nLab
subobject classifier

Idea

A subobject classifier in a topos is a morphism $\mathrm{true}:*\to \Omega$ such that every monomorphism $A\hookrightarrow B$ in the topos (hence every subobject) is the pullback of this morphism along a unique morphism (the characteristic morphism of $A$) $B\to \Omega$.

In this sense $\Omega$ is the classifying object for subobjects.

Definition

Definition

In a category $C$ with finite limits, a subobject classifier is a monomorphism $\mathrm{true}:*\to \Omega$ out of the terminal object, such that for every monomorphism $U\to X$ in $C$ there is a unique morphism ${\chi }_U:X\to \Omega$ such that there is a pullback diagram

$$\begin{array}{ccc}U& \to & *\\ \downarrow & & {\downarrow }^{\mathrm{true}}\\ X& \stackrel{{\chi }_U}{\to }& \Omega \end{array}.$$\array{ U &\to& * \\ \downarrow && \downarrow^{\mathrlap{true}} \\ X &\stackrel{\chi_U}{\to}& \Omega } \,.

See for instance (MacLane-Moerdijk, p. 22).

Remark

Some terminology:

If it exists, the object $\Omega$ is also called the object of truth values, a global element $K\to \Omega$ is called a truth value and the element $\mathrm{true}:*\hookrightarrow \Omega$ is the truth value true, where all these terms allude to the internal logic of the category $C$.

Note that the subobjects classified by the truth values are subterminal objects.

The morphism ${\chi }_U$ is also called the characteristic map or classifying map of the subobject $U\hookrightarrow X$.

Proposition

If $C$ has finite limits and is in addition a locally small category, then it has a subobject classifier precisely if the subobject-assigning presheaf

$$\mathrm{Sub}:C^{\mathrm{op}}\to \mathrm{Set}$$Sub : C^{op} \to Set

$$X\mapsto \{U\hookrightarrow X\}/\sim$$X \mapsto \{U \hookrightarrow X\}/\sim

is representable. In this case the representing object is the subobject classifier: there is a natural isomorphism

$$\mathrm{Sub}(X)\simeq C(X,\Omega )$$Sub(X) \simeq C(X, \Omega)

in $X\in C$.

Moreover, in this case $C$ is well powered.

This appears for instance as (MacLane-Moerdijk, prop. I.3.1).

In more detail: given a morphism $f:c\to d$ in $C$, the function

$$\mathrm{Sub}(f):\mathrm{Sub}(d)\to \mathrm{Sub}(c)$$Sub(f): Sub(d) \to Sub(c)

takes a subobject $i:t\hookrightarrow d$ to the subobject of $d$ obtained by pulling back $i$ along $f$. (Notice that monomorphisms, as discussed there, are stable under pullback.)

The representability of this functor means there is an object $\Omega$ together with a subobject $t:T\hookrightarrow \Omega$ which is universal, meaning that given any subobject $i:s\hookrightarrow c$, there is a unique morphism $f:c\to \Omega$ such that $i$ is obtained as the pullback of $t$ along $f$.

Proof

To see that a subobject classifier induces such a natural isomorphism, we need that the morphisms $\mathrm{Sub}(f)$ for $f\in \mathrm{Mor}(C)$ corresponds to the morphisms $C(f,\Omega )$. This is the pasting law for pullbacks.

Conversely, to see that a subobjects-representing object $\Omega$ is a subobject classifier, use that by naturality we have for each morphism $\varphi :X\to \Omega$ a commuting diagram

$$\begin{array}{ccc}\mathrm{Sub}(\Omega )& \stackrel{\simeq }{\to }& C(\Omega ,\Omega )\\ {}^{\mathrm{Sub}(\varphi )}\downarrow & & {\downarrow }^{C(\varphi ,\Omega )}\\ \mathrm{Sub}(X)& \stackrel{\simeq }{\to }& C(X,\Omega )\end{array}$$\array{ Sub(\Omega) &\stackrel{\simeq}{\to}& C(\Omega, \Omega) \\ {}^{\mathllap{Sub(\phi)}}\downarrow && \downarrow^{\mathrlap{C(\phi,\Omega)}} \\ Sub(X) &\stackrel{\simeq}{\to}& C(X, \Omega) }

whose commuativity says that every element of $\mathrm{Sub}(X)$ is the pullback along some $\varphi :X\to \Omega$ of the subobject of $\Omega$ corresponding under the natural isomorphism to $\mathrm{Id}:\Omega \to \Omega$.

By further playing around with this one finds that this latter subobject of $\Omega$ has to be a terminal object.

Examples

In $\mathrm{Set}$

In the category of sets, the 2-element set $2=\{f,t\}$ plays the role of $\Omega$; the morphism $t:1\to 2$ just names the element $t$. Given a subset $S\subseteq X$, the characteristic function ${\chi }_S:X\to 2$ is the function defined by ${\chi }_S(x)=t$ if $x\in S$, and ${\chi }_S(x)=f$ if $x\notin S$.

Remark

It is not usually true in toposes that $\Omega$ is the coproduct $2=1+1$; toposes where that occurs are called Boolean. Thus the category $\mathrm{Set}$ of sets is a Boolean topos, as is the presheaf topos ${\mathrm{Set}}^G$ when $G$ is a groupoid.

In a presheaf topos

The subobject classifier in a presheaf topos $\mathrm{PSh}(S)$ is the presheaf that sends each object $U\in S$ to the set $\mathrm{sieves}(U)$ of sieves on it, equivalently the set of subobjects of the representable presheaf $Y(U)$: $\Omega :U\mapsto \mathrm{sieves}(U)$.

The corresponding morphism $\mathrm{true}:*\to \Omega$ of presheaves is the natural transformation that picks over each object the maximal sieve ${\mathrm{true}}_U={\mathrm{maximal}}_{\mathrm{sieves}(U)}:*\to \mathrm{sieves}(U)$

In $G\mathrm{Set}$

As a special case of presheaf toposes, for $G$ a discrete group and $G\mathrm{Set}=[BG,\mathrm{Set}]$ the topos of permutation representations, there are precisely two sieves on the single object of the delooping groupoid $BG$: the trivial one and the empty one. Hence the subobject classifier here is the 2-element set as in Set, but now regarded as a $G$-set with trivial $G$-action.

In a non-boolean topos

An example of a non-Boolean topos is the category of sheaves over a “typical” topological space $X$ such as the real line $ℝ$ in its usual topology. In this case, $\Omega$ is the sheaf where the set of sections over an open subset $U$ is the set of open subsets of $U$, with the obvious restriction maps; the sheaf topos in this case is guaranteed to be non-Boolean provided there are some non-regular open sets in $X$ (a open set is regular if it is the interior of its closure). The “internal logic” of such a topos is intuitionistic.

In a slice topos

Proposition

Let $ℰ$ be a topos and $X\in ℰ$ any object. Write $ℰ/X$ for the corresponding over-topos.

The subobject classifier of $ℰ/X$ is $p_2:{\Omega }_{ℰ}\times X\to X$.

Proof

This follows for instance from the statement that the inverse image of any base change geometric morphism is a logical functor and hence preserves subobject classifiers: Here we are looking at the base change along $p:X\to *$ and hence $p^{*}{\Omega }_{ℰ}\simeq {\Omega }_{ℰ}\times X$.

But the statement is also easily directly checked.

Properties

The subobject classifier always comes with the structure of an internal poset; that is, a relation $\subseteq \hookrightarrow \Omega \times \Omega$ which is internally reflexive, antisymmetric, and transitive. This can be constructed directly, or obtained via the Yoneda lemma since the collection of subobjects of any object is an external poset.

In fact, this internal poset is an internal Heyting algebra; it's an internal Boolean algebra if and only if the topos is Boolean.

Generalizations: object classifier

In higher topoi the the subobject classifiers are the universal fibrations:

in the (n+1)-topos $n\mathrm{Cat}$ of n-categories the subobject classifier is the forgetful functor

$$n\mathrm{true}:(n-1){\mathrm{Cat}}_{*}\to (n-1)\mathrm{Cat}$$n true : (n-1)Cat_* \to (n-1)Cat

from the $n$-category of pointed $(n-1)$-categories to that of $(n-1)$-categories, which forgets the point.

This is described in more detail at generalized universal bundle. See also the discussion at stuff, structure, property.

In fact, using the notion of (-1)-category the subobject classifier in Set does fit precisely into this pattern:

the 2-element set $2$ may be regarded as the 0-category of (-1)-categories (of which there are two) and the one-element set $*$ is the 0-category of pointed (-1)-categories, of which there is one.

In the context of (∞,1)-topos theory subobject classifiers are discussed in section 6.1.6 of

• Jacob Lurie, Higher Topos Theory .

As pointed out there, from some perspective it is not so much a subobject classifier that matters in higher topos theory, but an

• object classifier.

Related concepts

• object classifier

• type of propositions, type of types

• classifying space, classifying stack, moduli space, moduli stack, derived moduli space

• classifying topos

• universal principal bundle, universal principal ∞-bundle

• classifying morphism

References

section I.3 and I.4 in

• Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic