category theory

# Contents

## Idea and definition

In the category Set of sets, for $X$ a set, an element $x \in X$ is equivalently a morphism in Set (namely a function of sets) $x : {*} \to X$, where “$*$” denotes the point – the set with a single element.

However, from the point of view of the category Set, most things that can be said and done about elements of a set $X$, can more generally be said and done for morphisms $x \colon U\to X$, for any other set $U$. The point is just that many constructions can be performed “elementwise”. For instance, the fact that elements of $X\times Y$ are exactly pairs $(x,y)$ of an element of $X$ and an element of $Y$, when performed “elementwise” for morphisms out of $U$, expresses the universal property of a product. In structural set theory such as ETCS, one sometimes (but not necessarily) takes this point of view for axiomatizing the structure of $Set$.

On the other hand, once elements of objects are regarded as morphisms into these objects, the same reasoning applies to every category $C$. Accordingly, for $C$ any category and $X$ an object of $C$, one may refer to a morphism $x \colon U \to X$ a generalized element of $X$. One says this is a generalized element with stage of definition given by $U$.

The perspective of generalized elements of objects of a category $C$ is related to regarding $C$ as its image under the Yoneda embedding

$Y : C \hookrightarrow [C^{op}, Set]$

into its presheaf category. Under this embedding, every object $X$ of $C$ is mapped to the functor – the representable functor represented by it –

$GenEl(X) : C^{op} \to Set$

that sends each object $U$ of $C$ to the set of generalized elements of $X$ at stage $U$.

It is also worth noting that the internal logic or type theory of a category provides us a way to go backwards formally. By reasoning about “abstract elements” in a set-theoretic style like ordinary elements, the interpretation then “compiles” such proofs to category-theoretic ones which actually apply to all generalized elements.

## Examples

### In $Set$

The primordial example is when $C$ is the category Set of sets and $I$ is a terminal object in $Set$ — that is, a set with one element. Then elements of any set $c$ are in one-to-one correspondence with functions $f: I \to c$. This correspondence works as follows: given any element of $c$ there is a unique function $f: I \to c$ with this element in its image, and conversely each function $f: I \to c$ has a unique element of $c$ in its image.

### In concrete categories

In the same way, in a concrete category whose underlying-set functor is represented by $I$, the $I$-elements of an object are the same as the elements of its underlying set. (The category of sets is actually a special case of this, since it is concrete, with the identity functor represented by a terminal object.)

### Global elements

Generalizing from $Set$ in another way, in any category with a terminal object $I$, we call a morphism $f : I \to c$ a global element of the object $c$.

### In monoidal categories

On the other hand, it is common to take $I$ to be the unit object whenever $C$ is a monoidal category. The generalized elements defined over this $I$ are important in enriched category theory).

### For generators

Arguably, the most general case where generalized elements defined at only one stage $I$ are “sufficient” when $I$ is some sort of generator of the category. However, not every category has a single object as any sort of generator! Instead, in arbitrary categories, generalized elements of all possible stages of definition must often be used to replace global elements. Thus while a set is determined by its global elements, an object of an arbitrary category is determined by all of its generalized elements (this is one way to state the Yoneda lemma).

### In presheaf categories

For $C = [D^{op}, Set]$ a presheaf category and for $I = \Delta_{pt} = (d \mapsto \{\bullet\})$ the presheaf constant at the singleton set, the generalized elements of a presheaf $F$ are the global sections of this presheaf, equivalently these are the elements in the limit set over $F$.

On the other hand, if $I=D(-,d)$ is a representable presheaf, then the generalized elements of $F$ at stage $d$ are precisely the elements of the set $F(d)$, by the Yoneda lemma.

### In abelian categories

An element in an abelian category is an equivalence class of generalised elements.

## Relationship to type theory

In the internal type theory of a category $C$, the generalized elements of $X$ at stage $U$ can be identified with terms of type $X$ in context $u\colon U$:

$u\colon U \vdash x(u) \colon X \,.$

See the references below.

The fact that all type-theoretic constructions can be performed in any context implies that we can manipulate ordinary elements, and end up speaking also about generalized elements defined at arbitrary stages.

## References

### In type theory

The interpretation of terms in type theory as generalized elements of objects in a category is discussed for instance on p. 8 of

• Steve Awodey, Andrej Bauer, Propositions as $[$Types$]$, Journal of Logic and Computation. Volume 14, Issue 4, August 2004, pp. 447-471 (pdf)

Revised on July 23, 2013 21:40:15 by Anonymous Coward (68.54.30.66)