nLab quasitopos



Topos Theory

topos theory



Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory




A quasitopos is a particular kind of category that has properties similar to that of a topos, but is not quite a topos. A major difference is that it need not be balanced: a morphism that is both monic and epic is not necessarily invertible. A quasitopos that is balanced is a topos.

Instead of the usual subobject classifier, it has a classifier only for strong subobjects. It satisfies the uniqueness, but not the existence, part of the sheaf axioms (Elephant A2.6).

Note that some of the literature definitions use the notion of a regular monomorphism. Since every regular monomorphism is a strong one, this article only uses strong monomorphism.



A quasitopos is a finitely complete, finitely cocomplete, locally cartesian closed category EE in which there exists an object Ω\Omega that classifies strong monomorphisms.

In particular, this means

  • Every finite limit and colimit exists;
  • For each morphism f:ABf: A \to B, the pullback functor between slice quasitoposes,
    f *:E/BE/A,f^*: E/B \to E/A,

    admits a right adjoint;

  • There is a map t:1Ωt: 1 \to \Omega such that every strong monomorphism i:AXi: A \to X occurs as the pullback of tt along some unique morphism χ i:XΩ\chi_i: X \to \Omega:
    A 1 i t X χ i Ω\array{ A & \to & 1\\ i \downarrow & & \downarrow t\\ X & \overset{\chi_i}{\to} & \Omega }

The object Ω\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:1Ωt\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. From this one can show that every morphism factors as an epimorphism followed by a regular monomorphism (see Wyler, proposition 12.5). It then follows that every strong monomorphism is regular, and therefore \mathcal{M} is precisely the class of strong monomorphisms.


Let CC be a category with two Grothendieck topologies JJ and KK such that JKJ\subseteq K. The full subcategory BiSep(C,J,K)PSh(C)BiSep(C,J,K) \hookrightarrow PSh(C) of the category of presheaves over CC consisting of the sheaves for JJ that are also separated for KK is a quasitopos. A category equivalent to such a category is called a Grothendieck quasitopos, by analogy with the notion of Grothendieck topos.

In particular, this includes the category of separated presheaves on a given site (if we take JJ to be the trivial topology), and also includes all Grothendieck toposes (if we take K=JK=J).

Equivalently, a Grothendieck quasitopos is a category of the form Sep k(E)Sep_k(\mathbf{E}), the category of kk-separated objects for a Lawvere-Tierney topology kk on a Grothendieck topos E\mathbf{E}.




(pushout of strong monos)

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 A.2.6.2, using the synonym cocover for strong monomorphism. Since a topos is a quasitopos in which all monomorphisms are strong, this implies that the pushout of a mono in a topos is again a mono and that the resulting square is a pullback. Together with the fact that colimits are universal in a topos, this implies that a topos is an adhesive category.


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 CC in a quasitopos is called coarse if for every monic epic? morphism f:ABf : A \to B every morphism ACA \to C factors uniquely through ff.

So the coarse objects are those that see monic epic morphisms as isomorphisms, hence that do not 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

Coarse(). Coarse(\mathcal{E}) \stackrel{\leftarrow}{\hookrightarrow} \mathcal{E} \,.

This is Elephant, prop 2.6.12.


If SepPSh(C)\mathcal{E} \simeq SepPSh(C) is a Grothendieck quasitopos of separated presheaves on a site CC, then Coarse()Sh(C)Coarse(\mathcal{E}) \simeq Sh(C) is the sheaf topos on CC.

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)

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 AA and BB, in the pullback

P B A A+B\array{P & \overset{}{\to} & B\\ \downarrow && \downarrow\\ A & \underset{}{\to} & A+B}

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

It is true, however, that such a PP is always a quotient of the initial object, i.e. the unique map 0P0\to P is epic. If the map 010\to 1 is strong monic, as it is in the “topological” examples, then 00 can have no proper epimorphic images, and so coproducts are disjoint. The converse also holds, since if coproducts are disjoint then 010\to 1 is an equalizer of the two injections 11+11\rightrightarrows 1+1. A quasitopos with this property is sometimes called solid.

More generally, in any quasitopos EE, we can factor 010\to 1 into an epic followed by a strong monic, 00¯10\to \bar{0} \to 1. One can prove that then the slice category E/0¯E/\bar{0} is a Heyting algebra (i.e. a posetal quasitopos), while the co-slice category 0¯/E\bar{0}/E is a solid quasitopos, and moreover EE itself is recoverable via Artin gluing from a particular functor E/0¯0¯/EE/\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.

Internal logic

Like a topos, a quasitopos has an internal logic, for which the usual choice is to represent propositions by strong subobjects. The resulting internal logic fails to satisfy the function comprehension principle, forcing one to distinguish between functions and anafunctions.


  • 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 following examples are categories of separated presheaves for the ¬¬\neg\neg-topology on various presheaf toposes:

    • The category Mono of monomorphisms between sets (as presheaves on the interval category - the Sierpinski topos).

    • The category EndoRel or Bin of sets equipped with a relation (as presheaves on Quiv

      G 1=(0ts1),G_1 = (0 \stackrel{\overset{s}{\to}}{\underset{t}{\to}} 1),

      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 11, 22).

    • The category of sets equipped with a reflexive symmetric relation (as presheaves on the full subcategory of finite sets consisting of just the objects 11, 22). See category of simple graphs.

  • 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.


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.

Textbook accounts:

Quasi-toposes of concrete sheaves are considered in

A review is in

  • John Baez, Alex Hoffnung, Convenient categories of smooth spaces, Trans. Amer. Math. Soc., Vol. 363 No. 11 (2011), 5789-5825, (arXiv)

More generally, quasi-sheaf toposes are discussed in

Last revised on November 12, 2023 at 22:22:32. See the history of this page for a list of all contributions to it.