nLab category of classes

Definition

The category of classes has classes as objects and maps of classes as morphisms.

In the Zermelo-Fraenkel set theory, a class is a proposition $P$ with a designated free variable $x$. We interpret $P(x)$ as saying that $x$ belongs to the class $P$. We also write $x\in P$, while understanding that $P$ is not a set, but $x$ is.

For example, any set $A$ is a class, whose proposition is $a\in A$ and the designated free variable is $a$.

The proposition $a=a$ defines the class of all sets, which does not arise via the above construction from any set.

Relations and maps

Given two classes $A$ and $B$, we can form their product $A\times B$ as the class $A\times B$ such that $P(x)$ means $x=(a,b)$ for some $a\in A$ and $b\in B$, where $(a,b)$ denotes the usual ordered pair constructed in Zermelo-Fraenkel set theory as $(a,b)=\{\{a\},\{a,b\}\}$ or $(a,b)=\{\{\emptyset,a\},\{b\}\}$ or $(a,b)=\{\{\{\emptyset\},a\},\{b\}\}$.

If $P(x)$ implies $Q(x)$, we say that $P$ is a subclass of $Q$.

A relation from $A$ to $B$ is a subclass of $A\times B$.

A map $f$ from $A$ to $B$ is a relation $R$ from $A$ to $B$ such that for any $a\in A$ there is a unique $b\in B$ such that $R(a,b)$. In this case we write $f(a)$ for this unique $b$, so $f(a)=b$ means $R(a,b)$.

Maps of classes can be composed in the usual manner, which produces a category.

Here a category is understood in the sense of a first-order logic with two sorts (objects and morphisms), but at no point we attempt to consider all classes as a single unified whole.

A family $f$ of classes indexed by a class $I$ is a map of classes $f\colon T\to I$. The class indexed by $i\in I$ is the preimage $f^*\{i\}$. Such families can be pulled back along maps of classes $J\to I$ and pushed forward along maps $I\to J$.

We can now define large categories as having a class of objects, a class of morphisms, together with composition and identities satisfying the usual axioms. The category of classes considered above is not a large category in this sense.

We do not require large categories to be locally small.

Diagrams

Suppose $I$ is a large category. An $I$-indexed diagram of classes is defined as follows. First, we have an $I$-indexed family of classes $f\colon T\to I$. Secondly, we have a transition map

$$tr\colon\{(t,h)\mid t\in T, h\in Mor(I), f(t)=dom(h)\}\to T,$$

which is a map of classes such that $f(tr(t,h))=codom(h)$. (The domain of $tr$ is a class because $t$ and $h$ are sets.) Finally, the transition map satisfies the usual axioms expected from a functor.

For any large category $I$ and a class $C$ we can define the constant diagram indexed by $I$ with value $C$. We take $T=I\times C$ with $f\colon T\to I$ being the projection map. The transition map sends $(t,h)\mapsto t$.

Limits and colimits as adjoint functors

For any large category $I$ we have a constant diagram functor that sends a class $C$ to the constant diagram on $C$.

We can talk about left and right adjoint functors to this functor in the sense of a first-order logic with two sorts (objects and morphisms), augmented with symbols for the functor, its adjoint, together with unit and counit maps that satisfy the triangle identities.

Limits

The category of classes admits all small limits.

First, it admits equalizers: the equalizer of $f,g\colon A\to B$ is the subclass $E$ of $A$ defined by $E(e)=(e\in A \wedge f(e)=g(e))$.

Secondly, it admits small products: the small product of an $I$-indexed family of classes $f\colon T\to I$, where $I$ is an arbitrary set (considered as a class when used with $f$) can be constructed as the class $P$ such that $p\in P$ if $p$ is a map of sets? whose domain is $I$ and for all $i\in I$ we have $p(i)\in T$ and $f(p(i))=i$. (Observe that $P$ is indeed a class.)

Finally, it admits all small limits because the usual reduction of small limits to equalizers of small products continues to work provided that we adhere to the above convention on the definition of families of classes.

Colimits

The category of classes admits all colimits indexed by arbitrary large categories $I$, i.e., large colimits.

First, the standard reduction of $I$-indexed colimits to a coequalizer of a pair of arrows between coproducts indexed by $Mor(I)$ and $Ob(I)$ still works in this context since class-indexed families of classes can be pulled back along source and target maps $Mor(I)\to Ob(I)$.

Secondly, class-indexed coproducts of classes can be computed simply by taking the total class $T$ of the corresponding class-indexed family $f\colon T\to I$ of classes.

Thirdly, coequalizers of classes exist by Scott's trick. Observe that given a pair of arrows $f,g:X\to Y$ between classes, we can define an equivalence relation on $Y$ by saying that $y~y'$ if there is a map $h:[0,n]\to Y$ such that $h(0)=y$, $h(n)=y'$ and for any $i\in[0,n)$ there is $x\in X$ such that $h(i)=f(x)$ and $h(i+1)=g(x)$ or $h(i)=g(x)$ and $h(i+1)=f(x)$. The quotient of $Y$ by this equivalence relation exists by Scott's trick and is precisely the desired coequalizer.

Other categorical properties

The category of classes is a regular category: the obvious notion of image factorization is stable under pullbacks. It is also a Barr-exact category: every equivalence relation $R$ on a class $C$ is induced by the quotient map $C\to C/R$.

It is also an infinitary extensive category, where “infinitary” means “class-indexed”. Indeed, pullbacks of coproduct injections along arbitrary maps of classes exist and class-indexed coproducts are disjoint and stable under pullback.

It is also well-pointed: for every two maps between classes $f,g:X\rightarrow Y$ and every element $x\in X$, if $f(x) = g(y)$, then $f = g$, and the category of classes is not the terminal category.

It likewise has all objects corresponding to large cardinals, most notably a natural numbers object. Otherwise, the category of finite sets FinSet is vacuously a category of classes, as the notions of ‘finitary’ and ‘class-indexed’/‘infinitary’ coincide.

As such, the category of classes is a well-pointed infinitary Heyting or Boolean pretopos, depending upon the external logic used, with a natural numbers object and other large cardinals, and where “infinitary” is used in the rather strong sense of “class-indexed”.

The category of classes is not cartesian closed or locally cartesian closed and does not have power objects. Indeed, the class of all sets $S$ does not have a power object, or, equivalently, there is no internal hom $Hom(S,\{0,1\})$.

Category with class structure

The category of classes is a primordial example of a category with class structure. Its open maps are precisely those maps of classes $f\colon T\to I$ such that $f^*\{i\}$ is a set for each $i\in I$. Small maps coincide with open maps. The powerclass of a class $C$ is the class of all sets $A$ such that $A$ is a subclass of $C$. The universal class is the class of all sets.

Related notions

• class

• type of classes

• proper class

• large category

• category with class structure

• algebraic set theory