The hyperconnected maps that are local

doi:10.1016/j.jpaa.2020.106596

Journal of Pure and Applied Algebra

Volume 225, Issue 5, May 2021, 106596

https://doi.org/10.1016/j.jpaa.2020.106596 Get rights and content

Abstract

A level $j : E_{j} \to E$ of a topos $E$ is said to have monic skeleta if, for every X in $E$ , the counit $j_{!} (j^{⁎} X) \to X$ is monic. For instance, the centre of a hyperconnected geometric morphism is such a level. We establish two related sufficient conditions for an adjunction to extend to a level with monic skeleta. As an application, we characterize the hyperconnected geometric morphisms that are local providing an interesting expression for the associated centres that suggests a generalization of open subtoposes. As a corollary, we obtain that a hyperconnected $p : E \to S$ is pre-cohesive if and only if $p_{⁎} : E \to S$ preserves coequalizers and $p^{⁎} : S \to E$ is cartesian closed.

Section snippets

Local maps and levels with monic skeleta

Recall that a geometric morphism $f : F \to E$ is local if the direct image $f_{⁎} : F \to E$ has an $E$ -indexed right adjoint. Alternatively, f is local if and only if $f_{⁎}$ has a fully faithful right adjoint, usually denoted by $f^{!} : E \to F$ . In this case, $f^{!}$ is the direct image of a subtopos $E \to F$ called the centre of f. See C3.6 in [6].

Local geometric morphisms were defined in [9] as an abstraction of the notion of ‘local topos’ introduced by Grothendieck and Verdier in [4, VI,8.4]. It is relevant to remark that, as the

Mono-coreflections

We first need to introduce an auxiliary piece of terminology.

Definition 2.1

Let $N : A \to B$ and $M : B \to C$ be functors. We say that M is N-faithful if M is faithful on morphisms whose codomain is in the image of N.

For the rest of the section let $L ⊣ M : B \to A$ be a coreflective subcategory with counit β. The (necessarily iso) unit will be denoted by α.

Lemma 2.2

Each of the following items implies the next:

1.
M is L-faithful.
2.
( $L ⊣ M$ is a mono-coreflection.) The counit β is monic.
3.
(The subcategory L is closed under ‘quotients’.) For every strong

Principal topologies

Let $E$ be a topos.

Lemma 3.1

For any Lawvere-Tierney topology j in $E$ , the following are equivalent:

1.
Every object X in $E$ has a least j-dense subobject.
2.
For every X in $E$ , the j-closure operation $Sub (X) \to Sub (X)$ has a left adjoint.

Proof

See Lemma 2.3 in [1]. □

The Lawvere-Tierney topologies satisfying the equivalent conditions of Lemma 3.1 are called principal in [1]. Theorem 4.2 in [10] implies that if $E$ is a presheaf topos then a subtopos $E_{j} \to E$ is a level if and only if j is principal. One implication of this

Topologies inducing levels with monic skeleta

In this section we characterize the principal topologies determined by levels with monic skeleta. The main tool is Lemma 4.1 below that we restate from [18].

Lemma 4.1

Let the following categories and functors on the left below be such that $L ⊣ M$ is a full coreflective subcategory with counit β and $F ⊣ G$ is a full reflective subcategory with unit η. If the natural $F β : F L M \to F$ is an isomorphism then the diagram on the right above commutes (up to that canonical iso). If, moreover, $M η_{L} : M L \to M G F L$ is also an

Idempotent comonads that have a right adjoint

Let $E$ be a topos and let $L ⊣ M : E \to E_{β}$ be a mono-coreflection with counit β. Again, we let $C = L M$ so that $β_{X} : C X \to X$ . Define the family $c = (c_{X} : Sub (X) \to Sub (X) | X \in E)$ of functions by declaring that $c_{X} u = β_{X} \Rightarrow u$ for any monic $u : U \to X$ .

Lemma 5.1

The following are equivalent:

1.
The family $c$ is a universal closure operator.
2.
For every $f : X \to Y$ and monic $u : U \to X$ , $β_{X} \Rightarrow f^{⁎} u \leq f^{⁎} (β_{Y} \Rightarrow u)$ .
3.
For every $f : X \to Y$ and monic $u : U \to X$ , $β_{X} \Rightarrow f^{⁎} u = f^{⁎} (β_{Y} \Rightarrow u)$ .

Proof

For any X, $c_{X}$ is easily seen to be monotone, inflationary and idempotent using the standard properties of Heyting

The hyperconnected maps that are local

Here we apply the results in earlier sections to characterize the hyperconnected geometric morphisms that are local. We also derive a characterization of the hyperconnected maps that are pre-cohesive and give an alternative proof of one of the main results in [18].

Corollary 6.1

A hyperconnected geometric morphism $p : E \to S$ is local if and only if $p_{⁎} : E \to S$ preserves coequalizers.

Proof

If p is local then $p_{⁎}$ preserves coequalizers because it is a left adjoint. For the converse let β be the counit of $p^{⁎} ⊣ p_{⁎}$ which is

Acknowledgements

Much of the work was done during a visit to the Università di Bologna in 2018, with the support of C. Smith and funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 690974.

References (19)

S. Awodey et al.
Elementary axioms for local maps of toposes
J. Pure Appl. Algebra
(2003)
F.W. Lawvere
Linearization of graphic toposes via Coxeter groups
J. Pure Appl. Algebra
(2002)
F. Borceux
Handbook of Categorical Algebra 1
(1994)
A. Grothendieck, Pursuing stacks, typescript,...
A. Grothendieck et al.
Théorie des topos (SGA 4, exposés I-VI)
(1972)
P.T. Johnstone
How general is a generalized space?
Aspects of Topology, Mem. H. Dowker
Lond. Math. Soc. Lect. Note Ser.
(1985)
P.T. Johnstone
Sketches of an Elephant: A Topos Theory Compendium
(2002)
P.T. Johnstone
Remarks on punctual local connectedness
Theory Appl. Categ.
(2011)
P.T. Johnstone
Calibrated toposes
Bull. Belg. Math. Soc. Simon Stevin
(2012)

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The hyperconnected maps that are local

Abstract

Section snippets

Local maps and levels with monic skeleta

Mono-coreflections

Principal topologies

Topologies inducing levels with monic skeleta

Idempotent comonads that have a right adjoint

The hyperconnected maps that are local

Acknowledgements

J. Pure Appl. Algebra

J. Pure Appl. Algebra

Handbook of Categorical Algebra 1

Théorie des topos (SGA 4, exposés I-VI)

How general is a generalized space?

Aspects of Topology, Mem. H. Dowker

Lond. Math. Soc. Lect. Note Ser.

Sketches of an Elephant: A Topos Theory Compendium

Remarks on punctual local connectedness

Theory Appl. Categ.

Calibrated toposes

Bull. Belg. Math. Soc. Simon Stevin