category theory

# Contents

## Idea

Category theory is a toolset for describing the general abstract structures in mathematics.

As opposed to set theory, category theory focuses not on elements $x,y, \cdots$ – called objects – but on the relations between these objects: the (homo)morphisms between them

$x \stackrel{f}{\to} y \,.$

Later this will lead naturally on to an infinite sequence of steps: first 2-category theory which focuses on relation between relations, morphisms between morphisms: 2-morphisms, then 3-category theory, etc. and to various variants, bicategories, Gray categories …. Eventually this leads to higher category theory, where one considers $k$-morphisms in all dimensions and to a wealth of interacting intuitions and concepts.

The concept that formalizes this is that of a category: a collection of arrows/morphism that can be composed if they are adjacent

$\left( \array{ x \\ \downarrow^{\mathrlap{f}} \\ y \\ \downarrow^{\mathrlap{g}} \\ z } \;\right) \;\;\;\; \mapsto \;\;\;\; \left( \array{ x \\ \downarrow^{\mathrlap{g \circ f}} \\ z } \;\;\;\right) \,.$
###### Examples

The archetypical example of a category is the category Set of sets and functions between sets.

The classical examples of categories are concrete categories whose objects are sets with extra structure and whose morphisms are structure preserving functions of sets, such as Top, Grp, Vect. These are the examples from which the term category derives: these categories literally categorize mathematical structures by packing structures of the same type (same category) and structure preserving mappings between them into a single whole structure, a category.

But by far not all categories are of this type and categories are much more versatile than these classical examples suggest. After all, a category is just a quiver (a directed graph) with a notion of composition of its edges. As such it generalizes the concepts of monoid and poset. If the category is a groupoid it generalizes the concept of group (in a sense called horizontal categorification). Thinking of a category as a generalized poset is particularly useful when studying limits and adjunctions.

Archetypical examples of non-concrete categories are the fundamental groupoid of a topological space and the fundamental category of a directed space.

###### Terminology

Categories were named after the examples of concrete categories. As Saunders Mac Lane writes

Now the discovery of ideas as general as these is chiefly the willingness to make a brash or speculative abstraction, in this case supported by the pleasure of purloining words from the philosophers: “Category” from Aristotle and Kant, “Functor” from Carnap (Logische Syntax der Sprache), and “natural transformation” from then current informal parlance.

However, the categories of category theory are way more general than these concrete categories, and the way Aristotle and Kant use the term is not particularly related to what Eilenberg & Mac Lane did with it.

###### The basic trinity of concepts

Category theory reflects on itself. Categories are about collections of morphisms. And there are evident morphisms between categories: functors. And there are evident morphisms between functors: natural transformations.

This trinity of concepts

is what category theory is built on.

In higher category theory this continues with

• $k$-transfors for all $k \in \mathbb{N}$.
###### Conceptual unification

A major driving force behind the development of category theory is its ability to abstract and unify concepts. General statements about categories apply to each specific concrete category of mathematical structures. The general notion of universal constructions in categories, such as representable functors, adjoint functors and limits, turns out to prevail throughout mathematics and manifest itself in myriads of special examples.

###### Abstract nonsense

This abstraction power of category theory has led Norman Steenrod to coin the term abstract nonsense or general abstract nonsense for it. It is being used as in “This property is not specific to this context, it already follows from general abstract nonsense”. Peter Freyd expressed a similar feeling by his witticism:

“Perhaps the purpose of categorical algebra is to show that which is trivial is trivially trivial.”

But abstract nonsense still tends to meet with some resistance. In the preface of his 1965 book Theory of Categories Barry Mitchell writes:

A number of sophisticated people tend to disparage category theory as consistently as others disparage certain kinds of classical music. When obliged to speak of a category they do so in an apologetic tone, similar to the way some say, “It was a gift – I’ve never even played it” when a record of Chopin Nocturnes is discovered in their possession. For this reason I add to the usual prerequisite that the reader have a fair amount of mathematical sophistication, the further prerequisite that he have no other kind.

###### The $n$-POV

The vast applicability and expressiveness of category theory leads to the observation that most structures in mathematics are best understood from a category theoretic or higher category theoretic viewpoint. This is the nPOV.

## The central constructions

### Presheaves

Much of the power of category theory rests in the fact that it reflects on itself. For instance that functors between two categories form themselves a category: the functor category.

This leads to the notion of presheaf categories and sheaf toposes. Much of category theory is topos theory.

Under Isbell duality this sets the stage for everything in mathematics related to space and algebra and their duality.

### Universal constructions

Elementary as it is, the definition of a category supports a powerful set of constructions: universal constructions. These include

All these are special cases of each other and thus reflect different aspect of one single phenomenon. Applying category theory means applying these constructions in specific situations and using general abstract theorems for deducing statements about concrete contexts.

## The central theorems

Category theory has a handful of central lemmas and theorems. Their proof is typically easy, sometimes almost tautological. Their power rests in the fact that they apply over and over again all over mathematics. Many concrete constructions get simplified by observing that they are but special realizations of these general abstract results in category theory. These central theorems are

## Applications

For a detailed list of applications see

### In pure mathematics

Apart from its general role in mathematics, category theory provides the high-level language for

### Outside of mathematics

Outside of pure mathematics, category theory finds major applications in

## Contrast to theories of other objects

### Category theory vs. set theory

Here set theory is assumed to be a theory of the usual concept of sets, that is material set theory.

No one of these is more fundamental than the other as a foundation of mathematics. Category theory is a holistic (structural) approach to mathematics that can (through such methods as Lawvere's ETCS) provide foundations of mathematics and (through algebraic set theory) reproduce all the different axiomatic set theories; elementary category theory does not need the concept of set to be formulated. Set theory is an analytic approach (element-wise) and can reproduce category theory by simply defining all the concepts in the usual way, as long as one include a technique to handle large categories (for instance by using classes instead of sets, or by including as an axiom that an uncountable inaccessible cardinal exists or even that Grothendieck universes exist).

Set theoryCategory theory
membership relation-
setscategories
elementsobjects
-morphisms
functionsfunctors
equations between elementsisomorphisms between objects
equations between setsequivalences between categories
equations between functionsnatural transformations between functors

Lawvere pointed out that set theory is axiomatized by a binary membership relation while category theory is axiomatized by a ternary composition relation.

The process of going from sets to categories is a special case of categorification and the reverse process is a special case of decategorification.

For a philosophical consideration of foundations covering and comparing sets, structuralism (a la Bourbaki?) and categories, see the article

• Sets, categories and structuralism - Costas Drossos

### Category theory vs. order theory

A category may be thought of as a categorification of a poset rather than of a set; much (but by no means all) of category theory also appears in order theory.

See category theory vs order theory for more discussion.

## Theorems

higher category theory

Some theorems in category theory are folklore.

## References

### History

Category theory was introduced in

The reason for introducing categories was to introduce functors, and the reason for introducing functors was to introduce natural transformations (more specifically natural equivalences) in order to define what natural means in mathematics:

If topology were pubically defined as the study of families of sets closed under finite intersection and infinite unions a serious disservice would be perpetrated on embryonic students of topology. The mathematical correctness of such a definition reveals nothing about topology except that its basic axioms can be made quite simple. And with category theory we are confronted with the same pedagogical problem. The basic axioms, which we will shortly be forced to give, are much too simple.

A better (albeit not perfect) description of topology is that it is the study of continuous maps; and category theory is likewise better described as the theory of functors. Both desciptions are logically inadmissible as initial definitions, but they more accurately reflect both the present and the historical motivations of the subjects. It is not too misleading, at least historically, to say that categories are what one must define in order to define natural transformations. (from Freyd 64, page 1)

The paper Eilenberg-Maclane 45 was a clash of ideas from abstract algebra (Mac Lane) and topology/homotopy theory (Eilenberg). It was first rejected on the ground that it had no content but was later published. Since then category theory has flourished into almost all areas of mathematics, has found many applications outside mathematics and even attempts to build a foundations of mathematics.

### Textbooks

#### Topos theory

The standard monographs on topos theory are

Other texts include