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A sufficiently cohesive topos is a cohesive topos that has enough connected objects in the sense that every object embeds into a connected object.
This can be viewed as a strong form of cohesiveness in the context of Lawvere's axiomatic approach to gros toposes. In fact, in Lawvere (1986) a big topos of spaces was defined as (one equivalent to) a sufficiently cohesive topos.
Sufficient cohesion is a relative concept and requires minimally the presence of an essential geometric morphism . Here we state it for an adjoint quadruple between toposes
such that (and hence ) is fully faithful and preserves finite products.
This (and in particular) is called a ‘cohesive topos’ (over ) at cohesive topos, and will be referred to as a ‘weakly cohesive topos’ in the present entry - a sufficiently cohesive topos in this context corresponds to the three axioms (0-2) for a ‘gros topos’ in Lawvere (1986) where the concept of sufficient cohesion was considered for the first time.
A weakly cohesive topos is called pre-cohesive if furthermore satisfies the Nullstellensatz i.e. the canonical map is an epimorphism. This is the situation explored in Menni (2014a, 2014b).
A pre-cohesive topos that moreover satisfies the continuity principle that natural in , , is called ‘cohesive’ in Lawvere (2007) where the term ‘sufficently cohesive’ occurs for the first time although the notion is defined in a more restricted environment than the earlier papers (cf. Lawvere 1986, 1992, 1999).
An object in a weakly cohesive topos is called contractible if for every object : .
In particular, a contractible object is connected: . Since exponentiation preserves in any Cartesian closed category, the terminal object is contractible in any weakly cohesive topos.
A weakly cohesive topos is called sufficiently cohesive if the subobject classifier is contractible i.e. for every object or, in other words, if all power objects are connected.
Since in general, every object embeds into its power object via the singleton map it follows that in a sufficiently cohesive topos every object embeds into a connected object i.e. has enough connected objects.
Furthermore, since in a Cartesian closed category
one sees that in a sufficiently cohesive topos power objects are not only connected but even contractible:
and hence it follows that in a sufficiently cohesive topos every object embeds into a contractible object.
Conversely, if every power object embeds into a connected object then the power objects will be connected themselves by proposition below since power objects are injective in general. Whence a (weakly) cohesive topos is sufficiently cohesive iff every object embeds into a connected object iff every object embeds into a contractible object. The last formulation is taken as the definition of sufficient cohesion in Lawvere (2007).
The definition of sufficient cohesion by ‘having enough contractible objects’ (Lawvere 2007) has of course the advantage that it even works in the wider context of Cartesian closed extensive categories that lack a subobject classifier. The definition by ‘having enough connected objects’ would be feasible for merely distributive categories that are not necessarily Cartesian closed (cf. Lawvere 1991, 1992) but Lawvere (1991, p.4) suggests that in this case the definition should be strenghtened to demand that every object is the equaliser of a pair of maps between two connected objects.
Let us first record some easy but useful facts concerning the interplay between connectedness and constancy.
Recall that in a category with a terminal object a morphism is called constant if factors through the terminal object :
An object is a terminal object iff is constant.
Proof. “”: By assumption . Since has a point there exists for every object at least one map . Suppose then that is some map :
But the righthand side does not depend on hence it is the only map .
Let be an object in a weakly cohesive topos and be a constant endomap such that . Then is connected i.e. .
Proof. Since by assumption , and, accordingly, are constants.
In a weakly cohesive topos, retracts of connected objects are connected themselves.
Proof. Let be a retract of with . Then factors as and applying shows that factors through the terminal object .
Since in general, injective objects are retracts of the objects that they embed into because such inclusions factor through by injectivity, it follows that in a weakly cohesive topos injective objects that embed into a connected object are connected themselves.
In particular, all injective objects are connected in a sufficiently cohesive topos. Since power objects are injective in general one gets the converse as well:
A weakly cohesive topos is sufficiently cohesive iff all injective objects are connected.
We can replace ‘connected’ with ‘contractible’ in this proposition as the following propositions show.
Let be a topos. Then all injective objects are connected iff all injective objects are contractible.
Proof. “”: by the following lemma.
Let be an injective object in a topos . Then is injective for any object , or in other words, exponentials preserve injective objects.
Proof. (cf. at injective object) We have to show that for every mono the induced function is onto. From the exponential adjunction one has the following commutative diagram
whence it suffices to show that is a mono since is injective. But this is easy to see:
which is equivalent to and .
Summing up we get:
A weakly cohesive topos is sufficiently cohesive iff all injective objects are contractible.
In a sufficiently cohesive topos the subobject classifier is obviously connected since . It is the aim of this section to prove that in a weakly cohesive topos the converse holds as well: is connected iff is contractible.
Before embarking on a proof let us consider two non-examples that display varied behaviors in respect to the connectedness of the subobject classifier and the exactness properties of the components functor :
The Sierpinski topos is weakly cohesive over since there exists a string of adjoint functors with
.
The Nullstellensatz fails as does the continuity principle. As a right adjoint preserves all limits and the terminal object in particular whence is connected in . Since the underlying category satisfies the Ore condition trivially, it follows then from a general result1 of Lawvere that is not connected and, accordingly, that the Sierpinski topos is not sufficiently cohesive!
On the other hand, consider the topos of directed graphs : it lacks the right adjoint to the section functor but has a connected components functor and the subobject classifier for directed graphs is connected.
Although does not preserve finite products in general, it nevertheless preserves finite products with the subobject classifier since has arcs between all nodes in both directions. This implies that the proof of proposition below still goes through and since this is the only argument that hinges on exactness properties of and nothing hinges on (the existence of) the right adjoint at all, the conclusions from hold as well: the subobject classifier of is not only connected but contractible! In other words, despite not being sufficiently cohesive nevertheless has enough contractible objects.
Recall that in any topos, the subobject classifier has two points fitting into the following pullback diagram (which is an equaliser diagram as well) due to the classifying property of for the monomorphism :
In a sufficiently cohesive topos is furthermore connected whence together with its Heyting algebra structure we can view it as a generalized (non linear) “interval” object:
An object in a weakly cohesive topos is called a (cohesive) connector if and has two points with empty equaliser: .
In a topos with a connected subobject classifier itself is a connector. Conversely the existence of a connector implies the connectedness of :
Let be weakly cohesive topos. has a connector iff .
Proof. “”: Let be a connector. Then is a subobject with characteristic map . Consider the two composites , :
For this simply yields by the definition of .
For we claim that since we have the following diagram:
Here the left square is a pullback since have empty equaliser by assumption. The right square is a pullback since it classifies whence the outer square is a pullback too. Therefore classifies which is exactly the definition of .
Since is terminal . Whence
This says that and are in the same connected component but a lattice whose top and bottom elements are in the same component is necessarily connected.
One can use connectors to define a (generalized) homotopy relation between maps that behaves well under taking connected components.
Let be a connector. Two parallel maps are called I-homotopic, in signs: , if there exists a map with the property that
(In this case, is also called an (-)homotopy between and .)
The following result brings together two ingredients for the equivalence between contractability and connectedness of , namely, the preservation of finite products by and .
Let and be a pair of parallel maps in a weakly cohesive topos with the property that . Then . In particular, implies .
Proof. Since preserves finite products it maps product diagrams to product diagrams whence
but these two maps coincide since by assumption.
Since for an I-homotopy and, by assumption, , , and these maps necessarily coincide since is terminal whence whence as claimed.
For the following the monoid structure of will become important. So let us briefly review the basics:
In a general topos, the Heyting algebra structure endows the subobject classifier with the structure of an internal monoid: The multiplication is given by conjunction
The conjunction is defined as the characteristic map of .
Importantly, the other truth value plays the role of a (multiplicative) zero with respect to this multiplication.
For the following we need
Let be a weakly cohesive topos whose subobject classifier is a connector i.e. . Then the conjunction is an -homotopy from to the constant map .
Proof. We have to show that and . This is more or less clear from the propositional structure of but let us spell out the details diagrammatically:
For the first equation, consider the commutative diagram:
This pullback pasted to the classifying pullback diagram for displays as the characteristic map of which of course is none other than .
For the second, consider the following pullback:
Chasing the arrows around in the second component yields the equation
but this implies since it corresponds to the pullback of and . Hence the pasted
is the classifying pullback of . Since it is easily seen that is also the characteristic map of the claim follows.
In order to show that the connectedness of implies its contractibility we will now lift this -homotopy between and a constant map to a -homotopy between and a constant map .
Let be a weakly cohesive topos with connector . Suppose that is a -homotopy between and a constant map .Then there exists for every object a -homotopy from to a constant endomap .
Proof. We get from as follows
The corresponding terms in the internal language are
respectively. The last term evaluates to i.e. to at and at evaluates to , i.e. the function mapping f to , the function constantly on , whence is a -homotopy from to the constantly map.
By prop. and the preceding the next is immediate:
Let be a weakly cohesive topos whose subobject classifier is a connector. The -homotopy lifts to a -homotopy between and a constant map .
In classical topology, a space with the property that is homotopic to a constant map is called a contractible space. Hence proposition can be viewed as a synthetic internal avatar of the classical fact, that any two parallel maps into a contractible space are homotopic.
Pursuing this analogy, the preceding result shows that in a weakly cohesive topos with connected subobject classifier, the power object of an object is akin to the cone of a topological space in providing a contractible object to embed into. In this perspective, one can think of a sufficiently cohesive topos as being equipped with a generalized cone construction.
Let be a weakly cohesive topos. Then the subobject classifier is connected iff is contractible. In other words, is sufficiently cohesive iff .
Proof. “”: The propositions and imply that for all is a constant map. By observation it then follows that for all which is just the definition of being contractible.
For convenience and summary let us collect all the equivalent formulations of sufficient cohesion in one place:
A weakly cohesive topos is sufficiently cohesive iff satisfies the following equivalent conditions:
The subobject classifier is contractible i.e. for every object . ‘power objects are connected’ or ‘truth is contractible’
The subobject classifier is connected i.e. . ‘truth is connected’
The subobject classifier is a connector. ‘truth is a connector’
has a connector.
Every object embeds into a contractible object. ‘ has enough contractible objects’
Every object embeds into a connected object. ‘ has enough connected objects’
All injective objects are connected.
All injective objects are contractible.
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