nLab Grothendieck topos

Definition

Classically, we have:

A Grothendieck topos $\mathcal{T}$ is a category that admits a geometric embedding

$$\mathcal{T} \stackrel{\stackrel{lex}{\leftarrow}}{\hookrightarrow} PSh(C)$$

in a presheaf category, i.e., a full and faithful functor that has a left exact left adjoint.

This is equivalently the category of sheaves (Set-valued presheaves satisfying the sheaf condition) over a small site.

See at sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes for discussion of this characterization.

Since smallness can be relative, we also have:

For a given fixed category of sets $S$, a Grothendieck topos over $S$ is a category of sheaves ($S$-valued presheaves satisfying the sheaf condition) over a site which is small relative to $S$, that is a site internal to $S$.

Note that a Grothendieck topos is a topos because (or if) $S$ is.

The site is not considered part of the structure; different sites may give rise to equivalent category of sheaves.

By the general theory of geometric morphisms, every Grothendieck topos sits inside a category of presheaves by a geometric embedding $Sh(S) \hookrightarrow PSh(S)$.

• This may be taken as an alternative definition of sheaf: since Lawvere-Tierney topologies are bijectively given by geometric embeddings, instead of explicitly defining a sheaf as a presheaf satisfying descent, one may define categories of sheaves as geometric embeddings into presheaf categories.

• For details on the relation between the two perspectives see geometric embedding.

• This perspective is useful for defining the vertical categorification of sheaves: stacks and ∞-stacks: the higher categories of these may be defined as geometric embeddings into higher categories of presheaves. This has been worked out in detail for (∞,1)-categories. See (∞,1)-category of (∞,1)-sheaves.

• Sometimes it is useful to distinguish between petit topos and gros topos.

Properties

General

Proposition

Every Grothendieck topos is a locally presentable category.

(Borceux 94, vol 3, prop. 3.4.16)

Proposition

Every Grothendieck topos is a total category and a cototal category.

Proof

From the page total category, totality follows from the fact that a Grothendieck topos is

• Cocomplete,
• Well-copowered (because quotient objects of an object $X$ are in bijection with equivalence relations on $X$, and there is a small set of these because a Grothendieck topos is well-powered), and
• Has a generator (e.g., the coproduct of all representables for a small site presentation).

Dually, a Grothendieck topos is

• Complete,
• Well-powered (because subobjects of an object $X$ are in bijection with elements of a small hom-set $\hom(X, \Omega)$), and
• Has a cogenerator (e.g., the product of power objects $\Omega^c$ where $c$ ranges over representables for a small site presentation).

Therefore a Grothendieck topos is also cototal.

Giraud's axiomatic characterization

Giraud characterized Grothendieck toposes as categories satisfying certain exactness and small completeness properties (where “small” is again relative to the given category of sets $S$). The exactness properties are elementary (not depending on $S$), and are satisfied in any elementary topos, or even a pretopos.

Giraud's theorem characterises a Grothendieck topos as follows:

1. a locally small category with a small generating set,
2. with all finite limits,
3. with all small coproducts, which are disjoint, and pullback-stable,
4. where all congruences have effective quotient objects, which are also pullback-stable.

These conditions are equivalent to

• a locally small infinitary pretopos with a small generating set.

See the Elephant, theorem C.2.2.8. (There, the assumption of local smallness is not stated explicitly, but it is included in the definition of $\infty$-pretopos by way of well-poweredness; on the nLab it is not so included, so we have to state it explicitly. To see that it is necessary, note that if $U$ and $V$ are Grothendieck universes with $U\in V$, then $Set_V$ satisfies all the other conditions relative to $Set_U$, but is not locally small and is not a Grothendieck topos.) See also Wikipedia.

Sometimes (3,4) are combined and strengthened to the statement that the category has all small colimits, which are effective and pullback-stable. However, this is a mistake for two reasons: it is a significantly stronger axiomatisation (since without the small generating set, not every infinitary pretopos has this property), and it is not valid in weak foundations (while the definition given above is).

Street’s axiomatic characterization

Augmenting the aforementioned Proposition that Grothendieck toposes are total categories, Street more or less characterizes Grothendieck toposes as lex total categories having the same “size” as $Set$.

In more detail: we take as background set theory ZFC + “there is a strongly inaccessible cardinal” $\kappa$; equivalently, the existence of a single Grothendieck universe $U$ (a set of “small sets”). Supposing given a model $V$ of $ZFC+universe$, “category” shall then refer to category theory interpreted in $V$. Let $Set$ be the category of small sets; note that the set of morphisms of $Set$ has size $\kappa$.

Recall that locally small category $E$ is lex total if the Yoneda embedding $y: E \to [E^{op}, Set]$ has a left exact left adjoint.

Theorem

(Street) A category $E$ is a Grothendieck topos iff it is lex total and the size of the set of isomorphism classes of objects is $\kappa$ or less.

Remark

This result is in the spirit of saying “every Grothendieck topos is the category of sheaves with respect to the canonical topology on itself”. Putting aside set-theoretic issues, it suggests that Grothendieck toposes be seen as analogous to frames, which may be defined as lex total objects in $\mathbf{2}$-$Cat$. In this setting, the appropriate morphisms are left exact left adjoints, so that Grothendieck toposes and geometric morphisms between them would be analogous to locales and continuous maps between them.

One can deduce formally that lex total categories are locally cartesian closed Heyting pretoposes.

As lex reflections of categories of presheaves

sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes

As localic groupoids

Every Grothendieck topos is equivalent to the classifying topos of a localic groupoid, every Grothendieck topos with enough points is equivalent to the classifying topos of a topological groupoid, and in fact the 2-category Topos of Grothendieck toposes is equivalent to a localization of that of localic groupoids.

See at classifying topos of a localic groupoid for more.

Internal logic

Being an elementary topos, a Grothendieck topos has an internal logic that can be taken to be higher-order logic or a form of dependent type theory. It is also useful to take its internal geometric logic, in particular because that is preserved by geometric morphisms and because every geometric theory has a Grothendieck classifying topos.

Although the internal logic of a Grothendieck topos is constructive logic, there are a number of principles that are true in every Grothendieck topos (at least, assuming the base topos Set is classical) but are not constructively provable (and in particular can fail in other toposes). Roughly speaking, many of these axioms assert that classicality fails “in only a small way”. These include:

• WISC: every set has a weakly initial set of covers.
• The (constructive) axiom of multiple choice (AMC).
• The axiom of stack completions.
• (allegedly) the local small cardinality selection axiom (SCSA).
• Every complete small category is a preorder.

2-monadicity over locally presentable categories

The 2-category of Grothendieck toposes and logical morphisms is 2-monadic over the 2-category of locally presentable categories and cocontinuous functors. This is proven in Bunge and Carboni 1995. See symmetric topos for more details.

Enrichment over accessible categories with directed colimits

The 2-category of Grothendieck toposes and geometric morphisms is enriched as a bicategory over the 2-category of accessible categories with directed colimits (note that accessible categories always have $\kappa$-filtered colimits for some cardinal $\kappa$, but not necessarily directed colimits unless they are $\omega$-accessible) and functors preserving directed colimits. This is proven in Di Liberti 2022.

In weak foundations

We have two definitions of a Grothendieck topos:

• the category of sheaves on some small site,
• a category that satisfies Giraud's axioms (as listed above).

The theorem that these are equivalent can be proved in quite weak foundations, whether finitist, predicative, or constructive (or all three at once), as long as we axiomatize correctly given the caveats listed in the previous section. Some hard-nosed predicativists (and even hard-nosed ZFC fundamentalists) may object to the language (on the ground that large categories such as $Set$ and other nontrivial Grothendieck toposes don't really exist), but they should accept the theorems when suitably phrased.

In predicative mathematics, however, we cannot prove that every Grothendieck topos is in fact a topos! In fact, it is immediate that the category of sets is a Grothendieck topos, but $Set$ is an elementary topos if and only if power sets are small, which is precisely what predicativists doubt. One can use the term Grothendieck pretopos to avoid implying that we have an elementary topos. On the other hand, since Grothendeick toposes came first, perhaps it is the definition of ‘elementary topos’ that is too strong.

Similarly, in finitist mathematics, we cannot prove that every Grothendieck topos has a natural numbers object; while in strongly predicative mathematics, we cannot prove that every Grothendieck topos is cartesian closed. In each case, once a property is accepted of $Set$ (the axiom of infinity and small function sets, in these examples), it can be proved for all Grothendieck toposes.

Constructivism as such is irrelevant; even in classical mathematics, most Grothendieck toposes are not boolean. However, for an analogous result, try the theorem that the category of presheaves on a groupoid (and hence any category of sheaves contained within it) is boolean. (Again, $Set$ itself is an example of this.)

The theorem that every Grothendieck topos is cocomplete is a subtle point; it fails only in finitist predicative mathematics. (The key point in the proof is to generate the transitive closure $\sim^*$ of a binary relation $\sim$. One proof defines $a \sim^* b$ to mean that $a \sim x_0 \sim \cdots \sim x_{n-1} \sim b$ for some $n$, which is predicative but infinitary; another defines $a \sim^* b$ to mean that $a \sim' b$ for every transitive relation $\sim'$ that contains $\sim$, which is finitary but impredicative.)

Examples and non-examples

By definition, every category of sheaves is a Grothendieck topos. Examples include:

• Every presheaf category $PSh(C) = Set^{C^{op}}$ is a Grothendieck topos.

• In particular Set is a Grothendieck topos, because it is equivalent to $PSh(\ast)$.

• Broadly, most of the toposes that people have worked with are Grothendieck.

On the other hand, there are (elementary) toposes that fail to be Grothendieck toposes:

• Because a Grothendieck topos has all small limits, $FinSet$ is not Grothendieck.

• More generally, the category $Set_{\lt \pi}$ of sets with cardinality below any bound $\pi$ is not a Grothendieck topos, even when $\pi$ is a strong limit cardinal so that $Set_{\lt \pi}$ is an (elementary) topos.

• Similarly, any category $FinSet^{C^{op}}$ of $FinSet$-valued presheaves is a topos but not a Grothendieck topos, and the same goes more generally for $Set_{\lt \pi}^{C^{op}}$ for $\pi$ a strong limit cardinal.

• The effective topos, the maps in which can be thought of as computable functions, is not Grothendieck.

• Other non-Grothendieck toposes occur in the topos-theoretic approach to nonstandard analysis.

Related concepts

• topos

  • Grothendieck topos, category of sheaves

    • geometric logic, geometric type theory

  • 2-topos

  • (∞,1)-sheaf (∞,1)-topos

• lex total category

Locally presentable categories: Cocomplete possibly-large categories generated under filtered colimits by small generators under small relations. Equivalently, accessible reflective localizations of free cocompletions. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a left exact localization.

$\phantom{A}$(n,r)-categories$\phantom{A}$	$\phantom{A}$toposes$\phantom{A}$	locally presentable	loc finitely pres	localization theorem	free cocompletion	accessible
(0,1)-category theory	locales	suplattice	algebraic lattices	Porst’s theorem	powerset	poset
category theory	toposes	locally presentable categories	locally finitely presentable categories	Gabriel–Ulmer’s theorem	presheaf category	accessible categories
model category theory	model toposes	combinatorial model categories		Dugger's theorem	global model structures on simplicial presheaves	n/a
(∞,1)-category theory	(∞,1)-toposes	locally presentable (∞,1)-categories		Simpson’s theorem	(∞,1)-presheaf (∞,1)-categories	accessible (∞,1)-categories

History

From Colin McLarty (sent to the Categories mailing list on Apr 14, 2018):

At the start of 1958 Grothendieck believed the correct Weil cohomology of a scheme $S$ would be the derived functor cohomology of some category $Ab(S)$ of sheaves of Abelian groups on $S$ —and $Ab(S)$ was likely to be the category of all Abelian group objects in some category of sheaves of sets on $S$. He had no name yet for such conjectural categories of sheaves of sets but for later reference I will call this category $T(S)$. It would require some new notion of sheaves of sets more general than the existing notion using topological spaces. He had no concrete idea what this new notion of sheaf might be. His experience in Kansas in 1955 suggested it was likely to come from some new notion of espace etale over a scheme. and it should have some good exactness properties.

On April 21, 1958 Grothendieck went to the Seminaire Chevalley to hear Serre describe isotrivial fiber bundles on a variety V, which are bundles that become trivial when pulled back to some surjective family of finite, unramified maps to V. Serre showed this gave the expected one-dimensional Weil cohomology groups of V. Grothendieck immediately told Serre it would work in all dimensions–which Serre found “very optimistic”“ at the time.

All of this is well documented in familiar sources.

Grothendieck’s 1973 topos lectures in Buffalo show that during Serre’s talk Grothendieck saw the Weil espaces etales over a scheme $S$ should be patched together from Serre’s unramified maps to $S$. That is, the sought category $T(S)$ of sheaves of sets on S should be generated by the unramified maps to S, and should have arbitrary colimits of these, and finite limits. He first sought to axiomatize this roughly the way his Tohoku paper had axiomatized sheaves of groups.

A few changes came soon: He realized finite unramified maps should be replaced by flat unramified maps (just as topological espaces etales are trivial locally in the fiber and not on the base). He abandoned the axiomatic approach to $T(S)$ as too vague and shifted to construction by sites. And for sites he began treating sheaves as a kind of functor, rather than as espaces etales. Only after the notion ot site developed would Giraud give his topos axioms, and no one has yet really taken a precise notion of espace etale much beyond the topological case.

But April 21 1958 was the birth of topos theory. The term topos came later. (Lots of people are not named on the day they are born.)

As of the summer of 1973 Grothendieck’s stated preferred definition of topos was still: a category with arbitrary colimits, finite limits, and a small generating set. He says over and over this is not quite adequate for proofs. He says proofs require the notion of site, or else the Giraud axioms, but he calls the vaguer idea more intuitive and says that is the way to think about a topos.

Also in 1973 Grothendieck says the objects in any topos should be seen as espaces etales over the terminal object of the topos, in a generalized sense that includes saying any orbit of a group action lies “etale” over a fixed point. Today, it is not obvious that this can work well in general. In SGA4 Grothendieck had already given evidence that it would work for petit topos but not gros topos. But still in 1973 he did say it. And it works perfectly for petit etale toposes—as long as we generalize the notion of espace etale to include a Galois orbit (in an extension field) lying over a single point (in the ground field).

References

Basic exposition:

• Michael Barr, Colin McLarty, Charles Wells, Variable set theory, prepared for Scientific American but unpublished (~1985) [pdf, pdf]

A quick introduction of the basic facts of sheaf-topos theory is chapter I, “Background in topos theory” in

• Ieke Moerdijk, Classifying Spaces and Classifying Topoi Lecture Notes in Mathematics 1616, Springer (1995)

Textbook accounts include

• Francis Borceux, vol 3 of Handbook of Categorical Algebra, Cambridge University Press (1994)

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

Grothendieck topoi appear around section III,4 there. A proof of Giraud’s theorem is in appendix A.

The proof of Giraud’s theorem for (∞,1)-topoi is section 6.1.5 of

• Jacob Lurie, Higher Topos Theory

Street’s characterization of Grothendieck toposes is given in

• Ross Street, Notions of topos, Bull. Australian Math. Soc. 23 (1981), 199-208; MR83a:18014. (pdf link)

These are also called Giraud frames in

• Steven Vickers, Generalized point-free spaces, pointwise [arXiv:2206.01113]

On 2-monadicity of the opposite of the 2-category of Grothendieck toposes:

• Marta Bunge, Aurelio Carboni, The Symmetric Topos , JPAA 105 (1995) pp.233-249.

On enrichment of the 2-category of Grothendieck toposes:

• Ivan Di Liberti, General facts on the Scott Adjunction, Applied Categorical Structures 30.3 (2022): 569-591.