Subobject classifier

In the mathematical field of category theory, a subobject classifier is a special object ${\displaystyle \Omega }$ of a category such that, informally, the subobjects of any object ${\displaystyle X}$ correspond to the morphisms from ${\displaystyle X}$ to ${\displaystyle \Omega }$. This provides an analogue of the set of Booleans ${\displaystyle \{0,1\}}$ in categories other than the category of sets.

The main use of subobject classifiers is in topos theory, where an elementary topos is defined as a category with a subobject classifier and certain additional requirements. In the internal language of an elementary topos, the subobject classifier is used to interpret truth values, hence the alternative name “object of truth values”.

Let ${\displaystyle X}$ be a set. A subset ${\displaystyle Y\subseteq X}$ can be equivalently described by its indicator function

$${\displaystyle {\begin{aligned}\chi _{Y}:X&\to \{0,1\}\\x&\mapsto {\begin{cases}1{\text{ if }}x\in Y\\0{\text{ if }}x\notin Y\end{cases}}\end{aligned}}}$$

Informally, subsets of ${\displaystyle X}$ can be identified with functions ${\displaystyle X\to \{0,1\}}$. A subobject classifier ${\displaystyle \Omega }$ of a category ${\displaystyle {\mathcal {C}}}$ is an object which plays a similar role as ${\displaystyle \{0,1\}}$ does in the category of sets: subobjects of an object ${\displaystyle X}$ can be identified with morphisms from ${\displaystyle X}$ to the subobject classifier. To recover the subset with indicator function ${\displaystyle \chi }$ in a “purely categorical way”, one can take a pullback

$${\displaystyle {\begin{array}{lcl}&Y&\rightarrow &\{1\}&\\&\downarrow &&\downarrow \\&X&{\underset {\chi }{\rightarrow }}&\{0,1\}&\\\end{array}}}$$

where the function from ${\displaystyle \{1\}}$ to ${\displaystyle \{0,1\}}$ is the inclusion map. Indeed, the subset ${\displaystyle Y:=\{x\in X\mid \chi (x)=1\}}$, equipped with the inclusion map ${\displaystyle Y\to X}$ (and the unique, constant map ${\displaystyle Y\to \{1\}}$) is such a pullback because it has the correct universal property since a map into ${\displaystyle X}$ which gives the constant function 1 when composed with ${\displaystyle \chi }$ is the same as a map into ${\displaystyle Y}$. A purely categorical characterization of “${\displaystyle Y}$ is the subset with characteristic function ${\displaystyle \chi }$” is that the diagram above is a pullback.

For the general definition, we start with a category C that has a terminal object, which we denote by 1. The object Ω of C is a subobject classifier for C if there exists a morphism

1 → Ω

with the following property:

For each monomorphism j: U → X there is a unique morphism χ_j: X → Ω such that the following commutative diagram

is a pullback diagram—that is, U is the limit of the diagram:

The morphism χ _j is then called the classifying morphism for the subobject represented by j.

The category of sheaves of sets on a topological space X has a subobject classifier Ω which can be described as follows: For any open set U of X, Ω(U) is the set of all open subsets of U. The terminal object is the sheaf 1 which assigns the singleton {*} to every open set U of X. The morphism η:1 → Ω is given by the family of maps η_U : 1(U) → Ω(U) defined by η_U(*)=U for every open set U of X. Given a sheaf F on X and a sub-sheaf j: G → F, the classifying morphism χ _j : F → Ω is given by the family of maps χ _j,U : F(U) → Ω(U), where χ _j,U(x) is the union of all open sets V of U such that the restriction of x to V (in the sense of sheaves) is contained in j_V(G(V)).

Roughly speaking an assertion inside this topos is variably true or false, and its truth value from the viewpoint of an open subset U is the open subset of U where the assertion is true.

Given a small category ${\displaystyle C}$, the category of presheaves ${\displaystyle \mathrm {Set} ^{C^{op}}}$ (i.e. the functor category consisting of all contravariant functors from ${\displaystyle C}$ to ${\displaystyle \mathrm {Set} }$) has a subobject classifer given by the functor sending any ${\displaystyle c\in C}$ to the set of sieves on ${\displaystyle c}$. The classifying morphisms are constructed quite similarly to the ones in the sheaves-of-sets example above.

Both examples above are subsumed by the following general fact: every elementary topos, defined as a category with finite limits and power objects, necessarily has a subobject classifier.¹ The two examples above are Grothendieck topoi, and every Grothendieck topos is an elementary topos.

A quasitopos has an object that is almost a subobject classifier; it only classifies strong subobjects.