nLab Topos

Definition

By $Topos$ (or $Toposes$) is denoted the category of toposes. Usually this means:

• objects are sheaf toposes;

• morphisms are geometric morphisms of toposes.

(In parts of the literature this is denoted merely “$Top$”, but that clashes with the common notation for the the category of topological spaces.)

This is naturally a 2-category, where

• 2-morphism are geometric transformations.

That is, a 2-morphism $f\to g$ is a natural transformation $f^* \to g^*$ (which is, by mate calculus, equivalent to a natural transformation $g_* \to f_*$ between direct images).

Thus, $Toposes$ is equivalent to both of

• the (non-full) sub-2-category of the 1-opposite $Cat^{op}$ of Cat on categories that are sheaf toposes and morphisms that are the inverse image parts of geometric morphisms, and

• the (non-full) sub-2-category of the 2-opposite $Cat^{co}$ of Cat on categories that are sheaf toposes and morphisms that are the direct image parts of geometric morphisms.

Remark

(elementary toposes and logical functors) We obtain a very different 2-category of toposes if we take objects to be elementary toposes and morphisms the logical functors; this 2-category is sometimes denoted $Log$ or $LogTopos$.

Properties

From topological spaces to toposes

The operation of forming categories of sheaves

$$Sh(-) \,\colon\, Top \longrightarrow Topos$$

embeds topological spaces into toposes. For $f \colon X \to Y$ a continuous map we have that $Sh(f)$ is the geometric morphism

$$Sh(f) \,\colon\, Sh(X) \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} Sh(Y)$$

with $f_*$ the direct image and $f^*$ the inverse image.

Strictly speaking, this functor is not an embedding if we consider $Top$ as a 1-category and $Toposes$ as a 2-category, since it is then not fully faithful in the 2-categorical sense—there can be nontrivial 2-cells between geometric morphisms between toposes of sheaves on topological spaces.

However, if we regard $Top$ as a (1,2)-category where the 2-cells are inequalities in the specialization ordering, then this functor does become a 2-categorically full embedding (i.e. an equivalence on hom-categories) if we restrict to the full subcategory $SobTop$ of sober spaces. This embedding can also be extended from $SobTop$ to the entire category of locales (which can be viewed as “Grothendieck 0-toposes”).

From toposes to higher toposes

There are similar full embeddings $Topos \hookrightarrow 2 Topos$ and $Topos \hookrightarrow (n,1)Topos$ of sheaf (1-)toposes into 2-sheaf 2-toposes and sheaf (n,1)-toposes for $2\le n\le \infty$.

Note that these embeddings are not the identity functor on underlying categories: a 1-topos is not itself an $n$-topos, instead we have to take $n$-sheaves on a suitable generating site for it.

From locally presentable categories to toposes

There is a canonical forgetful functor $U : Topos \to$ LocPrCat${}^{op}$ assigning underlying locally presentable categories.

This 2-functor has a right 2-adjoint (Bunge & Carboni 1995 p 235).

Limits and colimits

The 2-category $Topos$ is not all that well-endowed with limits, but its slice categories are finitely complete as 2-categories, and $ShTopos$ is closed under finite limits in $Topos/Set$. In particular, the terminal object in $ShToposes$ is the topos Set $\simeq Sh(*)$.

Colimits

The supply with colimits is better:

Proposition

All small (indexed) 2-colimits in $ShTopos$ exists and are computed as (indexed) 2-limits in Cat of the underlying inverse image functors.

This appears as Moerdijk, theorem 2.5

Proposition

Let

$$\array{ \mathcal{F} &\stackrel{p_2}{\to}& \mathcal{E}_2 \\ {}^{\mathllap{p_1}}\downarrow &\swArrow& \downarrow^{\mathrlap{f_2}} \\ \mathcal{E}_1 &\underset{f_1}{\to}& \mathcal{E} }$$

be a 2-pullback in $Topos$ such that

• $f_1, f_2$ are both pseudomonic morphisms

• $\mathcal{E}_1 \coprod \mathcal{E}_2 \to \mathcal{E}$ is an effective epimorphism;

then the diagram of inverse image functors

$$\array{ \mathcal{F} &\stackrel{p_2^*}{\leftarrow}& \mathcal{E}_2 \\ {}^{\mathllap{p_1^*}}\uparrow &\swArrow& \uparrow^{\mathrlap{f_2^*}} \\ \mathcal{E}_1 &\underset{f_1^*}{\leftarrow}& \mathcal{E} }$$

is a 2-pullback in Cat and so by the above the original square is also a 2-pushout.

This appears as theorem 5.1 in (BungeLack)

Proposition

The 2-category $Topos$ is an extensive category. Same for toposes bounded over a base.

This is in (BungeLack, proposition 4.3).

Products

Proposition

The Cartesian product of presheaf toposes $PSh(C_i)$ in $Topos$ is the presheaf topos on the product category of their sites:

(1)$$PSh(C_1) \times_{Topos} PSh(C_2) \;\simeq\; PSh(C_1 \times_{Cat} C_2 ) \,.$$

More generally, for $J_i$ a Grothendieck topology on $C_i$, the product in $Topos$ of the sheaf topos is the sheaf topos over the product site $(C_1 \times C_2, J_1 \times J_2)$, where $J_1 \times J_2$ is the Grothendieck topology on the product category generated from products of covering sieves from $J_1$ and $J_2$:

(2)$$Sh(C_1, J_1) \times_{Topos} PSh(C_2, J_2) \;\simeq\; PSh(C_1 \times C_2,\, J_1 \times J_2 ) \,.$$

(cf. Pitts 1985 proof of Thm. 2.3, Valero 2018 p 98-99)

Proof

Since every sheaf topos admits a site that has all finite limits, we first consider the topos-product of presheaf toposes (1) under the assumption that the sites $C_i$ have all finite limits. (We refer to these small categories $C_i$ as “sites” already, even though in the presheaf case they are as such equipped with the trivial Grothendieck topology).

To check the defining universal property of a Cartesian product, we use that geometric morphisms

$$f_\ast \,\colon\, \mathcal{X} \leftrightarrow \mathcal{Y} \,\colon\, f^\ast$$

between sheaf toposes (generally between locally presentable categories) correspond bijectively to finite limit-preserving cocontinuous functors

$$\mathcal{X} \overset{f^\ast}{\longleftarrow} \mathcal{Y}$$

(by the adjoint functor theorem for locally presentable categories).

Then writing

$$p_i \,\colon\, C_1 \times C_2 \longrightarrow C_i$$

for the canonical projection functors out of the product category of the given sites, and writing

$$p^\ast_i \,\colon\, PSh(C_i) \longrightarrow PSh( C_1 \times C_2 )$$

for the corresponding pre-composition functors on presheaves (which presverve all (co)limits since (co)limits of presheaves are computed objectwise), it is sufficient to show that for $\mathcal{X} \in Topos$ and

$$f^\ast_i \,\colon\, PSh(C_i) \longrightarrow \mathcal{X} \,, \;\;\;\; i \in \{1,2\}$$

a pair of cocontinuous and finitely continuous functors, there is a unique such functor

$$(f_1, f_2)^\ast \,\colon\, PSh(C_1 \times C_2) \longrightarrow \mathcal{X}$$

which makes the following diagram commute:

To this end, observe that for a representable presheaf

$$c_i \,\in\, C_i \xhookrightarrow{y} PSh(C_i)$$

(with the Yoneda embedding $y$ on the right)

we evidently have

(3)$$p^\ast_1\big( y(c_1) \big) \,\simeq\, y(c_1, \ast_2) \;\;\;\; \text{and} \;\;\;\; p^\ast_2\big( y(c_2) \big) \,\simeq\, y(\ast_1, c_2)$$

(where “$\ast_i$” denotes the terminal object of $C_i$). It follows that to be finite-limit preserving, and hence in particular finite product-preserving, the dashed map must send representables

$$(c_1, c_2) \,\in\, C_1 \times C_2 \xhookrightarrow{y} PSh(C_1 \times C_2)$$

to

$$\begin{array}{rcl} (f_1, f_2)^\ast \big( y(c_1, c_2) \big) &\simeq& (f_1, f_2)^\ast \big( y\big( (c_1, \ast_2) \times (\ast_1, c_2) \big) \\ &\simeq& (f_1, f_2)^\ast \big( y(c_1, \ast_2) \times y(\ast_1, c_2)) \big) \\ &\simeq& (f_1, f_2)^\ast \big( y(c_1, \ast_2) \big) \times (f_1, f_2)^\ast \big( y(\ast_1, c_2) \big) \\ &\simeq& (f_1, f_2)^\ast \circ p^\ast_1 \big( y(c_1) \big) \;\times\; (f_1, f_2)^\ast \circ p^\ast_2 \big( y(c_2) \big) \\ &\simeq& f^\ast_1\big(y(c_1)\big) \times f^\ast_2\big(y(c_2)\big) \,, \end{array}$$

where we have used, in order of appearance, that:

1. limits in a product category are computed via the limits in the factor categories,

2. the Yoneda embedding preserves limits,

3. $(f_1, f_2)^\ast$ is assumed to preserve finite limits,

4. the equivalences (3) hold,

5. the above triangles commute.

So this fixes the values of $(f_1, f_2)^\ast$ on all representables. But thereby it actually fixes its values on all of $PSh(C_1 \times C_2)$, since the latter is the free cocompletion of $C_1 \times C_2$ (all presheaves are colimits of representables) and since $(f_1, f_2)^\ast$ is also required to be cocontinuous and hence must preserve these colimits.

This establishes the claim

$$PSh(C_1) \times_{Topos} PSh(C_2) \,\simeq\, PSh(C_1 \times C_2)$$

for the case that the $C_i$ have all finite limits. To conclude, it is now sufficient to observe that this construction descends to sheaf toposes as claimed.

To that end, just to note that for a geometric morphism $f$ between presheaf toposes to descend to sheaves is equivalent to $f^\ast$ sending covering morphisms to local isomorphisms (cf. at morphism of sites, because, by adjunction, this is equivalently the condition for $f_\ast$ to take sheaves to sheaves, whence as such its left adjoint is $f^\ast$ followed by sheafification).

Example

A category $sSh(C)$ of simplicial sheaves is the Topos-product of the corresponding sheaf topos of Set-values sheaves with the topos $sSet$ of simplicial sets, the latter being the presheaf topos over the simplex category:

$$sSh(C) \,\equiv\, Sh(C,sSet) \;\simeq\; Sh(C \times \Delta) \;\simeq\; Sh(C) \times_{Topos} Sh(\Delta) \;\simeq\; Sh(C) \times_{Topos} sSet \,.$$

Pullbacks

Proposition

Let

$$\array{ && \mathcal{X} \\ && \downarrow^{\mathrlap{(g^* \dashv g_*)}} \\ \mathcal{Y} &\stackrel{(f^* \dashv f_*)}{\to}& \mathcal{Z} }$$

be a diagram of toposes. Then its pullback in the (2,1)-category version of $Topos$ is computed, roughly, by the pushout of their sites of definition.

More in detail: there exist sites $\tilde \mathcal{D}$, $\mathcal{D}$, and $\mathcal{C}$ with finite limits and morphisms of sites

$$\array{ && \mathcal{D} \\ && \uparrow^{\mathrlap{g}} \\ \tilde \mathcal{D} &\stackrel{f}{\leftarrow}& \mathcal{C} }$$

such that

$$\left( \array{ && \mathcal{X} \\ && ^{\mathllap{(g^* \dashv g_*)}} \Big\downarrow \\ \mathcal{Y} &\underset{(f^* \dashv f_*)}{\longrightarrow}& \mathcal{Z} } \right) \,\,\, \simeq \,\,\, \left( \array{ && Sh(\mathcal{D}) \\ && ^{\mathllap{(Lan_g \dashv (-)\circ g)}}\Big\downarrow \\ Sh(\tilde \mathcal{D}) &\underset{(Lan_f \dashv (-)\circ f)}{\longrightarrow}& Sh(\mathcal{C}) } \right) \,.$$

Let then

$$\array{ \widetilde{\mathcal{D}} \textstyle{\coprod_{\mathcal{C}}} \mathcal{D} &\overset{f'}{\longleftarrow}& \mathcal{D} \\ \mathllap{{}^{g'}}\big\uparrow &\swArrow_{\simeq}& \big\uparrow\mathrlap{^g} \\ \widetilde{\mathcal{D}} &\underset{f}{\longleftarrow}& \mathcal{C} } \,\,\,\,\, \in Cat^{lex}$$

be the pushout of the underlying categories in the non-full subcategory ${Cat}^{lex} \subset Cat$ of categories with finite limits and limit-preserving functors between them (cf. Lurie, prop. 6.3.4.3).

Let moreover

$$Sh\big( \widetilde{\mathcal{D}} \textstyle{\coprod_{\mathcal{C}}} \mathcal{D} \big) \hookrightarrow PSh\big( \widetilde{\mathcal{D}} \textstyle{\coprod_{\mathcal{C}}} \widetilde{\mathcal{D}} \big)$$

be the reflective subcategory obtained by localization at the class of morphisms generated by the inverse image $Lan_{f'}(-)$ of the coverings of $\mathcal{D}$ and the inverse image $Lan_{g'}(-)$ of the coverings of $\tilde \mathcal{D}$.

Then

$$\array{ Sh(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D}) &\to& \mathcal{X} \\ \downarrow && \downarrow^{\mathrlap{(g^* \dashv g_*)}} \\ \mathcal{Y} &\stackrel{(f^* \dashv f_*)}{\to}& \mathcal{Z} }$$

is a pullback square.

This appears as Lurie, prop. 6.3.4.6.

Remark

For localic toposes this reduces to the statement of localic reflection: the pullback of toposes is given by the of pullback the underlying locales which in turn is the pushout of the corresponding frames.

Free loop spaces

The free loop space object of a topos in Topos is called the isotropy group of a topos.

Related concepts

• Topos

• (∞,1)Topos

References

On Cartesian products and base change in $Topos$:

• Andrews Pitts: On product and change of base for toposes, Cahiers de topologie et géométrie différentielle catégoriques 26 1 (1985) 43-61 [numdam:CTGDC_1985__26_1_43_0, pdf]

On the characterization of colimits in $Topos$:

• Ieke Moerdijk, The classifying topos of a continuous groupoid. I Transaction of the American mathematical society 310 2 (1988) [pdf]

The fact that $Topos$ is extensive:

• Marta Bunge, Steve Lack: van Kampen theorem for toposes [ps]

General limits and colimits of toposes:

• Jacob Lurie, 6.3.2-6.3.4 of: Higher Topos Theory (2009)

  (discussed for (∞,1)-toposes, but the statements are verbatim true in the (2,1)-category category $Topos$)

The adjunction between toposes and locally presentable categories:

• Marta Bunge, Aurelio Carboni: The symmetric topos, Journal of Pure and Applied Algebra 105 (1995) 233-249 [doi;10.1016/0022-4049(94)00157-X]

See also:

• Juan Sebastian Arias Valero: Gesture Theory: Topos-Theoretic Perspectives and Philosophical Framework, PhD thesis (2018) [handle:63339, pdf]

category: category