A quasitopos is a particular kind of category that has properties similar to that of a topos, but is not quite a topos.
Instead of the usual subobject classifier, it has a classifier only for strong subobjects.
A quasitopos is a finitely complete, finitely cocomplete, locally cartesian closed category $E$ in which there exists an object $\Omega$ that classifies strong monomorphisms.
In particular, this means
admits a right adjoint;
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.
Equivalently, in addition to finite limits and colimits and local cartesian closure, one may ask only that there exists a classifier $t\colon 1\to\Omega$ as above for some class $\mathcal{M}$ of monomorphisms which contains the regular monomorphisms and is closed under composition and pullback. It then follows that $\mathcal{M}$ is precisely the class of strong monics, and also equal to the class of regular monics.
The full subcategory $SepPSh(C) \hookrightarrow PSh(C)$ of a category of presheaves over a site $C$, consisting of the separated presheaves on $C$, is a quasitopos. A category equivalent to such a separated presheaf category is called a Grothendieck quasitopos, by analogy with the notion of Grothendieck topos.
In a quasitopos the pushout of a strong monomorphism is again a strong mono, and the resulting square is also a pullback square.
This appears as Elephant, lemma 2.6.2.
A quasitopos that is also a balanced category is a topos.
This is Elephant, corollary 2.6.3.
A quasitopos has disjoint coproducts precisely if the unique morphism $\emptyset \to *$ from the initial object to the terminal object is a strong monomorphism.
This is Elephant, corollary 2.6.5.
An object $C$ in a quasitopos is called coarse if for every monic epic morphism $f : A \to B$ every morphism $A \to C$ factors uniquely through $f$.
So the coarse objects are those that see monic epic morphisms as isomorphisms, hence that do onot see the failure of the quasitopos to be a balanced category.
In a quasitopos $\mathcal{E}$ the full subcategory on coarse objects is a topos and a reflective subcategory
This is Elephant, prop 2.6.12.
If $\mathcal{E} \simeq SepPSh(C)$ is a Grothendieck quasitopos of separated presheaves on a site $C$, then $Coarse(\mathcal{E}) \simeq Sh(C)$ is the sheaf topos on $C$.
This is in Elephant, section A4.4.
There is a Giraud theorem characterizing Grothendieck quasitoposes:
Grothendieck quasitoposes are 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)
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
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, $0\to \bar{0} \to 1$. One can prove that then the slice category $E/\bar{0}$ is a Heyting algebra (i.e. a posetal quasitopos), while the co-slice category $\bar{0}/E$ is a solid quasitopos, and moreover $E$ itself is recoverable via Artin gluing from a particular functor $E/\bar{0} \to \bar{0}/E$. 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.
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.
Conversely, if a quasitopos is also a balanced category, then it is also a topos.
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 marked simplicial sets.
A category of concrete sheaves on a concrete site is a Grothendieck quasitopos. See local topos.
This includes the following examples:
The category of simplicial complexes.
The category of diffeological spaces.
The following examples are categories of separated presheaves for the $\neg\neg$-topology on various presheaf toposes:
The category of monomorphisms between sets (as presheaves on the interval category).
The category of sets equipped with a relation (as presheaves on
a truncation of the globular category).
The category of sets equipped with a reflexive relation (as presheaves on a truncated reflexive globular category).
The category of sets equipped with a symmetric relation (as presheaves on the full subcategory of finite sets and injections consisting of just the objects $1$, $2$).
The category of sets equipped with a reflexive symmetric relation (as presheaves on the full subcategory of finite sets consisting of just the objects $1$, $2$).
The category of bornological sets.
The category of assemblies of a partial combinatory algebra.
The category of Spanier’s quasi-topological spaces, the category of concrete sheaves on the site consisting of compact Hausdorff spaces with the finite covering topology. See Dubuc-Espanol.
quasitopos
Original articles include
Jacques Penon, Quasi-topos , C.R. Acad. Sci. Paris 276 Série A (1973) pp.237–240. (gallica)
Jacques Penon, Sur le quasi-topos , Cahiers Top. Géom. Diff. 18 (1977), 181–218.
Standard textbook references are
Oswald Wyler, Lecture Notes on Topoi and Quasitopoi , World Scientific Singapore 1991.
Peter Johnstone, Sketches of an Elephant, Oxford UP 2002. (section A2.6)
J. Adamek, H. Herrlich, G. E. Strecker, Abstract and Concrete Categories , Dover Mineola 2009. (Available online as TAC Reprint no.17 (2006) pp.1-507; section 28)
Quasi-toposes of concrete sheaves are considered in
Eduardo Dubuc, Concrete quasitopoi , Lecture Notes in Math. 753 (1979), 239–254
Eduardo Dubuc, L. Espanol, Quasitopoi over a base category (arXiv:math.CT/0612727)
A review is in
More generally, quasi-sheaf toposes are discussed in