham87, SixWingedSeraph, Rjwilmsi, Salix alba, Goclenius, Chris Pressey, John Z, Selvakumar.sarangan, YurikBot, Lau-rentius, Hairy Dude, Jimp, Grubber, Archelon, Mikeblas, Crasshopper, Robertbyrne, Ott2, Pred, Brentt, WikiWizard, Sardanaphalus,SmackBot, Unyoyega, XudongGuan~enwiki, BiT, Kmarinas86, Dan Hoey, RDBrown, Zoran.skoda, Gutworth, Michiexile, Nbarth, Gofor it!, Ashawley, Alriode, Cybercobra, Jon Awbrey, Lambiam, Azrael ezra, Physis, , CBM, Hga, Sam Staton, Blaisorblade,Julian Mendez, Thijs!bot, Headbomb, Lotte Monz, WinBot, Seaphoto, JAnDbot, Ensign beedrill, R'n'B, Popx, Maurice Carbonaro, The-Seven, Alexwright, SamStokes, Daniel5Ko, Seoneo, JohnBlackburne, LokiClock, Anonymous Dissident, Hesam7, Ontoraul, Magmi,Don4of4, Davin, Dmcq, SieBot, Gerakibot, DesolateReality, Curtdbz, Balrivo, Garyzx, Adrianwn, Tkeu, Auntof6, Tlepp, Cenarium,Hans Adler, Palnot, Topology Expert, Gzhanstong, Favonian, Htamas, Calculuslover, Loupeter, TeH nOmInAtOr, Legobot, Luckas-bot, Yobot, Amirobot, AnomieBOT, Minnecologies, Fotino, Bci2, ArthurBot, Xqbot, Txebixev, Luis Felipe Schenone, Omnipaedista,Point-set topologist, FrescoBot, Sawomir Biay, Tkuvho, Alidev, ElNuevoEinstein, Magmalex, Stanny32, EmausBot, Jmencisom, Quon-dum, Stwalczyk, Earldouglas, Chester Markel, Deer*lake, AvocatoBot, Brad7777, ChrisGualtieri, JYBot, Deltahedron, Makecat-bot,Lugia2453, Purnendu Karmakar, SakeUPenn, JMP EAX, Lanzdsey, KasparBot and Anonymous: 127

Mathematical structure Source: https://en.wikipedia.org/wiki/Mathematical_structure?oldid=657108115 Contributors: The Anome,Michael Hardy, Charles Matthews, Gandalf61, Giftlite, Gubbubu, Notinasnaid, Tompw, Rgdboer, Chalst, Tsirel, Msh210, Kinema, OlegAlexandrov, Linas, Isnow, Wbeek, Salix alba, Hillman, Trovatore, MalafayaBot, PieRRoMaN, Byelf2007, Lambiam, Ntsimp, HStel,Thijs!bot, Aadal, David Eppstein, Pavel Jelnek, DorganBot, VictorMak, Jorgen W, Addbot, LaaknorBot, Ginosbot, ., Luckas-bot, Pcap, JRB-Europe, JackieBot, ArthurBot, Xqbot, FrescoBot, Nicolas Perrault III, RedBot, Katovatzschyn, ,Dan morenus, Wcherowi, Helpful Pixie Bot, OCCullens and Anonymous: 24

3.4.2 Images File:Brain.png Source: https://upload.wikimedia.org/wikipedia/commons/7/73/Nicolas_P._Rougier%27s_rendering_of_the_human_

brain.png License: GPL Contributors: http://www.loria.fr/~{}rougier Original artist: Nicolas Rougier File:Cat-on-mat.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/4b/Cat-on-mat.svg License: Public domain Contrib-

utors: http://en.wikipedia.org/wiki/File:Cat-on-mat.GIF Original artist: w:Ademoor File:Commons-logo.svg Source: https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg License: ? Contributors: ? Origi-

nal artist: ? File:Commutative_diagram_for_morphism.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/ef/Commutative_diagram_

for_morphism.svg License: Public domain Contributors: Own work, based on en:Image:MorphismComposition-01.png Original artist:User:Cepheus

22 CHAPTER 3. MATHEMATICAL STRUCTURE

File:Folder_Hexagonal_Icon.svg Source: https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg License: Cc-by-sa-3.0 Contributors: ? Original artist: ?

File:Internet_map_1024.jpg Source: https://upload.wikimedia.org/wikipedia/commons/d/d2/Internet_map_1024.jpg License: CC BY2.5 Contributors: Originally from the English Wikipedia; description page is/was here. Original artist: The Opte Project

File:JerryFelix.JPG Source: https://upload.wikimedia.org/wikipedia/commons/b/bc/JerryFelix.JPG License: CC-BY-SA-3.0 Contrib-utors: ? Original artist: ?

File:Natural_transformation.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/f2/Natural_transformation.svg License:Public domain Contributors: I created this work entirely by myself. Original artist: Ryan Reich (talk)

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File:Portal-puzzle.svg Source: https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg License: Public domain Contributors:? Original artist: ?

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3.4.3 Content license Creative Commons Attribution-Share Alike 3.0

AbstractionOriginsThemesCompressionInstantiationMaterial processOntological statusPhysicalityReferencing and referringSimplification and orderingThought processes

As used in different disciplinesIn artIn computer scienceIn linguisticsIn mathematicsIn musicIn neurologyIn philosophyIn psychologyIn social theory

See also NotesReferencesExternal links

Category theoryAn abstraction of other mathematical concepts Utility Categories, objects, and morphisms Functors Natural transformations

Categories, objects, and morphismsCategoriesMorphisms

FunctorsNatural transformationsOther conceptsUniversal constructions, limits, and colimitsEquivalent categoriesFurther concepts and resultsHigher-dimensional categories

Historical notesSee also NotesReferencesFurther readingExternal links

Mathematical structureExample: the real numbersSee also ReferencesText and image sources, contributors, and licensesTextImagesContent license