nLab
quasitopos

A quasitopos is a particular kind of category that is not quite a topos. Instead of the usual subobject classifier, it has a classifier only for strong subobjects.

Definition

A quasitopos is a finitely complete, finitely cocomplete, locally cartesian closed category $E$ with an object $\Omega$ that classifies strong monomorphisms. In particular, this means

• Every finite limit and colimit exists;

• For each morphism $f:A\to B$, the pullback functor between slice quasitoposes,

  $$f^{*}:E/B\to E/A,$$f^*: E/B \to E/A,

  admits a right adjoint;

• There is a map $t:1\to \Omega$ such that every strong monomorphism $i:A\to X$ occurs as the pullback of $t$ along some unique morphism ${\chi }_i:X\to \Omega$:

  $$\begin{array}{ccc}A& \to & 1\\ i\downarrow & & \downarrow t\\ X& \stackrel{{\chi }_i}{\to }& \Omega \end{array}$$\array{A & \to & 1\\ i \downarrow & & \downarrow t\\ X & \overset{\chi_i}{\to} & \Omega }

Equivalently, in addition to finite limits and colimits and local cartesian closure, one may ask only that there exists a classifier $t:1\to \Omega$ as above for some class $ℳ$ of monomorphisms which contains the regular monomorphisms and is closed under composition and pullback. It then follows that $ℳ$ is precisely the class of strong monics, and also equal to the class of regular monics.

The object $\Omega$ above is sometimes called a strong-subobject classifier, since it classifies strong subobjects, but also sometimes called a weak subobject classifier, since it satisfies a weaker property than an ordinary subobject classifier.

Examples

• Any (elementary) topos is a quasitopos. The first two properties are known, and in a topos every monomorphism is strong, so the ordinary subobject classifier works.

• The category of separated presheaves with respect to a site is a quasitopos. Such a quasitopos is called a Grothendieck quasitopos, by analogy with the notion of Grothendieck topos. There is a Giraud theorem characterizing Grothendieck quasitoposes as those quasitoposes which are locally small, cocomplete, and have a generating set, or equivalently as the locally presentable categories which are locally cartesian closed and in which every strong congruence has a effective quotient; see C2.2.13 of the Elephant.

• Any Heyting algebra is a quasitopos. This is in notable contrast to the case of topoi, since no nontrivial poset is a topos. The crucial distinction is that every morphism in a poset is both monic and epic, but only the identities are strong monic or strong epic.

• The category of pseudotopological spaces is a quasitopos, as is the category of subsequential spaces. (The latter is Grothendieck, but not the former.) The category of topological spaces fails only to be locally cartesian closed. In such “topological” quasitopoi, the strong monics are the “subspace inclusions” (i.e. those monics whose source has the topology induced from the target), and the strong-subobject classifier is the two-point space with the indiscrete topology. (In particular, we cannot demand any sort of separation axiom and still have a quasitopos.)

• The category of simplicial complexes is a quasitopos.

• The category of diffeological spaces is a quasitopos.

Extensivity and exactness

A topos is always extensive and exact, but this is not the case for quasitopoi.

A quasitopos is a coherent category, since it has finite colimits which are stable under pullback (since it is locally cartesian closed), and so in particular its initial object is strict, and it has finite coproducts which are pullback-stable, but they need not be disjoint: for objects $A$ and $B$, in the pullback

$$\begin{array}{ccc}P& \stackrel{}{\to }& B\\ \downarrow & & \downarrow \\ A& \underset{}{\to }& A+B\end{array}$$\array{P & \overset{}{\to} & B\\ \downarrow && \downarrow\\ A & \underset{}{\to} & A+B}

the object $P$ need not be initial. This is easy to see from the fact that any Heyting category is a quasitopos, since then $A+B$ is the join $A\vee B$, and so the pullback is the meet $A\wedge B$, which is not in general the bottom element.

It is true, however, that such a $P$ is always a quotient of the initial object, i.e. the unique map $0\to P$ is epic. If the map $0\to 1$ is strong monic, as it is in the “topological” examples, then $0$ can have no proper epimorphic images, and so coproducts are disjoint. The converse also holds, since if coproducts are disjoint then $0\to 1$ is an equalizer of the two injections $1\rightrightarrows 1+1$. A quasitopos with this property is sometimes called solid.

More generally, in any quasitopos $E$, we can factor $0\to 1$ into an epic followed by a strong monic, $0\to \overline{0}\to 1$. One can prove that then the slice category $E/\overline{0}$ is a Heyting algebra (i.e. a posetal quasitopos), while the co-slice category $\overline{0}/E$ is a solid quasitopos, and moreover $E$ itself is recoverable via Artin gluing? from a particular functor $E/\overline{0}\to \overline{0}/E$. Thus, to a certain extent, the only interest in the theory of quasitoposes, above and beyond the theory of Heyting algebras, is in the solid ones.

By contrast, if a solid quasitopos is additionally exact, and hence a pretopos, then in particular it is balanced, which implies that it is in fact a topos. One can prove, however, that a quasitopos is always quasi-exact, meaning that every strong congruence has an effective quotient.

References

• Oswald Wyler, Lecture Notes on Topoi and Quasitopoi, World Scientific, 1991.

• Peter Johnstone, Sketches of an Elephant, A2.6 and C2.2.

Need to reference work by various combinations of authors including Baez, Dolan, Hoffnung, and others on diffeological spaces and simplicial complexes…