essential geometric morphism


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A geometric morphism f:EFf : E \to F between toposes is a functor of the underlying categories that is consistent with the interpretation of EE and FF as generalized topological spaces.

If F=Set=Sh(*)F = Set = Sh(*) is the terminal sheaf topos, then ESetE \to Set is essential if EE is a locally connected topos . In general, ff being essential is a necessary (but not sufficient) condition to ensure that ff behaves like a map of topological spaces whose fibers are locally connected: that it is a locally connected geometric morphism.



Given a geometric morphism (f *f *):EF(f^* \dashv f_*) : E \to F, it is an essential geometric morphism if the inverse image functor f *f^* has not only the right adjoint f *f_*, but also a left adjoint f !f_!:

(f !f *f *):Ef *f *f !F. (f_! \dashv f^* \dashv f_*) \;\;\; : \;\;\; E \stackrel{\overset{f_!}{\longrightarrow}}{\stackrel{\overset{f^*}{\longleftarrow}}{\underset{f_*}{\longrightarrow}}} F \,.

A point of a topos x:SetEx : Set \to E which is given by an essential geometric morphism is called an essential point of EE.


There are various further conditions that can be imposed on a geometric morphism:


Relation to morphisms of (co)sites

For CC and DD small categories write [C,Set][C,Set] and [D,Set][D,Set] for the corresponding copresheaf toposes. (If we think of the opposite categories C opC^{op} and D opD^{op} as sites equipped with the trivial coverage, then these are the corresponding sheaf toposes.)


This construction extends to a 2-functor

[,Set]:Cat small coTopos ess [-,Set] : Cat_{small}^{co} \to Topos_{ess}

from the 2-category Cat small{}_{small} with 2-morphisms reversed) to the sub-2-category of Topos on essential geometric morphisms, where a functor f:CDf : C \to D is sent to the essential geometric morphism

(f !f *f !):[C,Set]f *:=Ran ff *:=()ff !:=Lan f[D,Set], (f_! \dashv f^* \dashv f_!) : [C,Set] \stackrel{\overset{f_! := Lan_f}{\to}}{\stackrel{\overset{f^* := (-) \circ f}{\leftarrow}}{\underset{f_* := Ran_f}{\to}}} [D,Set] \,,

where Lan fLan_f and Ran fRan_f denote the left and right Kan extension along ff, respectively.


This 2-functor is a full and faithful 2-functor when restricted to Cauchy complete categories:

[,Set]:Cat CauchyComp coTopos ess. [-, Set] : Cat^co_{CauchyComp} \hookrightarrow Topos_{ess} \,.

For all small categories C,DC,D we have an equivalence of categories

Func(C¯,D¯) opTopos ess([C,Set],[D,Set]) Func(\overline{C},\overline{D})^{op} \simeq Topos_{ess}([C,Set], [D,Set])

between the opposite category of the functor category between the Cauchy completions of CC and DD and the the category of essential geometric morphisms between the copresheaf toposes and geometric transformations between them.

In particular, since every poset – when regarded as a category – is Cauchy complete, we have


The 2-functor

[,Set]:PosetTopos ess [-,Set] : Poset \to Topos_{ess}

is a full and faithful 2-functor.


Sometimes it is useful to decompose this statement as follows.

There is a functor

Alex:PosetLocale Alex : Poset \to Locale

which assigns to each poset a locale called its Alexandroff locale. By a theorem discussed there, a morphisms of locales f:XYf : X \to Y is in the image of this functor precisely if its inverse image morphism f *Op(Y)Op(X)f^* Op(Y) \to Op(X) of frames has a left adjoint in the 2-category Locale.

Moreover, for any poset PP the sheaf topos over AlexPAlex P is naturally equivalent to [P,Set][P,Set]

[,Set]ShAlex. [-,Set] \simeq Sh \circ Alex \,.

With this, the fact that [,Set]:PosetTopos[-,Set] : Poset \to Topos hits precisely the essential geometric morphisms follows with the basic fact about localic reflection, which says that Sh:LocaleToposSh : Locale \to Topos is a full and faithful 2-functor.

Some morphism calculus


Let f:EFf : E \to F be an essential geometric morphism.

For every ϕ:Xf *f *A\phi : X \to f^* f_* A in EE the diagram

X ϕ f *f *A f *f !X A \array{ X &\stackrel{\phi}{\to}& f^* f_* A \\ \downarrow && \downarrow \\ f^* f_! X &\stackrel{}{\to}& A }

commutes, where the vertical morphisms are unit and counit, respectively, and where the bottom horizontal morphism is the adjunct of ϕ\phi under the composite adjunction (f *f !f *f *)(f^* f_! \dashv f^* f_*).


The morphism ϕ:Xf *f *A\phi : X \to f^* f_* A is the component of a natural transformation

* X E A ϕ f * E f * F. \array{ *&&\overset{X}{\to}&& E \\ {}^{\mathllap{A}}\downarrow &\Downarrow^\phi&& {}^{\mathllap{f^*}}\nearrow \\ E &\underset{f_*}{\to}&F } \,.

The composite Xϕf *f *AAX \stackrel{\phi}{\to} f^* f_* A \to A is the component of this composed with the counit f *f *Idf^* f_* \Rightarrow Id.

We may insert the 2-identity given by the zig-zag law

=* X E = E A ϕ f * f ! f * E f * F = F. \cdots \;\;\; = \;\;\; \array{ *&&\overset{X}{\to}&& E && = && E \\ {}^{\mathllap{A}}\downarrow &\Downarrow^\phi&& {}^{\mathllap{f^*}}\nearrow &\Downarrow& \searrow^{f_!} &\Downarrow& \nearrow_{\mathrlap{f^*}} \\ E &\underset{f_*}{\to}&F &&=&& F } \,.

Composing this with the counit f *f *Idf^* f_* \Rightarrow Id produces the transformation whose component is manifestly the morphism Xf *f !XAX \to f^* f_! X \to A.


Etale geometric morphisms

For any morphism f:ABf\colon A\to B in a topos EE, the induced geometric morphism f:E/AE/Bf\colon E/A \to E/B of overcategory toposes is essential.

For the case B=*B = * the terminal object, the geometric morphism

π:E/AE \pi : E/A \to E

is also called an etale geometric morphism.

Locally connected toposes

A locally connected topos EE is one where the global section geometric morphism Γ:ESet\Gamma : E \to Set is essential.

(f !f *f *):EΓLConstΠ 0Set. (f_! \dashv f^* \dashv f_*) \;\;\; : \;\;\; E \stackrel{\overset{\Pi_0}{\longrightarrow}}{\stackrel{\overset{LConst}{\longleftarrow}}{\underset{\Gamma}{\longrightarrow}}} Set \,.

In this case, the functor Γ !=Π 0:ESet\Gamma_! = \Pi_0 : E \to Set sends each object to its set of connected components. More on this situation is at homotopy groups in an (∞,1)-topos.

Note, though that if p:ESp\colon E\to S is an arbitrary geometric morphism through which we regard EE as an SS-topos, i.e. a topos “in the world of SS,” the condition for EE to be locally connected as an SS-topos is not just that pp is essential, but that the left adjoint p !p_! can be made into an SS-indexed functor (which is automatically true for p *p^* and p *p_*). This is automatically the case for SetSet-toposes (at least, when our foundation is material set theory—and if our foundation is structural set theory, then our large categories and functors all need to be assumed to be SetSet-indexing anyway). For more see locally connected geometric morphism.

Tiny objects

The tiny objects of a presheaf topos [C,Set][C,Set] are precisely the essential points Set[C,Set]Set \to [C,Set]. See tiny object for details.


As many other things, it all started as an exercise in

Speaking of exercises, consider the results of Roos reported in exercise 7.3 of

  • Peter Johnstone, Topos Theory , Academic Press New York 1977. (Dover reprint 2014, pp.254f)

The case of sheaves valued in FinSet is considered in

  • J. Haigh, Essential geometric morphisms between toposes of finite sets , Math. Proc. Phil. Soc. 87 (1980) pp.21-24.

The standard reference for essential localizations 1, aka levels, is

  • G. M. Kelly, F. W. Lawvere, On the Complete Lattice of Essential Localizations , Bull. Soc. Math. de Belgique XLI (1989) pp.261-299.

The definition of geometric morphism appears before Lemma A.4.1.5 in

Connected surjective and local geometric morphisms are discussed in

Further refinements are in

  • Bill Lawvere, Axiomatic cohesion Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41–49. (pdf)

  1. See at Aufhebung for further references on essential localizations.

Revised on September 4, 2015 18:20:52 by Urs Schreiber (