nLab
sufficiently cohesive topos

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Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Cohesion

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Backround

Definition

Presentation over a site

Structures in a cohesive (,1)(\infty,1)-topos

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?

Models

Contents

Idea

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.

Terminology

Sufficient cohesion is a relative concept and requires minimally the presence of an essential geometric morphism p:𝒮p:\mathcal{E}\to\mathcal{S}. Here we state it for an adjoint quadruple between toposes

p !p *p *p !:𝒮p_!\dashv p^* \dashv p_* \dashv p^!\;\colon\;\mathcal{S}\to\mathcal{E}

such that p !p^! (and hence p *p^\ast) is fully faithful and p !p_! preserves finite products.

This (and \mathcal{E} in particular) is called a ‘cohesive topos’ (over 𝒮\mathcal{S}) 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 pp furthermore satisfies the Nullstellensatz i.e. the canonical map θ:p *p !\theta:p_\ast\to p_! is an epimorphism. This is the situation explored in Menni (2014a, 2014b).

A pre-cohesive topos that moreover satisfies the continuity principle that p !(X p *(Y))p !(X) Yp_!(X^{p^\ast(Y)})\simeq p_!(X)^{Y} natural in XX\in\mathcal{E}, Y𝒮Y\in\mathcal{S}, 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).

Definitions

Definition

An object XX in a weakly cohesive topos p:𝒮p:\mathcal{E}\to\mathcal{S} is called contractible if for every object YY\in\mathcal{E}: p !(X Y)=1p_!(X^Y)=1.

Remark

In particular, a contractible object is connected: p !(X)=p !(X 1)=1p_!(X)=p_!(X^1)=1. Since exponentiation () Y(-)^Y preserves 11 in any Cartesian closed category, the terminal object is contractible in any weakly cohesive topos.

Definition

A weakly cohesive topos p:𝒮p:\mathcal{E}\to\mathcal{S} is called sufficiently cohesive if the subobject classifier Ω\Omega\in\mathcal{E} is contractible i.e. p !(Ω X)=1p_!(\Omega^X)=1 for every object XX\in\mathcal{E} or, in other words, if all power objects are connected.

Remark

Since in general, every object XX embeds into its power object via the singleton map {}:XΩ X\{\}:X\rightarrowtail\Omega^X it follows that in a sufficiently cohesive topos \mathcal{E} every object embeds into a connected object i.e. \mathcal{E} has enough connected objects.

Furthermore, since in a Cartesian closed category

(X Y) ZX (Y×Z)X (Z×Y)(X Z) Y(X^Y)^Z\simeq X^{(Y\times Z)}\simeq X^{(Z\times Y)}\simeq (X^Z)^Y

one sees that in a sufficiently cohesive topos power objects Ω X\Omega^X are not only connected but even contractible:

p !((Ω X) Z)=p !(Ω X×Z)=1p_!((\Omega^X)^Z)=p_!(\Omega^{X\times Z})=1

and hence it follows that in a sufficiently cohesive topos every object embeds into a contractible object.

Conversely, if every power object Ω X\Omega^X embeds into a connected object then the power objects Ω X\Omega^X 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).

Remark

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.

Properties

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 c:XYc:X\to Y is called constant if cc factors through the terminal object 11:

XcY=X! X1c *Y.X\overset{c}{\to} Y=X\overset{!_X}{\to}1\overset{c_\ast}{\to} Y\quad .
Observation

An object XX is a terminal object iff id Xid_X is constant.

Proof. “\Leftarrow”: By assumption id X=(id X) *! Xid_X=(id_X)_\ast\circ !_X. Since XX has a point (id X) *(id_X)_\ast there exists for every object ZZ at least one map ZXZ\to X. Suppose then that ff is some map ZXZ\to X:

f=id Xf=(id X) *! Xf=(id X) *! Z.f=id_X\circ f=(id_X)_\ast\circ !_X\circ f=(id_X)_\ast\circ !_Z\quad .

But the righthand side does not depend on ff hence it is the only map ZXZ\to X. \qed

Observation

Let XX be an object in a weakly cohesive topos and c:XXc:X\to X be a constant endomap such that p !(c)=p !(id X)p_!(c)=p_!(id_X). Then XX is connected i.e. p !(X)=1p_!(X)=1.

Proof. Since by assumption p !(1)=1p_!(1)=1, p !(c)p_!(c) and, accordingly, p !(id X)=id p !(X)p_!(id_X)=id_{p_!(X)} are constants. \qed

Observation

In a weakly cohesive topos, retracts of connected objects are connected themselves.

Proof. Let XX be a retract of YY with p !(Y)=1p_!(Y)=1. Then id Xid_X factors as XYXX\rightarrowtail Y\to X and applying p !p_! shows that id p !(X)id_{p_!(X)} factors through the terminal object p !(Y)p_!(Y). \qed

Since in general, injective objects II are retracts of the objects XX that they embed into because such inclusions IXI\rightarrowtail X factor through id Iid_I 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:

Proposition

A weakly cohesive topos is sufficiently cohesive iff all injective objects are connected. \qed

We can replace ‘connected’ with ‘contractible’ in this proposition as the following propositions show.

Proposition

Let \mathcal{E} be a topos. Then all injective objects are connected iff all injective objects are contractible.

Proof. “\Rightarrow”: by the following lemma. \qed

Lemma

Let II be an injective object in a topos \mathcal{E}. Then I XI^X is injective for any object XX , or in other words, exponentials ( ) X(_-)^X preserve injective objects.

Proof. (cf. at injective object) We have to show that for every mono m:SBm:S\rightarrowtail B the induced function m *:Hom(B,I X)Hom(S,I X)m^\ast:Hom(B,I^X)\to Hom(S,I^X) is onto. From the exponential adjunction one has the following commutative diagram

Hom(B,I X) m * Hom(S,I X) Hom(X×B,I) (m×id X) * Hom(X×S,I) \array{ Hom(B,I^X) &\overset{m^\ast}{\rightarrow} & Hom(S,I^X) \\ \simeq \parallel& &\parallel \simeq \\ Hom(X\times B,I) &\underset{(m\times id_X)^\ast}{\rightarrow} &Hom(X\times S,I) }

whence it suffices to show that m×id X:S×XB×Xm\times id_X:S\times X\to B\times X is a mono since II is injective. But this is easy to see:

(m×id X)f 1,f 2 =(m×id X)g 1,g 2 mf 1,f 2 =mg 1,g 2 \begin{aligned} (m\times id_X)\circ \langle f_1,f_2\rangle &= (m\times id_X)\circ\langle g_1,g_2\rangle \\ \langle m\circ f_1, f_2\rangle &=\langle m\circ g_1,g_2\rangle \end{aligned}

which is equivalent to mf 1=mg 1\m\circ f_1=m\circ g_1 and f 2=g 2f_2=g_2. \qed

Summing up we get:

Proposition

A weakly cohesive topos \mathcal{E} is sufficiently cohesive iff all injective objects are contractible. \qed

Cohesively connected truth

In a sufficiently cohesive topos the subobject classifier is obviously connected since Ω=Ω 1\Omega=\Omega^1. It is the aim of this section to prove that in a weakly cohesive topos the converse holds as well: Ω\Omega is connected iff Ω\Omega 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 p !p_!:

Example

The Sierpinski topos Set Set^\to is weakly cohesive over SetSet since there exists a string of adjoint functors LΠΔΓB:SetSet L\dashv\Pi\dashv\Delta\dashv\Gamma\dashv B: Set\to Set^\to with

  • L(Z)=ZL(Z)=\emptyset \to Z

  • Π(XY)=Y\Pi(X\to Y) = Y

  • Δ(Z)=ZidZ\Delta(Z)=Z\overset{id}{\to} Z

  • Γ(XY)=X\Gamma (X\to Y) = X

  • B(Z)=Z1B(Z)=Z\to 1.

The Nullstellensatz fails as does the continuity principle. As a right adjoint Π\Pi preserves all limits and the terminal object in particular whence 11 is connected in Set Set^\to. Since the underlying category \to satisfies the Ore condition trivially, it follows then from a general result1 of Lawvere that Ω\Omega is not connected and, accordingly, that the Sierpinski topos is not sufficiently cohesive!

Example

On the other hand, consider the topos of directed graphs Set Set^\rightrightarrows: it lacks the right adjoint p !p^! to the section functor p *p_* but has a connected components functor p !p_! and the subobject classifier for directed graphs Ω\Omega is connected.

Although p !p_! does not preserve finite products in general, it nevertheless preserves finite products with the subobject classifier Ω\Omega since Ω\Omega 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 p !p_! and nothing hinges on (the existence of) the right adjoint p !p^! at all, the conclusions from hold as well: the subobject classifier of Set Set^\rightrightarrows is not only connected but contractible! In other words, despite not being sufficiently cohesive Set Set^\rightrightarrows nevertheless has enough contractible objects.

Recall that in any topos, the subobject classifier Ω\Omega has two points true,false\mathsf{true},\mathsf{false} fitting into the following pullback diagram (which is an equaliser diagram as well) due to the classifying property of Ω\Omega for the monomorphism 010\to 1:

0 1 true 1 false Ω \array{ 0 &\to & 1 \\ \downarrow & &\downarrow\mathsf{true} \\ 1 &\underset{\mathsf{false}}{\to} &\Omega }

In a sufficiently cohesive topos Ω\Omega is furthermore connected whence together with its Heyting algebra structure we can view it as a generalized (non linear) “interval” object:

Definition

An object TT in a weakly cohesive topos is called a (cohesive) connector if p !(T)=1p_!(T)=1 and TT has two points t 0,t 1:1Tt_0,t_1:1\to T with empty equaliser: 0e1t 1t 0T0\overset{e}{\to} 1\overset{t_0}{\underset{t_1}{\rightrightarrows}} T.

In a topos with a connected subobject classifier Ω\Omega itself is a connector. Conversely the existence of a connector implies the connectedness of Ω\Omega:

Proposition

Let \mathcal{E} be weakly cohesive topos. \mathcal{E} has a connector TT iff p !(Ω)=1p_!(\Omega)=1.

Proof. “\Rightarrow”: Let 1t 1t 0T1\overset{t_0}{\underset{t_1}{\rightrightarrows}} T be a connector. Then t 1:1Tt_1:1\to T is a subobject with characteristic map χ 1:TΩ\chi_1:T\to\Omega. Consider the two composites χ 1t i\chi_1\circ t_i , i=0,1i=0,1:

For i=1i=1 this simply yields true\mathsf{true} by the definition of χ 1\chi_1.

For i=0i=0 we claim that χ 1t 0=false\chi_1\circ t_0=\mathsf{false} since we have the following diagram:

0 1 1 t 1 true 1 t 0 T χ 1 Ω \array{ 0 &\to & 1 &\to & 1 \\ \downarrow & &t_1\downarrow& & \downarrow \mathsf{true} \\ 1&\underset{t_0}{\to} &T&\underset{\chi_1}{\to}&\Omega }

Here the left square is a pullback since t o,t 1t_o,t_1 have empty equaliser by assumption. The right square is a pullback since it classifies t 1t_1 whence the outer square is a pullback too. Therefore χ 1t 0\chi_1\circ t_0 classifies 010\to 1 which is exactly the definition of false\mathsf{false}.

Since p !(T)p_!(T) is terminal p !(t 0)=p !(t 1)p_!(t_0)=p_!(t_1). Whence

p !(true)=p !(χ 1t 1)=p !(χ 1)p !(t 1)=p !(χ 1)p !(t 0)=p !(χ 1t 0)=p !(false)p_!(\mathsf{true})=p_!(\chi_1\circ t_1)=p_!(\chi_1)\circ p_!(t_1)=p_!(\chi_1)\circ p_!(t_0)=p_!(\chi_1\circ t_0)=p_!(\mathsf{false})\quad

This says that true\mathsf{true} and false\mathsf{false} are in the same connected component but a lattice whose top and bottom elements are in the same component is necessarily connected. \qed

One can use connectors to define a (generalized) homotopy relation between maps that behaves well under taking connected components.

Definition

Let II be a connector. Two parallel maps f,g:ABf,g:A\to B are called I-homotopic, in signs: f Igf\sim_I g, if there exists a map h:A×IBh:A\times I\to B with the property that

f=hid A,t 0! Aandg=hid A,t 1! A.f=h\circ\langle id_A, t_0\circ !_A\rangle\quad and\quad g=h\circ\langle id_A, t_1\circ !_A\rangle\quad.

(In this case, hh is also called an (II-)homotopy between ff and gg.)

The following result brings together two ingredients for the equivalence between contractability and connectedness of Ω\Omega, namely, the preservation of finite products by p !p_! and p !(Ω)=1p_!(\Omega)=1.

Proposition

Let f=hi,k 1f=h\circ\langle i, k_1\rangle and g=hi,k 2g=h\circ\langle i, k_2\rangle be a pair of parallel maps in a weakly cohesive topos with the property that p !(k 1)=p !(k 2)p_!(k_1)=p_!(k_2). Then p !(f)=p !(g)p_!(f)=p_!(g). In particular, f Igf\sim_I g implies p !(f)=p !(g)p_!(f)=p_!(g).

Proof. Since p !p_! preserves finite products it maps product diagrams to product diagrams whence

p !(hi,k j)=p !(h)p !(i,k j)=p !(h)p !(i),p !(k j),j{1,2}p_!(h\circ\langle i, k_j\rangle)=p_!(h)\circ p_!(\langle i, k_j\rangle)=p_!(h)\circ \langle p_!( i), p_!(k_j)\rangle\quad,\quad j\in\{1,2\}

but these two maps coincide since p !(k 1)=p !(k 2)p_!(k_1)=p_!(k_2) by assumption.

Since for an I-homotopy k j=t j! A:AIk_j=t_j\circ !_A:A\to I and, p !(I)=1p_!(I)=1 by assumption, p !(k j):p !(A)1p_!(k_j):p_!(A)\to 1, j{1,2}j\in\{1,2\}, and these maps necessarily coincide since 11 is terminal whence p !(t 0! A)=p !(t 1! A)p_!(t_0\circ !_A)=p_!(t_1\circ !_A) whence p !(f)=p !(g)p_!(f)=p_!(g) as claimed. \qed

For the following the monoid structure of Ω\Omega 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

Ω×ΩΩ, and the unit by 1trueΩ.\Omega\times\Omega\overset{\wedge}{\to}\Omega\quad , \text{ and the unit by }\quad 1\overset{\mathsf{true}}{\to}\Omega\quad .

The conjunction \wedge is defined as the characteristic map of 1true,trueΩ×Ω1\xrightarrow{\langle\mathsf{true},\mathsf{true}\rangle}\Omega\times\Omega.

Importantly, the other truth value 1falseΩ1\xrightarrow{\mathsf{false}}\Omega plays the role of a (multiplicative) zero with respect to this multiplication.

For the following we need

Proposition

Let \mathcal{E} be a weakly cohesive topos whose subobject classifier is a connector i.e. p !(Ω)=1p_!(\Omega)=1. Then the conjunction :Ω×ΩΩ\wedge :\Omega\times\Omega{\to}\Omega is an Ω\Omega-homotopy from id Ωid_{\Omega} to the constant map false! Ω\mathsf{false}\circ !_\Omega.

Proof. We have to show that id Ω,true! Ω=id Ω\wedge\circ\langle id_\Omega, \mathsf{true}\circ !_\Omega\rangle=id_\Omega and id Ω,false! Ω=false! Ω\wedge\circ\langle id_\Omega, \mathsf{false}\circ !_\Omega\rangle=\mathsf{false}\circ !_\Omega. This is more or less clear from the propositional structure of Ω\Omega but let us spell out the details diagrammatically:

For the first equation, consider the commutative diagram:

1 1 true true,true Ω id Ω,true! Ω Ω×Ω \array{ 1 &\to & 1 \\ {}_\mathsf{true}\downarrow & &\downarrow_{\langle\mathsf{true},\mathsf{true}\rangle} \\ \Omega &\xrightarrow{\langle id_\Omega,\mathsf{true}\circ !_\Omega\rangle}&\Omega\times\Omega }

This pullback pasted to the classifying pullback diagram for true,true\langle\mathsf{true},\mathsf{true}\rangle displays id Ω,true! Ω\wedge\circ\langle id_\Omega, \mathsf{true}\circ !_\Omega\rangle as the characteristic map of true\mathsf{true} which of course is none other than id Ωid_\Omega.

For the second, consider the following pullback:

X ! X 1 true! X true,true Ω id Ω,false! Ω Ω×Ω \array{ X &\xrightarrow{!_X} & 1 \\ {}_{\mathsf{true}\circ !_X}\downarrow & &\downarrow_{\langle\mathsf{true},\mathsf{true}\rangle} \\ \Omega &\xrightarrow{\langle id_\Omega,\mathsf{false}\circ !_\Omega\rangle }&\Omega\times\Omega }

Chasing the arrows around in the second component yields the equation

true! X=false! Ωtrue! X=false! X\mathsf{true}\circ !_X=\mathsf{false}\circ !_\Omega\circ\mathsf{true}\circ !_X=\mathsf{false}\circ !_X

but this implies X=0X=0 since it corresponds to the pullback of true\mathsf{true} and false\mathsf{false}. Hence the pasted

0 ! 0 1 1 true,true true Ω id Ω,false! Ω Ω×Ω Ω \array{ 0 &\xrightarrow{!_0} & 1 &\to & 1 \\ \downarrow & &\downarrow_{\langle\mathsf{true},\mathsf{true}\rangle}& &\downarrow _\mathsf{true} \\ \Omega &\xrightarrow{\langle id_\Omega,\mathsf{false}\circ !_\Omega\rangle }&\Omega\times\Omega&\xrightarrow{\wedge}&\Omega }

is the classifying pullback of 0Ω0\rightarrowtail\Omega. Since it is easily seen that false! Ω\mathsf{false}\circ !_\Omega is also the characteristic map of 0Ω0\rightarrowtail\Omega the claim follows. \qed

In order to show that the connectedness of Ω\Omega implies its contractibility we will now lift this Ω\Omega-homotopy between id Ωid_\Omega and a constant map ΩΩ\Omega\to\Omega to a Ω\Omega-homotopy between id Ω Xid_{\Omega^X} and a constant map Ω XΩ X\Omega^X\to\Omega^X.

Proposition

Let \mathcal{E} be a weakly cohesive topos with connector 1t 1t 0T1\overset{t_0}{\underset{t_1}{\rightrightarrows}} T. Suppose that m:X×TXm:X\times T\to X is a TT-homotopy between id Xid_X and a constant map c:XXc:X\to X\,.Then there exists for every object YY a TT-homotopy μ:X Y×TX Y\mu:X^Y\times T\to X^Y from id X Yid_{X^Y} to a constant endomap X YX YX^Y\to X^Y.

Proof. We get μ\mu from X×TmXX\times T \overset{m}{\to} X as follows

X X Tby transposal X Y (X T) Yby application of the endofunctor (-) Y X Y (X Y) Tby using rules for powers X Y×T μX Yby reversing transposal . \begin{aligned} X& \to X^{T}\quad\text{by transposal} \\ X^Y&\to (X^{T})^{Y}\quad\text{by application of the endofunctor (-)}^Y \\ X^Y &\to (X^Y)^{T}\quad\text{by using rules for powers} \\ X^Y\times T&\overset{\mu}{\to}X^Y\quad \text{by reversing transposal .} \end{aligned}

The corresponding terms in the internal language are

m=λ(x,t).m(x,t) λxλt.m(x,t) λfλyλt.m(f(y),t) λfλtλy.m(f(y),t) μ=λ(f,t)λy.m(f(y),t) \array{ m=\lambda (x,t).m(x,t) \\ \lambda x\lambda t.m(x,t) \\ \lambda f\lambda y\lambda t. m(f(y),t) \\ \lambda f\lambda t \lambda y. m(f(y),t) \\ \mu =\lambda (f,t)\lambda y. m(f(y),t) }

respectively. The last term evaluates to λfλy.f(y)\lambda f\lambda y.f(y) i.e. to id X Yid_{X^Y} at t 0t_0 and at t 1t_1 evaluates to λfλy.c\lambda f\lambda y.c , i.e. the function mapping f to c Yc^Y , the function constantly cc on YY, whence μ\mu is a TT-homotopy from id X Yid_{X^Y} to the constantly c Yc^Y map. \qed

By prop. and the preceding the next is immediate:

Corollary

Let \mathcal{E} be a weakly cohesive topos whose subobject classifier is a connector. The Ω\Omega-homotopy :Ω×ΩΩ\wedge :\Omega\times\Omega{\to}\Omega lifts to a Ω\Omega-homotopy between id Ω Xid_{\Omega^X} and a constant map Ω XΩ X\Omega^X\to\Omega^X. \qed

Remark

In classical topology, a space XX with the property that id Xid_X 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 XX are homotopic.

Pursuing this analogy, the preceding result shows that in a weakly cohesive topos with connected subobject classifier, the power object Ω X\Omega^X of an object XX is akin to the cone CX\mathbf{C}X of a topological space XX 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.

Theorem

Let \mathcal{E} be a weakly cohesive topos. Then the subobject classifier Ω\Omega is connected iff Ω\Omega is contractible. In other words, \mathcal{E} is sufficiently cohesive iff p !(Ω)=1p_!(\Omega)=1.

Proof. “\Rightarrow”: The propositions and imply that for all XX p !(id Ω X)=id p !(Ω X)p_!(id_{\Omega^X})=id_{p_!(\Omega^X)} is a constant map. By observation it then follows that p !(Ω X)=1p_!(\Omega^X)=1 for all XX which is just the definition of Ω\Omega being contractible. \qed

For convenience and summary let us collect all the equivalent formulations of sufficient cohesion in one place:

Theorem (Lawvere)

A weakly cohesive topos \mathcal{E} is sufficiently cohesive iff \mathcal{E} satisfies the following equivalent conditions:

  • The subobject classifier Ω\Omega\in\mathcal{E} is contractible i.e. p !(Ω X)=1p_!(\Omega^X)=1 for every object XX\in\mathcal{E}. ‘power objects are connected’ or ‘truth is contractible

  • The subobject classifier Ω\Omega\in\mathcal{E} is connected i.e. p !(Ω)=1p_!(\Omega)=1. ‘truth is connected

  • The subobject classifier Ω\Omega\in\mathcal{E} is a connector. ‘truth is a connector

  • \mathcal{E} has a connector.

  • Every object XX\in\mathcal{E} embeds into a contractible object. ‘\mathcal{E} has enough contractible objects

  • Every object XX\in\mathcal{E} embeds into a connected object. ‘\mathcal{E} has enough connected objects

  • All injective objects are connected.

  • All injective objects are contractible. \qed

References

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  1. (Theorem 12.2.3 in La Palme Reyes et al. (2004, p.221)). Of course, this can also easily be proved directly or read off the concrete objects and properties worked out in La Palme Reyes et al. (2004) where the Sierpinski topos is called the category of bouquets.

Last revised on January 7, 2019 at 16:09:01. See the history of this page for a list of all contributions to it.