nLab big and little toposes

Idea

True to their proverbial multifacetedness, there are two different ways in which toposes (by which here we understand Grothendieck sheaf toposes, as usual) encode spaces (and the same comments apply to higher topoi):

1. On the one hand, every topos $\mathcal{X}$ may itself be regarded as a generalized topological space (specifically a generalized locale), in which case the maps between these generalized spaces correspond to the geometric morphisms between the toposes. Actual topological spaces/locales $X$ are embodied in this manner by the category of sheaves $\mathcal{X} = Sh(X) \coloneqq Sh(Op_X)$ over (the site of open subsets) of $X$.

2. On the other hand, a topos $\mathcal{T}$ may also be regarded as a category of generalized geometric spaces, namely regarding each of its objects as a geometric space with the maps between these being the morphisms in the topos.

Broadly and informally, one speaks of a big topos (original French: gros topos) when thinking of a category of spaces and of a little topos when thinking of a category as a space.

Beware that every topos always has both of these aspects to it, and that how it is addressed depends on the context.

For example, in the archetypical case of the the category Set of sets, when regarded as a little topos it is the point space $\ast$, since $Set \simeq Sh(\ast)$, while when regarded as a big topos it is the category all discrete topological spaces. These two perspectives are related: Discrete topological spaces are precisely those whose structure is detected by their probes by points, namely by the system of ways of mapping the point space into them.

More generally, the little topos $Sh(X)$ of a topological space $X$ is equivalently the “big” category of all étale spaces fibered over $X$. Again, these two perspective are related: étale spaces over $X$ are just those spaces over $X$ whose structure is detected by their probes by open subsets of $X$, namely by the systems of ways of mapping open subsets of $X$ into them, over $X$.

Of course, the actual category of all topological spaces, Top, fails to be a topos. But it may be completed to a topos in a straightforward way, by regarding it as a (large) site (whose coverings are families of jointly surjective open embeddings), and then passing to the sheaf topos $Sh(Top)$ over that (of which Top is a full subcategory, via the Yoneda embedding).

This $Sh(Top)$ is the original gros topos according to SGA4 (1960s-70s, cf. pp. 161 here), where the terminology originates. Or rather, more generally and in the spirit of the relative point of view, these original authors speak for any $X \in Top$

• of the slice site $Top_{/X}$ as the “gros site”, and

• of the slice topos $Sh(Top_{/X})$ as the “gros topos”

of $X$, the latter being a category of “generalized topological spaces fibered over $X$”. This is the “gros topos” of $X$ in contrast to its “petit topos” $Sh(X)$.

Directly analogous constructions may be considered starting with other categories of spaces. Notably when replacing Top here with the category Sch? of all schemes (over some ground field), then for any $X \in Sch$ we have its gros topos $Sh(Sch_{/X})$.

This raises the question for characterization of the relation between little and big toposes more generally and in the abstract.

Early exploration along these lines led Lawvere 1986, 1989 to the notion of cohesive toposes. ¹ These are very big toposes in that their petit topos aspect is negligible: being strongly connected and local, the spatial aspect of a cohesive topos is just that of a “thickened point”. Hence cohesive toposes are variants of $Sh(Top)$ (which itself is not even cohesive either, but for instance the topos of “D-topological sets” $Sh(CartSp_{top})$ is), but not of $Sh(Top_{/X})$ for extended $X$. But the general notion of gros/petit topoi may be formalized within differentially cohesive topoi $\mathbf{H}$ (see there) which allows to speak for every object $X \in \mathbf{H}$ of its gros topos $\mathbf{H}_{/X}$ and its petit topos $\mathbf{H}^{ét}_{/H} \subset \mathbf{H}_{/X}$ (Schreiber 2017 §5.3.4, Sati & Schreiber 2026 §9.1.2.6).

More recently, the notion of fractured topoi (Lurie 2018, following Carchedi 2020) aims for a formalization of the gros/petit distinction closer in spirit to the original construction from SGA4: Here the axioms aim to directly capture the inclusion functor between little/big topoi of an object $X$, which is induced by a non-full functor of sites that behaves like the inclusion of étale maps among all ambient maps. This axiomatics allows again to systematically speak, for any object $X$ in a fractured topos, of its gros and its petit topos, see below.

Definitions

Via fractured topoi

A definition of the notion of gros and petit toposes via fractured topoi or analogously via fractured (∞,1)-topoi (due to Carchedi 2020, and Lurie 2018, going back to Joyal & Moerdijk and Dubuc):

Recall that a fractured (∞,1)-topos is a left adjoint (∞,1)-functor $j_!\colon E\to F$ between (∞,1)-toposes such that $F$ is generated by the image of $E$ under (∞,1)-colimits, the right adjoint of $j_!$ preserves (∞,1)-colimits, for every $U\in E$, the induced left adjoint $(j_!)_{/U}\colon E_{/U}\to F_{/j_!U}$ is fully faithful, and maps in the image of $j_!$ are stable under base changes along maps with domain in the image of $j_!$.

Now for any object $U\in E$, we define the petit topos of $U$ as $E_{/U}$ and the gros topos of $U$ as $F_{/j_! U}$.

Via differential cohesive topoi

For the time being, see here at differentially cohesive topos, or Sati & Schreiber 2026 §9.1.2.6.

Examples

For $X$ a topological space, the little topos that it defines is the category of sheaves $Sh(X) \coloneqq Sh(Op(X))$ on the category of open subsets of $X$. A general object in this topos can be regarded as an etale space over $X$. The space $X$ itself is incarnated as the terminal object $X = * \in Sh(X)$.

On the other hand, a big topos in which $X$ is incarnated is a category of sheaves on a site of test spaces with which $X$ may be probed. For instance for $C =$ Top, or Diff or CartSp with their standard coverages, $Sh(C)$ is such a big topos. See for instance, topological topos and the quasi-topos of quasitopological spaces.

In good cases, the intrinsic properties of $X$ do not depend on whether one regards it as a little topos or as an object of a gros topos. For instance at cohomology in the section Nonabelian sheaf cohomology with constant coefficients it is discussed how the nonabelian cohomology of a paracompact manifold $X$ with constant coefficients gives the same answer in each case.

Further remarks

The following is some leftover material from a previous version of this page, which is now partially redundant.

Remark

Objects in a big topos $Sh(S)$ may be thought of as spaces modeled on $S$, in the sense described at motivation for sheaves, cohomology and higher stacks and at space.

On the other hand, the objects of a petit topos, such as $Sh(X)$, can also be regarded as a kind of generalized spaces, but generalized spaces over $X$ on which the rigid structure of morphisms in $Op(X)$ (only inclusions of subsets, no more general maps) induces a correspondingly rigid structure so that they are not all that general. In fact, $Sh(Op(X))$ is equivalent to the category of etale spaces over $X$—i.e. spaces “modeled on $X$” in a certain sense. More generally, for any topos $E$, the objects of $E$ can be identified with local homeomorphisms of toposes into $E$.

From the “little topos” perspective, it can be helpful to think of a “big topos” as a “fat point,” which is not “spread out” very much spatially itself, but contains within that point lots of different types of “local data,” so that even spaces which are “rigidly” modeled on that point can have a lot of interesting cohesion and local structure. (One should not be misled by this into thinking that a big topos has only one point, although it is usually a local topos and hence has an initial point.)

Remark

If $X$ is a topological space, then the canonical little topos associated to $X$ is the sheaf topos $Sh(X)$. On the other hand, if $S$ is a site of probes enabling us to regard $X$ as an object of a big topos $H = Sh(S)$, then we can also consider the topos $H/X$ as a representative of $X$. These two toposes are often called the little topos of $X$ (or petit topos of $X$) and the big topos of $X$ (or gros topos of $X$) respectively.

There might be some debate about whether $H/X$ is, itself, “a little topos” or “a big topos.” While it certainly contains information about the space $X$ specifically, its objects are not “spaces locally modeled on $X$” but rather spaces locally modeled on the big site $S$ which happen to have a map to $X$. The standard phrase “the big topos of $X$” is the most descriptive.

Note that if $X$ is actually an object of the site $S$, then $H/X$ can be identified with the topos of sheaves on the slice site $S/X$ (and otherwise, it can be identified with the topos of sheaves on the category of elements of $X\in Sh(S)$). This site $S/X$ is often referred to as the big site of $X$, as compared to the little site, which is $Op(X)$ (or appropriate replacement). The topos $Sh(S/X)$ can thus be viewed as spaces modelled on $S$, but parameterised by the representable sheaf $X$.

Note that when $S=Top$ with its local-homeomorphism topology, there is a canonical functor $Op(X) \to S/X$ which preserves finite limits and both preserves and reflects? covering families. Therefore, it induces both a geometric morphism $H/X \to Sh(X)$ and one $Sh(X) \to H/X$, of which the latter is the left adjoint of the former in Topos. In other words, the geometric morphism $H/X \to Sh(X)$ is local, and in particular a homotopy equivalence of toposes. This fact relating the big and little toposes of $X$ also holds in other cases.

Related entries

• functorial geometry, space and quantity

• topological site, continuous truth

• cohesive topos

  • sufficiently cohesive topos

  • infinitesimal cohesive (infinity,1)-topos

  • quality type

• étendue

• locally decidable topos

References

The notion of a gros topos of a topological space is due to Jean Giraud, with early discussion in:

• M. Artin, A. Grothendieck, J. L. Verdier, Théorie des Topos et Cohomologie Etale des Schémas (SGA4), Springer LNM 269 (1972). (exposé IV, 2.5 pp.316-318, 4.10 pp.358-365, pp. 161 here)

In this context see also

• Saunders Mac Lane, I. Moerdijk, pp. 113, 325, 416 in: Sheaves in Geometry and Logic, Springer (1994) [doi:10.1007/978-1-4612-0927-0 ]

In the context of a discussion of the big Zariski topos Lawvere (1976, p. 110) calls the gros-petit distinction ‘a surprising twist of logic that is not yet fully clarified’:

• William Lawvere, Variable quantities and variable structures in topoi, in Alex Heller, Myles Tierney (eds.), Algebra, Topology and Category Theory – A Collection of Papers in Honor of Samuel Eilenberg, Academic Press New York (1976) 101-131 [doi:10.1016/C2013-0-10841-0]

The suggestion that a general notion of gros topos is needed goes back to some remarks in Pursuing Stacks. A precise axiom system capturing the notion is first proposed in

• William Lawvere, Categories of spaces may not be generalized spaces, as exemplified by directed graphs, Revista Colombiana de Matematicas XX (1986) 179-186, reprinted as: Reprints in Theory and Applications of Categories, 9 (2005) 1-7 [tac:tr9]

“Axiom 0” (locality) used in Lawvere 1986 for gros toposes is argued in Lawvere 1994 to be essentially an insight due to Georg Cantor and is called the Cantorian Contrast (namely between discrete spaces and codiscrete spaces) in Lawvere & Rosebrugh (2003), p. 245.

• William Lawvere, Robert Rosebrugh, p. 245 in: Sets for Mathematics, Cambridge University Press (2003) [doi:10.1017/CBO9780511755460, book homepage, pdf]

The axioms 0 and 1 for toposes of generalized spaces given in Lawvere 1986 later became called the axioms for a cohesive topos

• William Lawvere, Cohesive Toposes and Cantor's "lauter Einsen", Philosophia Mathematica 2 1 (1994) 5-15 [doi:10.1093/philmat/2.1.5, pdf]

together with axiom 2 they make out a sufficiently cohesive topos.

Further discussion of this axiomatics for gros toposes is in

• Bill Lawvere, Categories of space and quantity in: J. Echeverria et al (eds.), The Space of mathematics, de Gruyter, Berlin, New York (1992) [pdf]

where a proposal for a general axiomatization of homotopy/homology-like “extensive quantities” and cohomology-like “intensive quantities”) as covariant and contravariant functors out of a distributive category are considered.

The following two papers contain Lawvere’s early view of a trichotomy between big toposes vs. étendue and locally decidable toposes as paradigmatic “generalized spaces” with “infinitesimally cohesive” in between, with the latter subsumed into the fine structure of cohesion in more recent versions

• F. W. Lawvere, Qualitative Distinctions between some Toposes of Generalized Graphs, in Categories in Computer Science and Logic, Cont. Math. 92 (1989) 261-299 [doi:10.1090/conm/092, pdf]

• F. W. Lawvere, Some Thoughts on the Future of Category Theory, 1-13 in Springer LNM 1488 (1991).

The left adjoint in a cohesive topos is also mentioned in

• Bill Lawvere, page 14 of: Taking categories seriously, Revista Colombiana de Matematicas XX (1986) 147-178, Reprints in Theory and Applications of Categories, 8 (2005) 1-24. [tac:tr8, pdf]

Under the term categories of cohesion these axioms are discussed in

• Bill Lawvere, Axiomatic cohesion, Theory and Applications of Categories 19 3 (2007) 41-49 [tac:19-03, pdf]

A formalization of gros/petit toposes of objects in differentially cohesive topoi:

• Urs Schreiber; §5.3.4 (pp. 537) in version 2 of: Differential Cohomology in a Cohesive ∞-Topos (2017)

• Hisham Sati, Urs Schreiber: Étale toposes, §9.1.2.6 in: Geometric Orbifold Cohomology, CRC Press (2026) [ISBN:9781041147510]

Another definition of gros vs petit toposes and remarks on applications in Galois theory:

• Nick Duncan: Gros and petit toposes [pdf]

The notion of fractured topoi:

• David Carchedi, Higher Orbifolds and Deligne-Mumford Stacks as Structured Infinity Topoi, Memoirs of the American Mathematical Society 264 1282 (2020) [doi:10.1090/memo/1282, arXiv:1312.2204]

• Jacob Lurie: Fractured $\infty$-Topoi, chapter 20 of: Spectral Algebraic Geometry (2018) [pdf]

See also:

• Mathieu Anel: Grothendieck topologies from unique factorization systems [arXiv:0902.1130]

• Mamuka Jibladze, Homotopy types for “gros” toposes, thesis, pdf

• Peter Johnstone, Calibrated Toposes, Bull. Belgian Math. Soc. - Simon Stevin 19 5 (2012) 889-907. [euclid:1354031555]

A discussion and comparison of big vs little approaches to $(\infty,1)$-topos theory began at these blog entries:

• Cohesive (∞,1)-toposes and Petit (∞,1)-toposes.

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1. In fact, these early articles (Lawvere 1986, 1989) proposed one further axiom for big toposes, the contractibility of the subobject classifier. This extra condition makes a cohesive topos be “sufficiently cohesive”. ↩