conic section

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## Definition

Let $k$ be a [[field]]. A _conic section_ over $k$ is the [[zero locus|zero set]] of a degree 2 [[polynomial]] $p(x, y) \in k[x, y]$ in the [[affine space|affine plane]] $\mathbb{A}^2(k)$, or better yet the zero set of a [[homogeneous polynomial]] $p(x, y, z)$ of degree 2 in the [[projective space|projective plane]] $\mathbb{P}^2(k)$.

## Over the real numbers

In the classical case $k = \mathbb{R}$ of [[real numbers]], conic sections may be pictured in terms of [[intersections]] of a standard [[cone]] $x^2 + y^2 - z^2 = 0$ in affine 3-space with various affine hyperplanes (hence the name, "conic section"). In this picture, _nonsingular_ conic sections are classified (up to [[automorphisms]] of the affine plane) by the sign of the [[discriminant]] of $p$. In other words, if we write $p(x, y) = a x^2 + b x y + c y^2 + \lambda(x, y)$ where $deg(\lambda) = 1$ and put $D = b^2 - 4 a c$, then an isomorphism class is one of three types: ellipses (when $D \lt 0$), parabolas ($D = 0$), and hyperbolas ($D \gt 0$). Of course when we admit possibly singular conic sections, we get further isomorphism classes involving some degree of degeneracy (e.g., two lines, a double line, etc.).

The distinctions between [[ellipse]], [[parabola]], and [[hyperbola]] are artifacts of affine geometry: if we instead consider conic sections as projective [[algebraic variety|subvarieties]] of $\mathbb{P}^2(\mathbb{R})$, then considered up to projective transformations (automorphisms of the projective plane), these distinctions evaporate and there is really only one kind of nonsingular conic section. Put differently: if we fix a representation $\mathbb{P}^2(\mathbb{R}) = \mathbb{A}^2(\mathbb{R}) \sqcup L$ where $L$ is a chosen "line at infinity", then in the original classification up to affine transformations, i.e., the subgroup of projective transformations which take $L$ to itself, ellipses are those conic sections which do not intersect $L$, parabolas are those which intersect $L$ in a double point, and hyperbolas are those which intersect $L$ in two points. By enlarging to the group of all projective transformations, we can move $L$ to a line which *does* intersect an "ellipse" in two points, making it a "hyperbola" with respect to the new coordinate system, etc.

## Stereographic projection

Considered in terms of projective geometry, all *pointed*[^1] nonsingular conic sections $C \subset \mathbb{P}^2(k)$ are isomorphic and can be identified explicitly with a projective line $\mathbb{P}^1(k)$ by means of a [[stereographic projection]].

Geometrically, if $p$ is the chosen basepoint of $C$ and $L \subset \mathbb{P}^2(k)$ is a line not incident to $p$, then for any other point $q$ of $C$ the unique line $L(p, q)$ incident to $p$ and $q$ intersects $L$ in exactly one point, denoted $\phi(q)$. (Here $\phi(p)$ to be the intersection of the _tangent_ to $p$ at $C$ with $L$; this can be considered the basepoint of $L$.) In the opposite direction, to each point $x$ of $L$, the line $L(p, x)$ intersects $C$ in $p$ and (since a quadratic with one root will also have another root) another point $q$ (which might be the same as $p$; this happens precisely when $L(p, x)$ is the tangent at $p$); this gives the inverse $\phi^{-1}(x) = q$. In this way we obtain an isomorphism $\phi: C \to L$ of subvarieties.

Working over an [[algebraically closed field]] $k$, where every nonsingular conic $C$ has a point, we may conclude that $C$ is isomorphic (as a projective variety) to $\mathbb{P}^1(k)$. Hence $C$ is a curve of [[genus]] $0$.

[^1]: We took care to say *pointed*: note that depending on the field $k$, there might not even be a solution point on the conic $C$, even if $C$ is by all rights nonsingular. For example, consider $p(x, y) = x^2 + y^2 + 1$ over $\mathbb{R}$. Of course we can relax again if $k$ is algebraically closed.

## Projective duality

Working over an [[algebraically closed field]] $k$ (let us assume the [[characteristic]] is not $2$), all nondegenerate [[quadratic forms]] on a vector space $V$ are isomorphic and we may fix one as standard. For example, for $V = k^3$, we may fix attention on the quadratic form $q(x, y, z) = x^2 + y^2 + z^2$, which determines a conic section $C \subset \mathbb{P}^2(k)$ and an accompanying nondegenerate symmetric [[bilinear form]] $\langle-, -\rangle_q: V \times V \to k$.

Projective duality relative to $C$ is the projectivization of linear duality with respect to $\langle -, - \rangle$, which takes a linear subspace $L$ to its [[orthogonality|orthogonal dual]] $L^\perp = \{v \in V: (\forall w \in L)\; \langle v, w \rangle = 0\}$. We note that the orthogonal dual is an involution that takes joins of subspaces to meets and vice-versa. This construction descends through the quotient $k^3 \backslash \{0\} \to \mathbb{P}^2(k)$ to give an operation that takes points in $\mathbb{P}^2(k)$ (lines in $k^3$) to lines in $\mathbb{P}^2(k)$ (hyperplanes in $k^3$), and vice-versa, and moreover takes a join of two distinct points (the line incident to them) to the meet of their dual lines (the point of their intersection).

This duality may be visualized thus: given a nondegenerate conic $C$ and a point $p$ off of $C$, draw the two lines incident to $p$ that are tangent to $C$, and pass to the line incident to the tangent points. (This is easier to visualize by imagining $k = \mathbb{R}$ and considering a point $p$ exterior to say an ellipse $C$.) This defines the line that is projectively dual to $p$ (dual with respect to the conic $C$); if $p$ is *on* $C$, then the same procedure works by considering the two tangent points as infinitesimally close to $C$, so that the line between them is the tangent line at $p$: the projective dual of $p$ on $C$ is its tangent line (this is the case where the line in $k^3$ corresponding to $p$ is isotropic with respect to the bilinear form).

The entire procedure can be reversed and gives an anti-[[involution]] on the [[poset]] of flats of $\mathbb{P}^2(k)$, interchanging points and lines and interchanging meets and joins.

More generally, projective duality can be described in terms of an orthogonality map $\mathbb{P}(V) \to \mathbb{P}(V^\ast)$, where $V^\ast$ is the linear dual of $V$, which maps a subvariety to a corresponding dual "envelope" subvariety. This is explored in [GKZ](#GKZ). The role of the conic section (or generally a conic hypersurface) is simply to set up an explicit self-duality $\mathbb{P}(V^\ast) \cong \mathbb{P}(V)$.

## Pascal's "mystic hexagon"

(...)

## References

* I.M. Gel'fand, M. Kapranov, A. Zelevinsky, _Discriminants, Resultants, and Multidimensional Determinants_, Birkhäuser 2008 (paperback edition).
 {#GKZ}

[[!redirects conic sections]]
[[!redirects conic]]
[[!redirects conics]]


conical limit

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## Idea

A _conical limit_ is an ordinary [[limit]] as opposed to a more general [[weighted limit]].

When the [[base of enrichment]] is $Set$, every weighted limit can be expressed as a conical limit. However, it is not true that completeness under a class of weights can always be expressed as completeness under a class of diagrams. For instance, every [[power]] $A \times A$ is a [[product]], but the class of categories admitting powers cannot be expressed as the class of categories admitting $D$-indexed limits for some class of categories $D$.

## References

* Michael Albert, and [[Max Kelly]]. _The closure of a class of colimits_, Journal of Pure and Applied Algebra 51.1-2 (1988): 1-17. ([doi](https://doi.org/10.1016/0022-4049(88)90073-4))

[[!redirects conical limits]]

[[!redirects conical colimit]]
[[!redirects conical colimits]]


conical space

# Conical spaces
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## Idea

A **conical space** is a [[set]] equipped with a notion of taking [[real number|real]]-[[linear combinations]] with [[nonnegative number|nonnegative]] [[coefficients]] of its elements.

Conical spaces appear prominently in [[microlocal analysis]], where [[wave front sets]] of [[distributions]] are [[disjoint unions]] of (open-)conical spaces; also in [[Lorentzian geometry]] in the guise of open [[past]]/[[future]] [[causal cones]] and [[light cones]].

## Definition

### Conical sets

A _conical set_ is a [[module]] of the [[rig]] $\mathbb{R}^+ = {[{0,\infty}[}$: that is, the rig of [[nonnegative number|nonnegative]] [[real numbers]], with ordinary [[addition]] and [[multiplication]] as the rig operations.

Similarly one may consider "open conical sets" which are [[modules]] over the [[rig]] of [[positive number|positive]] [[real numbers]].

### Extended conical sets

An _extended conical space_ is a [[module]] over the [[rig]] $\bar{\mathbb{R}}^+ = [0,\infty]$ of [[nonnegative number|nonnegative]] [[extended real numbers]], with $0 \cdot \infty \coloneqq 0$.

For purposes of [[constructive mathematics]], one should take $\bar{\mathbb{R}}^+$ to be the space of nonnegative [[lower real numbers]].

Since we have a [[rig]] [[homomorphism]] $\mathbb{R}^+ \hookrightarrow \bar{\mathbb{R}}^+$, every extended conical space has an underlying conical space, and any conical space [[free construction|freely generates]] an extended conical space.  However, there is no direct relationship between [[vector spaces]] and extended conical spaces.

## Examples

### General

The  [[extended positive cone]] of any [[ordered group|ordered]] real [[vector space]] is an extended conical space.

* The [[positive cone]] of any [[ordered group|ordered]] real [[vector space]] is a conical space.

### Relation to vector spaces

Any [[rig]] [[homomorphism]] $A \to B$ gives a '[[restriction of scalars]]' [[functor]] $R\colon B Mod \to A Mod$ and '[[extension of scalars]]' [[functor]] $L \colon A Mod \to B Mod$.  In particular,  the rig homomorphism

$$
  \mathbb{R}^+ \hookrightarrow \mathbb{R}
$$

produces a pair of [[adjoint functors]] between the [[category of modules]] over $\mathbb{R}$ (that is,  [[real vector spaces]]) and the [[category of modules]] over $\mathbb{R}^+$ (that is, conical spaces).

In simple English: any real vector space has an [[forgetful functor|underlying]] conical space, and any conical space [[free construction|freely generates]] a real vector space.

## Related concepts

* [[affine space]]

* [[projective space]]

* [[conical singularity]]

## References

See also

* Wikipedia, _[Conical combination](https://en.wikipedia.org/wiki/Conical_combination)_

[[!redirects conical spaces]]

[[!redirects conical set]]
[[!redirects conical sets]]

[[!redirects extended conical space]]
[[!redirects extended conical spaces]]


conifold > history

see _[[conifold]]_


conifold transition

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## Idea

(...)

## Related concepts

* [[conical singularity]]

* [[flop transition]]

* [[ADE singularity]]

* [[Sasakian manifold]]

## References

Discussion in the context of [[gauged supergravity]] includes

* J. D. Edelstein, A. Paredes, A. V. Ramallo, _Singularity Resolution in Gauged Supergravity and Conifold Unification_, Phys. Lett. B554 (2003) 197-206 ([arXiv:hep-th/0212139](http://arxiv.org/abs/hep-th/0212139))

Discussion in [[M-theory]]

* {#Acharya00} [[Bobby Acharya]], _On Realising $N=1$ Super Yang-Mills in M theory_ ([arXiv:hep-th/0011089](http://arxiv.org/abs/hep-th/0011089))

* {#AtiyahMaldacenaVafa00} [[Michael Atiyah]], [[Juan Maldacena]], [[Cumrun Vafa]], _An M-theory Flop as a Large N Duality_, J. Math. Phys. 42:3209-3220, 2001 ([arXiv:hep-th/0011256](http://arxiv.org/abs/hep-th/0011256))

* {#Witten01} [[Edward Witten]], _Anomaly Cancellation On Manifolds Of $G_2$ Holonomy_ ([arXiv:hep-th/0108165](http://arxiv.org/abs/hep-th/0108165))

* [[Sergei Gukov]], [[James Sparks]], [[David Tong]], *Conifold Transitions and Five-Brane Condensation in M-Theory on $Spin(7)$ Manifolds*, Class. Quant. Grav. **20** (2003) 665-706 [[arXiv:hep-th/0207244](https://arxiv.org/abs/hep-th/0207244)]

* [[Kenneth Intriligator]], Hans Jockers, [[Peter Mayr]], [[David Morrison]], M. Ronen Plesser, _Conifold Transitions in M-theory on Calabi-Yau Fourfolds with Background Fluxes_, Adv. Theor. Math. Phys. 17 (2013) 601-699 ([arXiv:1203.6662](http://arxiv.org/abs/1203.6662))

Review includes

* {#AcharyaGukov04} [[Bobby Acharya]], [[Sergei Gukov]], _M theory and Singularities of Exceptional Holonomy Manifolds_, Phys. Rept. 392:121-189, 2004 ([arXiv:hep-th/0409191](http://arxiv.org/abs/hep-th/0409191))

[[!redirects conifold transitions]]

[[!redirects conifold]]
[[!redirects conifolds]]

[[!redirects conifold tansition]]


conjecture

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## Idea

In [[mathematics]], a *conjecture* is a [[proposition]] which is *expected* to be [[true]], hence expected to have a [[proof]], but for which no proof is (currently) known.

Hence being a conjecture is a sociological aspect of a proposition, not a mathematical aspect: Once a proof (or else a [[counterexample]]) is found, the conjecture ceases to be a conjecture and instead becomes a [[theorem]].

It happens that conjectures remain unproven while being perceived as trustworthy enough that further [[theorems]] are proven *assuming* the conjectures -- in this case the conjecture plays the role of a [[hypothesis]] in the sense of [[formal logic]].

For example, the "[[standard conjectures]]" in [[algebraic geometry]] serve as hypotheses in a wealth of [[theorems]] which are all proven (only) "assuming the standard conjectures" (cf. e.g. [arXiv:9804123](https://arxiv.org/abs/math/9804123)).

In other cases the term "[[hypothesis]]" is used synonymously with "conjecture" -- e.g. for the *[[homotopy hypothesis]]* (key cases of which have long become [[theorems]]) or the *[[cobordism hypothesis]]* (on which a proof has famously been claimed but not universally accepted).

## Examples

* [[Weil conjecture]]

* [[standard conjectures on algebraic cycles]]

* [[Beilinson conjecture]]

* [[Mumford conjecture]]

* [[geometric Langlands correspondence]]-conjecture

* [[section conjecture]]

* [[Sen's conjecture]]

## Related concepts

[[!include mathematical statements --- contents]]

## References

* Wikipedia, _[Conjecture](http://en.wikipedia.org/wiki/Conjecture)_

[[!redirects conjectures]]


conjugacy class

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## Definition

For $G$ a [[group]] and $g \in G$ an element, the _conjugacy class_ of $g$ is  the [[orbit]] of $g$ under the [[adjoint action]] of $G$ on itself, hence the subset $\{ h g h^{-1} | h \in G \} \subset G$ of all elements in $G$ obtained from $g$ by [[conjugation]] with another group element.

Similarly there is the conjugacy classes of larger [[subsets]] of a group, and in particular the _[[conjugacy classes of subgroups]]_.

## Examples

* The conjugacy class of the neutral element consists of precisely the neutral element itself.
* In an abelian group, conjugacy classes are singletons, one for each element of the group.

## Properties

### For discrete groups

For any [[finite group]] the number of its conjugacy classes is equal to the number of its [[irreducible representations]]. For finite [[groups of Lie type]] this result can be strengthened to show that, in heuristic terms, there is a canonical way to match conjugacy classes of a group $G$ to the irreducible representations of its dual group $G^{\ast}$. The dual group is defined in terms of the [[root datum]] of $G$ in Deligne-Lusztig theory.

Taking [[symmetric groups]] as [[general linear groups]] over the [[field with one element]], we see they are self-dual, and so there is a correspondence between their conjugacy classes and their irreducible representations. This correspondence is encoded by [[Young diagrams]].

### For Lie groups

For $G$ a connected [[compact Lie group]], then its conjugacy classes are in bijection with the [[quotient]] $T/W(G,T)$ of its [[maximal torus]] by the action of the [[Weyl group]]. See at _[maximal torus -- Properties](maximal+torus#Properties)_.

The conjugacy classes of a [[Lie group]] with binary invariant pairing are the leaves of a [[Dirac structure]] on the Lie group, the _[[Cartan-Dirac structure]]_.

Regarding the Lie group as the [[target space]] of the [[WZW model]], the conjugacy classes correspond to the [[D-branes]] of the model.

## Related concepts

* [[coset space]]

* [[character]]

* [[conjugacy class of subgroups]]

[[!redirects conjugacy classes]]


conjugate subgroup

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## Definition

For $G$ a [[group]] and $H_1, H_2 \hookrightarrow$ two [[subgroups]], one says that they are _conjugate subgroups_ if there [[existential quantifier|exists]] an [[element]] $g \in G$ such that the [[conjugation action]] by $g$ takes $H_1 \to H_1$:

$$
  H_1 \sim_{conj} H_2
  \;\;\;
  \coloneqq
  \;\;\;
  H_2 = Ad_g(H_1) = g H_1 g^{-1}
  \,.
$$

The [[equivalence classes]] under this [[relation]] are hence called the _[[conjugacy classes]] of subgroups_ of $G$. These play a key role in much of [[group theory]] and [[representation theory]], for instance as parameters for [[group characters]].

## Related concepts

* [[conjugacy class]]

* [[subgroup lattice]]

* [[HNN-extension]]

## References

See also

* Wikipedia, _[Conjugacy of subgroups and general subsets](https://en.wikipedia.org/wiki/Conjugacy_class#Conjugacy_of_subgroups_and_general_subsets)_

[[!redirects conjugate subgroups]]

[[!redirects conjugacy class of subgroups]]
[[!redirects conjugacy classes of subgroups]]


conjugate transpose matrix

If $A = (a_{i j})$ is a [[matrix]] with [[coefficients]] in a [[star algebra]] (such as the [[complex numbers]] under [[complex conjugation]]), then its _conjugate transpose_ $A^\dagger$ is the matrix $A^\dagger \coloneqq (a^\ast_{j i})$, hence the composite of passing to the [[transpose matrix]] and applying the star-operation

$$
  A^\dagger \coloneqq \left(A^t\right)^\ast = \left(A^\ast\right)^t
$$

Identifying matrices with [[linear maps]] and with respect to the standard [[inner product]] this operation represents passing to the [[adjoint operator]]. Therefore one speaks also of _adjoint matrices_.

## Related concepts

* [[transpose matrix]]

* [[hermitian matrix]]

* [[bilinear form]]

[[!redirects conjugate transpose matrices]]

[[!redirects adjoint matrix]]
[[!redirects adjoint matrices]]


conjugation action

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## Idea

For $G$ a [[group]] and $V_1$ and $V_2$ two objects equipped with $G$-[[action]], the **conjugation action** on morphisms $f : V_1 \to V_2$ (not necessarily respecting the $G$-action) is for $g \in G$

$$
  f \mapsto g \circ f \circ g^{-1}
  \,.
$$

The [[invariants]] of the conjugation action are the $G$-action [[homomorphism]].

In the case that the $G$-action on $V_2$ is trivial, this is the _precomposition action_ and in the case that the action on $V_1$ is trivial this is the _postcomposition action_.

In [[matrix calculus]] conjugation actions are also known as _[[similarity transformations]]_.

## Definition
 {#Definition}

+-- {: .num_defn #ConjugationActionForDiscrete1Groups}
###### Definition

Given a [[discrete group]] $G$ and two $G$-actions $\rho_1$ and $\rho_2$ on [[sets]] $S_1$ and $S_2$, respectively, then the [[function set]] $[S_1, S_2]$ is naturally equipped with the [[conjugation action]]

$$
  Ad \;\colon \;  [S_1, S_2] \times G \longrightarrow [S_1,S_2]
$$

which takes $((S_1 \stackrel{f}{\to} S_2), g)$ to

$$
  \rho_2(-)(g)\circ f \circ  \rho_1(-)(g^{-1})
  \;\colon\;
  S_1 \stackrel{\rho_1(-)(g^{-1})}{\longrightarrow} S_1 \stackrel{f}{\longrightarrow} S_2\stackrel{\rho_2(-)(g)}{\longrightarrow} S_2
  \,.
$$

=--

+-- {: .num_prop }
###### Proposition

The [[conjugation action]] construction of def. \ref{ConjugationActionForDiscrete1Groups} is the [[internal hom]] in the [[category]] of actions.

=--

+-- {: .proof}
###### Proof

We need to show that for any three [[permutation representations]],  [[functions]]

$$
  \phi \;\colon\; S_3 \longrightarrow [S_1,S_2]
$$

which [[intertwiner|intertwine]] the $G$-action on $S_3$ with the conjugation action on $[S_1,S_2]$ are in [[natural bijection]] with functions

$$
  \tilde \phi \;\colon\; S_3 \times S_1 \longrightarrow S_2
$$

which intertwine the diagonal action on the [[Cartesian product]] $S_3 \times S_1$ with the action on $S_2$.

The condition on $\phi$ means that for all $g\in G$ and $s_3 \in S_3$ it sends

$$
  \phi
   \;\colon\;
  \rho_3(s_3)(g)
  \mapsto
  \left(
    s_1 \mapsto
    \rho_2\left(
     \phi\left(s_3\right)\left( \rho_1\left(s_1\right)\left(g^{-1}\right) \right)\right)\left(g\right)
  \right)
 \,.
$$

This is equivalently a function $\tilde \phi$ of two variables which sends

$$
  \tilde \phi
  \;\colon\;
  (\rho_3(s_3)(g), s_1)
  \mapsto
  \rho_2
  (
     \phi(s_3)(
       \rho_1(s_1)(g^{-1})
     )
  )(g)
 \,.
$$

Since this has to hold for all values of the variables, it has to hold when substituing $s_1$ with $\rho_1(s_1)(g)$. After this substitution the above becomes

$$
  \tilde \phi
  \;\colon\;
  (\rho_3(s_3)(g),
  \rho_1(s_1)(g))
  \mapsto
   \rho_2(\phi(s_3)(s_1 ))(g)
  \,.
$$

This is the intertwining condition on $\tilde \phi$. Conversely, given $\tilde \phi$ satisfying this for all values of the variables, then running the argument backwards shows that its hom-[[adjunct]] $\phi$ satisfies its required intertwining condition.

=--

The following is immediate but conceptually important:

+-- {: .num_prop }
###### Proposition

The [[invariants]] of the conjugation action on $[S_1,S_2]$ is the set of action [[homomorphisms]]/[[intertwiners]].

=--

Hence the inclusion of invariants into the conjugation action gives the inclusion of the external [[hom set]] of the category of $G$-action into the set underlying the [[internal hom]]

$$
  G Act(\rho_1,\rho_2)\hookrightarrow [\rho_1,\rho_2]
  \,.
$$

+-- {: .num_remark }
###### Remark

Regarding the conjugation action as the [[internal hom]] of actions immediately gives the generalization of this concept to more general kinds of actions, notably to [[infinity-actions]] in general [[(infinity,1)-toposes]]. See at _[infinity-action -- Conjugation action](infinity-action#CartesianClosedMonoidalStructure)_ for more on this.

=--

## Related concepts

* [[inner automorphism]]

* [[adjoint action]], [[free loop space of classifying space]]

* [[conjugate subgroup]]

[[!redirects conjugation actions]]

[[!redirects precomposition action]]
[[!redirects precomposition actions]]

[[!redirects postcomposition action]]
[[!redirects postcomposition actions]]

[[!redirects pre-composition action]]
[[!redirects pre-composition actions]]

[[!redirects post-composition action]]
[[!redirects post-composition actions]]


conjunction

> This page is about conjunctions in double (or higher) categories; see [[logical conjunction]] for the meet of truth values.

***

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## Idea

A **conjunction** in a [[double category]] is a way of saying that a horizontal arrow and a vertical arrow are [[adjunction|adjoint]], even though they do not live in the same 2-category.

## Definition

Let $f\colon A\to B$ be a vertical arrow and $g\colon B\to A$ a horizontal arrow in a double category.  These arrows are said to be a **conjunction** if they come equipped with 2-cells
$$
\array{ A & \overset{id}{\to} & A \\
^f\downarrow & ^{\eta}\swArrow & \downarrow^{id} \\
B & \underset{g}{\to} & A}
\qquad and\qquad
\array{ B & \overset{g}{\to} & A \\
^{id} \downarrow & ^{\varepsilon}\swArrow & \downarrow^f \\
B & \underset{id}{\to} & B }
$$
such that $\varepsilon \circ_h \eta = id_g$ and $\eta \circ_v \varepsilon = id_{f}$, where $\circ_h$ and $\circ_v$ denote horizontal and vertical composition of 2-cells.

Given such a conjunction, we say that $f$ and $g$ are **conjoints** of each other, and that $g$ is the **right conjoint** of $f$ and that $f$ is the **left conjoint** of $g$.

## Examples

* In the double category $\mathbf{Sq}(K)$ of squares ([[quintets]]) in any 2-category $K$, a conjunction is simply an internal [[adjunction]] in $K$.

* In the [[double category of algebras | double category $T$-$\mathbf{Alg}$ of algebras, lax morphisms, and colax morphisms]] for a [[2-monad]] $T$, an conjunction is precisely a [[doctrinal adjunction]] between a colax morphism an a lax morphism.

## Remarks

* The horizontal (or vertical) dual of a conjunction is a [[companion pair]].

* Conjunctions (and companion pairs) have a [[mate]] correspondence generalizing the calculus of mates in 2-categories.

* If every vertical arrow in some double category $D$ has a right conjoint, then the functor $f\mapsto g$ is a [[pseudofunctor]] $V D^{op}\to H D$ from the vertical 2-category to the horizontal one, which is the identity on objects, and [[locally fully faithful 2-functor|locally fully faithful]] by the mate correspondence.  If every vertical arrow also has a [[companion]], then this makes $D$ into a [[proarrow equipment]], or equivalently a [[framed bicategory]].

## References

* [[Marco Grandis]] and [[Robert Pare]], *Adjoints for double categories*

* [[Robert Dawson]] and [[Robert Pare]] and [[Dorette Pronk]], *The Span construction*, [TAC](http://www.tac.mta.ca/tac/volumes/24/13/24-13abs.html).

* [[Michael Shulman]], *Framed bicategories and monoidal fibrations*, [TAC](http://www.tac.mta.ca/tac/volumes/20/18/20-18abs.html)

[[!redirects conjunction]]
[[!redirects conjunctions]]
[[!redirects conjunctionss]]
[[!redirects conjoint]]
[[!redirects conjoint pair]]
[[!redirects conjoints]]
[[!redirects conjoint pairs]]
[[!redirects conjunction in a double category]]
[[!redirects conjunctions in a double category]]
[[!redirects left conjoint]]
[[!redirects right conjoint]]
[[!redirects left conjoints]]
[[!redirects right conjoints]]
[[!redirects orthogonal adjunction]]
[[!redirects orthogonal adjunctions]]


connected category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea
 {#Idea}

A [[groupoid]] is *[[connected]]* if it is [[inhabited set|inhabited]] and every [[object]] is connected by a [[morphism]] to every other object.

So the [[skeleton|skeletal]] connected groupoids are precisely the [[delooping groupoids]] of [[groups]].

In [[homotopy theory]], [[groupoids]] model exactly the [[homotopy 1-types]] and connected groupoids model the [[connected homotopy type|connected]] homotopy 1-types. For instance the [[fundamental groupoid]] of a [[connected topological space]] is a connected groupoid.

Every [[category]] $C$ induces a [[free groupoid]] $F(C)$ by freely inverting all its [[morphisms]]. A category is called connected if the groupoid $F(C)$ is.

## Definition

A [[category]] $C$ is **connected** if it is [[inhabited set|inhabited]] and the following equivalent conditions hold:

* any two [[objects]] can be connected by a [[zigzag]] of [[morphisms]].

* the $\infty$-groupoidification of the category (the [[Kan fibrant replacement]] of its [[nerve]]) is a [[connected]] [[∞-groupoid]].

* the [[geometric realization]] of its [[nerve]] is a [[connected topological space]].

* the [[localization]] $C[C_1^{-1}]$ of $C$ at all its morphisms is a connected groupoid.

Note that the [[empty category]] is not connected.  For other purposes, one can argue about whether the [[empty set]] should be called "connected" (see [[connected space]]), but for the applications of connected categories, the empty category should definitely not be called connected.  In particular, a [[terminal object]] is not a [[connected limit]].

## Related concepts

* A *[[connected limit]]* is a [[limit]] whose domain [[diagram]] category is connected.

* [[connected homotopy type]]

## References

The notion of connected groupoids was originally defined in

* {#Brandt27} [[Heinrich Brandt]], *Über eine Verallgemeinerung des Gruppenbegriffes*, Mathematische Annalen **96** 1 (1927) 360-366 [[doi:10.1007/BF01209171](https://doi.org/10.1007%2FBF01209171)]

whence some authors also speak of *[[Brandt groupoids]]*.

For more see the references at *[[groupoids]]*.

[[!redirects connected category]]
[[!redirects connected categories]]
[[!redirects connected groupoid]]
[[!redirects connected groupoids]]


connected colimit

the [[formal duality|dual]] notion to *[[connected limit]]*, see there.


connected filtered space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
=--
=--

# Connected filtered spaces
* table of contents
{: toc}

## Definition

A [[filtered space]] $X_*$ is called a **connected filtered space** if it satisfies:

1. $(\phi)_0$: The function $\pi_0X_0  \to \pi_0X_r$ induced by   inclusion is surjective for all $r \gt 0$; and,

2. for all $i \geq 1$, $(\phi_i): \pi_i(X_r,X_i,v)=0$  for all $r \gt i$  and $ v \in  X_0$.

Another equivalent form is:

1. $(\phi_0')$: The function $\pi_0X_s  \to \pi_0X_r$ induced by   inclusion is surjective for all $0=s \lt r$ and bijective for all $1 \leq s \leq   r$; and,

2. for all $i \geq 1$, $(\phi_i'): \, \pi_j(X_r,X_i,v)=0$  for all  $v \in X_0$ and all $ j,r$ such that $ 1 \leq j \leq i \lt r$.

## Examples

Examples of connected filtered spaces are:

1. The skeletal filtration of a CW-complex.

2. The word length filtration of the James construction for a space with base point such that $\{x\} \to X$ is a closed cofibration.

3. The filtration $(B C)_*$ of the classifying space of a crossed complex, filtered using skeleta of $C$.

This condition occurs in the [[higher homotopy van Kampen theorem]] for [[crossed complex]]es.

[[!redirects connected filtered space]]
[[!redirects connected filtered spaces]]


connected graph

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Graph theory
+-- {: .hide}
[[!include graph theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An [[inhabited]] [[graph]] has two notions of being connected:

*  **weak connected** - it cannot be decomposed as the [[disjoint union]] of two inhabited graphs.

* **strong or path connected** - each [[pair]] of [[vertices]] is connected by a sequence of [[edges]] or in other words every [[homset]] of its [[free category]] is inhabited.

If a graph's edges are [[graph#undirected_graphs|undirected]] these two notions coincide while if [[graph#directed_graphs|directed]] the stronger condition implies the weaker.

For the various categories of graphs, which of their objects are _weakly connected_ correspond to the general notion of [[connected objects]] in these categories.

## Examples

* In [[perturbative quantum field theory]] the [[formal power series]] of [[Feynman amplitudes]] labeled by those [[Feynman diagrams]] whose underlying [[graphs]] are connected is called the _[[effective action]]_ $S_{eff}$. The [[exponential]] of the effective action then yields the full [[Feynman perturbation series]] over all Feynman diagrams. See at _[[S-matrix]]_ for more.

## References

* Wikipedia, _Connectivity (graph theory)_

[[!redirects connected graphs]]


connected homotopy type

#Contents#
* table of contents
{:toc}

## Definition

A [[homotopy type]] is _connected_ if it is a [[n-connected object of an (infinity,1)-topos|0-connected]].

## Related concepts

[[!include notions of type]]

[[!redirects connected homotopy types]]

[[!redirects connected type]]
[[!redirects connected types]]


connected limit

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A connected limit is the [[limit]] of a [[diagram]] that has just one [[connected component]].  For example, the [[product]] of two objects is not a connected limit because the relevant diagram has two connected components.  A [[pullback]], on the other hand, is a connected limit.

## Definition

A **connected limit** is a [[limit]] over a [[connected category]]. Similarly, a **connected colimit** is a colimit over a connected category.

## Examples

The following are all connected limits:

* [[pullbacks]]

* [[wide pullbacks]]

* [[equalizers]]

* [[cofiltered limits]]

* [[split idempotents]]

## Properties

### Construction from pullbacks and equalizers

+-- {: .num_theorem #PbEqToFinConnLim}
###### Theorem
If a category $C$ has [[pullbacks]] and [[equalizers]], then it has all [[finite limit|finite]] connected limits.
=--
+-- {: .proof}
###### Proof
Let $I$ be a finite connected category and $F\colon I\to C$ a functor.  Since $I$ is connected, it is inhabited; choose some object $x_0\in I$.  For each object $x\in I$, let $\ell(x)$ be the length of the shortest [[zigzag]] from $x_0$ to $x$.  Now order the objects of $I$ as $x_0, x_1, \dots, x_n$ such that for all $i$ we have $\ell(x_i) \le \ell(x_{i+i})$.

Now we inductively define objects $P_i\in C$, for $0\le i\le n$, with projections $p_{i j}\colon P_i \to F(x_j)$ for $j\le i$.  We begin with $P_0 = x_0$ and $p_{0 0}= 1_{x_0}$.  Assuming $P_i$ and $p_{i j}$ defined, choose a zigzag from $x_0$ to $x_{i+1}$ of minimal length, say
$$ x_0 \leftrightarrow y_1 \leftrightarrow \dots \leftrightarrow y_k \leftrightarrow x_{i+1}. $$
By our choice of the ordering of the objects of $I$, we have $y_k = x_j$ for some $j\le i$, and thus we have $q = p_{i j}\colon P_i \to y_k$.

If the final arrow $y_k \leftrightarrow x_{i+1}$ in the zigzag is directed as $y_k \to x_{i+1}$, then let $P_{i+1} = P_i$, let $p_{i+1, i+1}$ be the composite $P_i \xrightarrow{q} F(y_k) \to F(x_{i+1})$, and keep the other $p_{i j}$ unchanged.  On the other hand, if $y_k \leftrightarrow x_{i+1}$ is directed as $y_k \leftarrow x_{i+1}$, let $P_{i+1}$ be the pullback
$$ \array{
& P_{i+1} & \to & P_i \\
^{p_{i+1,i+1}} & \downarrow & & \downarrow^q \\
& F(x_{i+1}) & \to & F(y_k)
}$$
and define $p_{i+1,j}$ for $j\le i$ by composition with $P_{i+1}\to P_i$.

At the end of this procedure, we have an object $P_n$ with projections $p_{n,j}\colon P_n \to F(x_j)$ for all objects $x_j\in I$.  Now we order the morphisms in $I$ as $g_0,\dots,g_m$ and define, inductively, an object $Q_i$ with a morphism $q_i\colon Q_i \to P_n$.  We begin with $Q_0 = P_n$ and $q_0 = 1_{P_n}$.  Given $Q_i$ and $q_i$, we let $e\colon Q_{i+1} \to Q_{i}$ be the equalizer of $F(g_{i+1}) \circ p_{n,?} \circ q_i$ and $p_{n,?} \circ q_i$, where the ?s denote the indices of the objects that are the source and target of $g_{i+1}$.  We then set $q_{i+1} = q_i \circ e$.

At the end of this procedure, we have an object $Q_{m+1}$ together with a cone over the diagram $F$, which is easily verified to be a limit of $F$.
=--

Similarly, arbitrary connected limits may be built from [[wide pullbacks]] and equalizers.

+-- {: .num_theorem #internalsum}
###### Theorem
Let $C$ be (finitely) complete, and let $X$ be an object of $C$. Then the forgetful functor

$$\sum_X: C/X \to C$$

preserves and reflects (finite) connected limits.
=--

+-- {: .proof}
###### Proof
The wide pullback of the diagram $f_i: X_i \to X$ where $f_i = 1_X$ for all $i$ is clearly $X$, as is the equalizer of the pair of identity arrows. Since the product functor $C \times C \to C$ preserves arbitrary limits, we see that

$$X \times lim (c_i \stackrel{g_i}{\to} c) = (lim X_i \stackrel{f_i = 1_X}{\to} X) \times (lim c_i \stackrel{g_i}{\to} c) = lim X \times c_i \stackrel{1 \times g_i}{\to} X \times c,$$

i.e., $X \times -$ preserves wide pullbacks. The same line of argument shows $X \times -$ preserves equalizers, so $X \times -$ preserves connected limits.

The functor $X \times -$ carries a canonical [[comonad]] structure, whose category of coalgebras is $C/X$. The forgetful functor $C/X \to C$ is comonadic, and thus preserves and reflects any class of limits preserved by the comonad $X \times -$. Thus $\sum_X: C/X \to C$ preserves and reflects all connected limits.
=--

### Preservation from wide pullbacks

Recall that a [[wide pullback]] is a limit over a [[diagram]] whose underlying shape is the [[poset]] obtained by [[free functor|freely]] adjoining a [[top element|terminal element]] to a [[set|discrete poset]], which is certainly connected.

It is not true that if $C$ has wide pullbacks then it has connected limits.  The [[saturated class of limits|saturation]] of the class of wide pullbacks is the class of connected and "simply connected" limits (limits over categories $C$ whose groupoid reflection $\Pi_1(C)$ is trivial).

However, the following is true.

+-- {: .num_theorem #PreservesWidePbToConnected}
###### Theorem
Let $C$ be a complete category, and let $D$ be [[locally small category|locally small]]. Then a functor $G\colon C \to D$ preserves connected limits if and only if it preserves wide pullbacks.
=--

+-- {: .proof}
###### Proof
The forward direction is clear since wide pullbacks are examples of connected limits. Now suppose $G\colon C \to D$ preserves wide pullbacks. Then
\[C \stackrel{G}{\to} D \stackrel{hom(d, -)}{\to} Set \label{Gwithhom}  \]
preserves wide pullbacks for every object $d$ of $D$. Put $I = hom(d, G 1)$. The underlying functor
$$\sum\colon Set/I \to Set$$
reflects and preserves connected limits and in particular wide pullbacks, so that the evident lift
$$\hom(d, G-)\colon C \to Set/I$$
preserves wide pullbacks. It also preserves the terminal object, hence by [this proposition](/nlab/show/wide+pullback#WidePbToComplete) it preserves arbitrary limits. Therefore the composite
$$C \stackrel{\hom(d, G-)}{\to} Set/I \stackrel{\sum}{\to} Set$$
preserves connected limits, for every object $d$. Since this is the same composite as in \eqref{Gwithhom}, and since the representables $\hom(d, -)$ jointly reflect arbitrary limits, we conclude that $G$ preserves connected limits.
=--

The analogous argument works for finite limits. In particular, for $C$ finitely complete, a functor $G\colon C \to D$ that preserves pullbacks also preserves [[equalizers]].

## Related pages

* [[parametric right adjoint]]

* [[connected category]]

* [[wide pullback]]

* [[wide pushout]]

## References

* [[Robert Paré]], *Simply connected limits*.  Can. J. Math., Vol. XLH, No. 4, 1990, pp. 731-746, [CMS](http://math.ca/10.4153/CJM-1990-038-6)

[[!redirects connected limit]]
[[!redirects connected limits]]


connected object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context###
#### Compact objects
+-- {: .hide}
[[!include compact object - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

A connected object is a generalisation of the concept of [[connected space]] from [[Top]] to an arbitrary [[extensive category]].

## Definitions

Let $\mathcal{C}$ be an [[extensive category]].

+-- {: .num_defn #ConnectedObject}
###### Definition

An [[object]] $X$ of a [[category]] $\mathcal{C}$ is called *connected* if the [[hom-functor]] out of $X$

$$
  \mathcal{C}(X, -) \,\colon\, \mathcal{C} \to Set
  \,.
$$

[[preserved limit|preserves]] all [[coproducts]].

=--

+-- {: .num_remark}
###### Remark

By definition, $\mathcal{C}(X,-)$ preserves binary coproducts if the canonically defined morphism $\mathcal{C}(X,Y) + \mathcal{C}(X,Z) \to \mathcal{C}(X,Y + Z)$ is a [[natural bijection]].

=--

\begin{remark}
\label{InitialObjectIsNotConnected}
**(initial object is not connected)**
\linebreak
By Def. \ref{ConnectedObject}, the [[initial object]] $\varnothing$ of a category $\mathcal{C}$ is *not* connected:

Because, $\mathcal{C}(\varnothing, -) \,\colon\, \mathcal{C} \to Set$ is, by definition of [[initial objects]], the [[constant functor]] with value the [[singleton]] $\ast \,\in\, Set$, so that for $X , Y \,\in\, \mathcal{C}$ we have

$$
  \mathcal{C}
  \big(
    \varnothing
    ,\,
    X \sqcup Y
  \big)
  \;\simeq\;
  \ast
  \;\;
  \neq
  \;\;
  \ast \sqcup \ast
  \;\simeq\;
  \mathcal{C}
  (\varnothing,\, X)
  \sqcup
  \mathcal{C}
  (\varnothing,\, Y)
  \;\;\;
  \in
  \;
  Set
  \,.
$$

One may understand this as part of the pattern by which degenerate objects may be "[[too simple to be simple]]".  For example, also the [[empty space]] should not be considered connected, see [this remark](connected+space#ConnectedEmptySpace) at *[[connected space]]* .

\end{remark}

+-- {: .num_remark}
###### Remark

If $C$ is a [[infinitary extensive category]] then for $X \in Ob(\mathcal{C})$ to be connected it is enough to require that $\mathcal{C}(X,-)$ preserves *binary* coproducts. This is theorem \ref{RespectForBinaryCoproductsIsSufficient} below.

=--

## Properties

### Characterization in terms of coproducts

Let $C$ be an infinitary [[extensive category]], then

\begin{prop}
\label{RespectForBinaryCoproductsIsSufficient}
 An [[object]] $X$ of $C$ is connected, def. \ref{ConnectedObject}, if and only if the [[hom-functor]] $\hom(X, -) \colon C \to Set$ [[preserved limit|preserves]] binary [[coproducts]].
\end{prop}

\begin{proof}
The "only if" is clear, so we just prove the "if".

We first show that $\hom(X, -)$ preserves the [[initial object]] $0$. Indeed, if $\hom(X, -)$ preserves the binary product $X + 0 = X$, then the canonical map

$$\hom(X, X) + \hom(X, 0) \to \hom(X, X)$$

is a bijection of sets, where the restriction to $\hom(X, X)$ is also a bijection of sets $id: \hom(X, X) \to \hom(X, X)$. This forces the set $\hom(X, 0)$ to be empty.

Now let $\{Y_\alpha: \alpha \in A\}$ be a set of objects of $C$. We are required to show that each map
$$ f\colon X \to \sum_\alpha Y_\alpha $$
factors through a unique inclusion $i_\alpha: Y_\alpha \to \sum_\alpha Y_\alpha$. By infinite extensivity, each pullback $U_\alpha \coloneqq f^\ast (Y_\alpha)$ exists and the canonical map $\sum_\alpha U_\alpha \to X$ is an isomorphism. We will treat it as the identity.

Now all we need is to prove the following.

* Claim: $X = U_\alpha$ for exactly one $\alpha$. For the others, $U_\beta = 0$.

Indeed, for each $\alpha$, the identity map factors through one of the two summands in

$$ id\colon X \to U_\alpha + \sum_{\beta \neq \alpha} U_\beta $$

because $\hom(X, -)$ preserves binary coproducts. In others words, either $X = U_\alpha$ or $X = \sum_{\beta \neq \alpha} U_\beta$ (and the other is $0$). We cannot have $U_\alpha = 0$ for every $\alpha$, for then $X = \sum_\alpha U_\alpha$ would be $0$, contradicting the fact that $\hom(X, 0) = 0$. So $X = U_\alpha$ for at least one $\alpha$. And no more than one $\alpha$, since we have $U_\alpha \cap U_\beta = 0$ whenever $\alpha \neq \beta$.
\end{proof}

\begin{remark}
This above proof of Prop. \ref{RespectForBinaryCoproductsIsSufficient} is not [[constructive mathematics|constructive]], as we have no way to construct a particular $\alpha$ such that $X = U_\alpha$.  (It is constructive if [[Markov's principle]] applies to $A$.)
\end{remark}

From the proof above, we extract a result useful in its own right, giving an alternative definition of connected object.

\begin{proposition}
\label{InExtensiveCategoryCOnnectedObjectsArePrimitiveUnderCoproduct}
**(in extensive categories connected objects are primitive under coproduct)**
\linebreak
An object $X$ in an [[extensive category]] is connected, def. \ref{ConnectedObject}, if and only if in any [[coproduct]] decomposition $X \simeq U + V$, exactly one of $U$, $V$ is not the [[initial object]].
\end{proposition}

\begin{proof}
\label{ProofThatInExtensiveCategoryCOnnectedObjectsArePrimitiveUnderCoproduct}
In one direction, assume that $X$ is connected and consider an [[isomorphism]] $X \xrightarrow{\sim} X_1 \sqcup X_2$ to a [[coproduct]]. By assumption of connectedness, this morphism factors through one of the summands, say through $X_1$, as shown in the bottom row of the following diagram:

\begin{tikzcd}
    X_2
    \ar[r]
    \ar[d]
    \ar[dr, phantom, "\mbox{\tiny (pb)}"]
    &
    \varnothing
    \ar[r]
    \ar[d]
    \ar[dr, phantom, "\mbox{\tiny (pb)}"]
    &
    X_2
    \ar[d, "i_2"{right}]
    \\
    X
    \ar[r]
    \ar[
      rr,
      rounded corners,
      to path={
        -- ([yshift=-7pt]\tikztostart.south)
        -- node[below]{\scalebox{.7}{$\sim$}}
           ([yshift=-6pt]\tikztotarget.south)
        -- ([yshift=-0pt]\tikztotarget.south)
      }
    ]
    &
    X_1
    \ar[r, "i_1"{below}]
    &
    X_1 \sqcup X_2
    \mathrlap{\,.}
\end{tikzcd}
Consider then the [[pullback]] of the total bottom morphism along the inclusion $i_2$ of the other summand. Since pullbacks of isomorphisms are isomorphisms, the resulting top left object must be (isomorphic to) $X_2$, as shown. On the other hand, by the [[pasting law]] this pullback factors into two pullback squares, as shown above. But the pullback on the right gives the [[initial object]], since [[disjoint coproducts|coproducts are disjoint]] in an extensive category (see [here](extensive+category#DisjointAndPullbackStableCoproducts)). This exhibits a morphisms $X_2 \to \varnothing$. But since the initial objects in extensive categories are [[strict initial objects]], this must be an isomorphism, $X_2 \simeq \varnothing$. By the same argument, $X_1$ cannot be an initial object, since otherwise $X$ would be too, which it is not by assumption of connectedness (Rem. \ref{InitialObjectIsNotConnected}). Hence we have shown that exactly one of the two summands in $X \simeq X_1 \sqcup X_2$ is initial.

In the other direction, assume that $X$ has non non-trivial coproduct decomposition and consider any [[morphism]] $f \colon X \to Y + Z$ into a coproduct. By [[extensive category|extensivity]],
this implies (see [here](extensive+category#eq:HomIntoCoproductIsPullbackIfDomainIsCoproduct))
a coproduct decomposition $X = U + V$
with $U \coloneqq f^\ast(i_Y)$ and $V \coloneqq f^\ast(i_Z)$.
But, by assumption, either $U$ or $V$ is initial, meaning that $X$ is [[isomorphism|isomorphic]] to either $V$ or $U$, respectively, so that $f$ factors through either $Y$ or $X$, respectively. In other words, $f$ belongs to exactly one of the two subsets $\hom(X, Y) \hookrightarrow \hom(X, Y + Z)$ or $\hom(X, Z) \hookrightarrow \hom(X, Y + Z)$.
\end{proof}

### General properties

+-- {: .num_prop}
###### Proposition

A [[colimit]] of connected objects over a [[connected category|connected]] [[diagram]] is itself a connected object.

=--

+-- {: .proof}
###### Proof

Because coproducts in $Set$ commute with _limits_ of connected diagrams.

=--

+-- {: .num_prop}
###### Proposition

If $X \in Ob(C)$ is connected and $X \to Y$ is an [[epimorphism]], then $Y$ is connected.

=--

+-- {: .proof}
###### Proof

Certainly $Y$ is not [[initial object|initial]], because initial objects in extensive categories are [[strict initial object|strict]]. Suppose $Y = U + V$ (see prop. \ref{InExtensiveCategoryCOnnectedObjectsArePrimitiveUnderCoproduct} above), so that we have an epimorphism $X \to U + V$. By connectedness of $X$, this epi factors through one of the summands, say $U$. But then the inclusion $i_U: U \hookrightarrow U + V$ is epic, in fact an epic equalizer of two maps $U + i_1, U + i_2: U + V \rightrightarrows U + V + V$. This means $i_U$ is an isomorphism; by disjointness of coproducts, this forces $V$ to be initial. Of course $U$ is not initial; otherwise $Y$ would be initial.

=--

+-- {: .num_remark}
###### Remark

It need not be the case that products of connected objects are connected. For example, in the topos $\mathbb{Z}$-Set, the product $\mathbb{Z} \times \mathbb{Z}$ decomposes as a countable coproduct of copies of $\mathbb{Z}$. (For more on this topic, see also [[cohesive topos]].)

=--

We do have the following partial result, generalizing the case of $Top$.

+-- {: .num_theorem #prod}
######Theorem

Suppose $C$ is a cocomplete $\infty$-[[extensive category]] with
[[finite products]]. Assume that the [[terminal object]] is a [[separator]], and that [[Cartesian product]] functors $X \times (-) : C \to C$ preserve [[epimorphisms]]. Then a product of finitely many connected objects is itself connected.

=--

+-- {: .proof}
######Proof

In the first place, $1$ is connected. For suppose $1 = U + V$ where $U$ is not initial. The two coproduct inclusions $U \to U + U$ are distinct by disjointness of sums. Since $1$ is a separator, there must be a map $1 \to U$ separating these inclusions. We then conclude $U \cong 1$, and then $V \cong 0$ by disjointness of sums.

Now let $X$ and $Y$ be connected. The two inclusions $i_1, i_2: X \to X + X$ are distinct, so there exists a point $a \colon 1 \to X$ separating them. For each $y \colon 1 \to Y$, the +-shaped object $T_y = (X \times y) \cup (a \times Y)$ is connected (we define $T_y$ to be a sum of connected objects $X \times y$, $a \times Y$ amalgamated over the connected object $a \times y = 1 \times 1 = 1$, i.e., to be a connected colimit of connected objects). We have a map

$$\phi: \bigcup_{y \colon 1 \to Y} T_y \to X \times Y$$

where the union is again a sum of connected objects $T_y$ amalgamated over $a \times Y$, so this union is connected. The map $\phi$ is epic, because the evident map

$$\sum_{y \colon 1 \to Y} X \times y \cong X \times \sum_{y \colon 1 \to Y} 1 \to X \times Y$$

is epic (by the assumptions that $1$ is a [[separator]] and $X \times -$ preserves epis), and this map factors through $\phi$. It follows that the codomain $X \times Y$ of $\phi$ is also connected.

=--

+-- {: .num_remark #prod2}
###### Remark
The same method of proof shows that for an arbitrary family of [[connected spaces]] $\{X_\alpha\}_{\alpha \in A}$, the connected component of a point $x = (x_\alpha)$ in the product space $X = \prod_{\alpha \in A} X_\alpha$ contains at least all those points which differ from $x$ in at most finitely many coordinates. However, the set of such points is dense in $\prod_{\alpha \in A} X_\alpha$, so $\prod_{\alpha \in A} X_\alpha$ must also be connected.
=--

## Examples

* Connected objects in [[Top]] are precisely [[connected topological spaces]].

* For a [[group]] $G$, connected objects in the category $G Set$ of [[permutation representation]]s are precisely the ([[inhabited set|inhabited]]) [[transitive action|transitive]] $G$-sets.

* Objects in a [[locally connected topos]] are [[coproducts]] of connected objects.

  (This includes categories such as $G Set$, [[permutation representations]] of a [[group]] $G$.)

* A [[scheme]] is connected as a scheme (which by definition means that its underlying topological space is connected) if and only if it is connected as an object of the category of schemes. See [here](http://gausyu.github.io/MathBlog/2016/02/13/Connectedness-of-Schemes/).

* [[connected graph]]

## Related concepts

* [[connected topos]], [[locally connected topos]]

* [[disjoint coproduct]]

* [[coproduct-preserving representable]]

* [[n-connected object in an (∞,1)-topos]]

## References

* R. E. Hoffmann. *A categorical concept of connectedness*. Résumés Coll. Amiens. [[Cahiers]]. XIV-2 (1973). ([pdf](http://www.numdam.org/item/CTGDC_1973__14_2_153_0.pdf) page 34)

[[!redirects connected object]]
[[!redirects connected objects]]

[[!redirects connected]]


connected relation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
=--
=--

# Connected relations
* table of contents
{: toc}

## Definitions

A (binary) [[relation]] $\sim$ on a [[set]] $A$ is __connected__ if any two [[elements]] that are related in neither order are [[equality|equal]]:
$$ \forall (x, y: A),\; x \nsim y \;\wedge\; y \nsim x \;\Rightarrow\; x = y .$$
This is a basic property of [[strict total orders]].

Every [[tight relation]] is a connected relation. Every connected [[symmetric relation]] is a tight relation.

## Constructive aspects

Using [[excluded middle]], it is equivalent to say that every two elements are related in some order or equal:
$$ \forall (x, y: A),\; x \sim y \;\vee\; y \sim x \;\vee\; x = y .$$
However, this version is too strong for the intended applications to [[constructive mathematics]].  (In particular, $\lt$ on the located Dedekind [[real numbers]] satisfies the first definition but not this one.)

On the other hand, there is a stronger notion that may be used in [[constructive mathematics]], if $A$ is already equipped with a [[tight apartness]] $\#$.  In that case, we say that $\sim$ is __strongly connected__ if any two distinct elements are related in one order or the other:
$$ \forall (x, y: A),\; x \# y \;\Rightarrow\; x \sim y \;\vee\; y \sim x .$$
Since $\#$ is connected itself, every strongly connected relation is connected; the converse holds with [[excluded middle]] (through which every set has a unique tight apartness).

We can do a similar thing if $A$ is equipped with any [[inequality]] $\#$, except that in the general case, this is not necessarily stronger than being connected, and so we should call it __$\#$-connected__.

## In dependent type theory

In dependent type theory, an [[irreflexive relation]] $\sim$ with terms $a:A \vdash \mathrm{irr}(a):(a \sim a) \to \emptyset$ is connected if the canonical function
$$\mathrm{idtosymnotrel}(a, b):(a =_A b) \to (((a \sim b) \to \emptyset) \times ((b \sim a) \to \emptyset))$$
inductively defined by
$$\beta(a):\mathrm{idtosymnotrel}(a, a)(\mathrm{refl}_A(a)) =_{((a \sim a) \to \emptyset) \times ((a \sim a) \to \emptyset)} (\mathrm{irr}(a), \mathrm{irr}(a))$$
is an [[equivalence of types]]
$$\mathrm{conn}(a, b):\mathrm{isEquiv}(\mathrm{idtosymnotrel}(a, b))$$

## See also

* [[strict total order]]
* [[tight relation]]

[[!redirects connected relation]]
[[!redirects connected relations]]

[[!redirects strongly connected relation]]
[[!redirects strongly connected relations]]


connected site

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A *connected site* is a [[site]] satisfying sufficient conditions to make its [[topos of sheaves]] into a [[connected topos]].

## Definition

+-- {: .un_prop}
###### Proposition

Let $C$ be a [[locally connected site]]; we say it is a **connected locally connected site** if it is also has a [[terminal object]].

=--

## Properties

+-- {: .un_prop}
###### Proposition

If $C$ is connected locally connected site, then the [[sheaf topos]] $Sh(C)$ is a [[locally connected topos]] and [[connected topos]].

=--

+-- {: .proof}
###### Proof

Being a [[locally connected site]], we already know that we have a [[locally connected topos]] $(\Pi_0 \dashv \Delta \Delta \Gamma)  : Sh(C) \to Set$.
By the discussion there we need to check that $\Pi_0$ preserves the [[terminal object]].

The terminal object in the site [[representable functor|represents]] the terminal presheaf on $C$, which is the presheaf constant on the point. By the discussion at [[locally connected site]] we have that every constant presheaf is a [[sheaf]] over $C$, hence the terminal object of $Sh(C)$ is also represented by the terminal object in the site, and we just write "$*$" for all these terminal objects.

By the discussion there, the left adjoint $\Pi_0$ in the [[sheaf topos]] over a [[locally connected site]] is given by the [[colimit]] functor $\lim_\to : [C^{op}, Set]\to Set$. The colimit over a [[representable functor]] is always the point (this is the ([[co-Yoneda lemma|co]])-[[Yoneda lemma]] in slight disguise), hence indeed $\Pi_0 * = *$.

=--

## Examples

* The [[category of open subsets]] $Op(X)$ of a [[locally connected topological space|locally connected]] and [[connected topological space]] $Op(X)$ with the standard [[open cover]]-[[coverage]] is a connected site. The terminal object is $X$ itself.

* [[CartSp]], [[ThCartSp]].

## Related concepts

* [[locally connected topos]] / [[locally ∞-connected (∞,1)-topos]]

  * [[connected topos]] / [[∞-connected (∞,1)-topos]]

  * [[strongly connected topos]] / [[strongly ∞-connected (∞,1)-topos]]

  * [[totally connected topos]] / [[totally ∞-connected (∞,1)-topos]]

* [[local topos]] / [[local (∞,1)-topos]].

* [[cohesive topos]] / [[cohesive (∞,1)-topos]]

and

* [[locally connected site]] / [[locally ∞-connected site]]

  * **connected site** / [[∞-connected site]]

  * [[strongly connected site]] / [[strongly ∞-connected site]]

  * [[totally connected site]] / [[totally ∞-connected site]]

* [[local site]] / [[∞-local site]]

* [[cohesive site]], [[∞-cohesive site]]

[[!redirects connected sites]]


connected space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context###
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

A [[topological space]] is __connected__ if it can not be split up into two independent parts.

Every topological space may be decomposed into disjoint maximal connected [[subspaces]], called its __[[connected components]]__.  The [[underlying]] [[set]] of a topological space is the [[disjoint union]] of the underlying sets of its connected components, but the space itself is not necessarily the [[coproduct]] of its connected components in the [[category]] [[Top]] of topological spaces.

One often studies topological ideas first for connected spaces and then generalises to general spaces.  This is especially true if one is studying such [[nice topological spaces]] that every space *is* a coproduct of connected components (such as for example _[[locally connected topological spaces]]_).

## Definitions

### Classical definition
 {#ClassicalDefinition}

+-- {: .num_defn #ConnectedSpace}
###### Definition
**(connected topological space)**

A [[topological space]] $(X, \tau)$ is _connected_ if the following equivalent conditions hold:

1. For all [[pairs]] of topological spaces $(X_1, \tau_1), (X_2, \tau_2)$ such that $(X, \tau)$ is [[homeomorphism|homeomorphic]] to their [[disjoint union space]]

   $$
     (X,\tau)
       \,\simeq\,
     (X_1,\tau_1)
       \sqcup
     (X_2,\tau_2)
     \,,
   $$

   exactly one of the two spaces is the [[empty space]].

1. For all [[pairs]] of [[open subsets]] $U_1, U_2 \subset X$, if

   $$
     U_1
       \cup
     U_2
       \,=\,
     X
     \phantom{A}\text{and} \phantom{A}
     U_1 \cap U_2
       \,=\,
     \emptyset
   $$

   then exactly one of the two subsets is the [[empty set]].

1. If a [[subset]] $CO \subseteq X$ is [[clopen set|clopen]] (both [[closed subset|closed]] and [[open subset|open]]), then $ CO = X$ if and only if $CO$ is [[inhabited set|inhabited]].

=--

+-- {: .num_remark #ConnectedEmptySpace}
###### Remark

According to def. \ref{ConnectedSpace} the [[empty topological space]] is not regarded as connected. Some authors do want the empty space to count as a connected space. This means to change in the two first items of def. \ref{ConnectedSpace} the "exactly one" to "at least one" and in the third item "if and only if" to "if".

=--

+-- {: .num_prop}
###### Proposition

The conditions in def. \ref{ConnectedSpace} are indeed equivalent.

=--

+-- {: .proof}
###### Proof

First consider the equivalence of the first two statements:

Suppose that in every disjoint union decomposition of $(X,\tau)$ exactly one summand is empty. Now consider two disjoint open subsets $U_1, U_2 \subset X$ whose union is $X$ and whose intersection is empty. We need to show that exactly one of the two subsets is empty.

Write $(U_1, \tau_{1})$ and $(U_2, \tau_2)$ for the corresponding [[topological subspaces]]. Then observe that from the definition of [[subspace topology]] and the [[disjoint union space]] we have a [[homeomorphism]]

$$
  X \simeq (U_1, \tau_1) \sqcup (U_2, \tau_2)
$$

because by assumption every open subset $U \subset X$ is the disjoint union of open subsets of $U_1$ and $U_2$, respectively:

$$
  U = U \cap X = U \cap (U_1 \sqcup U_2) = (U \cap U_1) \sqcup (U \cap U_2)
  \,,
$$

which is the definition of the disjoint union topology.

Hence by assumption exactly one of the two summand spaces is the [[empty space]] and hence the underlying set is the empty set.

Conversely, suppose that for every pair of open subsets $U_1, U_2 \subset U$ with $U_1 \cup U_2 = X$ and $U_1 \cap U_2 = \emptyset$ then exactly one of the two is empty. Now consider a homeomorphism of the form $(X,\tau) \simeq (X_1, \tau_1) \sqcup (X_2,\tau_2)$. By the nature of the [[disjoint union space]] this means that $X_1, X_2 \subset X$ are disjoint open subsets of $X$ which cover $X$. So by asumption precisely one of the two subsets is the empty set and hence precisely one of the two topological spaces is the empty space.

Now regarding the equivalence to the third statement:

If a subset $CO \subset X$ is both closed and open, this means equivalently that it is open and that its [[complement]] $X \setminus CO$ is also open, hence equivalently that there are two open subsets $CO, X \setminus CO \subset X$ whose [[union]] is $X$ and whose [[intersection]] is [[empty space|empty]]. This way the third condition is equivalent to the second.

=--

### Constructive definition
 {#ConstructiveDefinition}

In [[constructive mathematics]] the conditions in Def. \ref{ConnectedSpace} no longer need to be equivalent. See [Taylor 2010, Def. 13.2](#Taylor10), who essentially speaks of "compact connectedness" for the first item in Def. \ref{ConnectedSpace} and of "overt connectedness" for the second.

Related discussion in terms of [[cohesive homotopy type theory]] (and for the special case of the [[Dedekind real numbers]]) is in [Shulman 2018, Thm. 11.1, 11.3](#Shulman18).

In this language of [[homotopy type theory]] one might then say:

A [[0-truncated]] [[cohesive infinity-groupoid]] is "compact connected" (in the sense of [Taylor 2010, Def. 13.2](#Taylor10)), if for [[subspaces]] $A \subseteq S$ and $B \subseteq S$ of $S$ with [[embeddings]] $i_{A,S}:A \hookrightarrow S$ and $i_{B,S}:B \hookrightarrow S$ such that the embedding $i_{A \cup B,S}:A \cup B \hookrightarrow S$ is an [[equivalence in homotopy type theory|equivalence]] $(i_{A \cup B,S}:A \cup B \simeq S)$ and the embedding $i_{\emptyset,A \cap B}:\emptyset \hookrightarrow A \cap B$ is an equivalence $(i_{\emptyset,A \cap B}:\emptyset \simeq A \cap B)$, either $i_{A,S}$ is an equivalence ($i_{A,S}:A \simeq S$) or $i_{B,S}$ is an equivalence ($i_{B,S}:B \simeq S$).

### Category-theoretic definition

In the language of [[category theory]] def. \ref{ConnectedSpace} may be rephrased as follows:

Let [[Top]] denote the [[category]] of all [[topological space]].

Then a topological space $X$ is connected precisely if the [[representable functor]]

$$
  hom(X, -)
    \;\colon\;
  Top \longrightarrow Set
$$

[[preserved limit|preserves]] all [[coproducts]].

This, in turn, is equivalent to just requiring that the functor preserves binary coproducts (a detailed proof of this, in a more general setting, is given at _[[connected object]]_ in [this proposition](connected+object#RespectForBinaryCoproductsIsSufficient). In that case, notice that we always have a map

$$
  hom(X,Y)
    +
  hom(X,Z)
  \longrightarrow
  hom(X,Y + Z)
  ,\,
$$

so $X$ is connected if this is always a [[bijection]]. This definition generalises to the notion of [[connected objects]] in an [[extensive category]].

The variant of the definition according to remark \ref{ConnectedEmptySpace}, with regards to the empty space, means, in terms of category theoretic language, that one
requires only that the maps
$$ hom(X,Y) + hom(X,Z) \to hom(X,Y + Z) $$
be [[surjections]].

However, many results come out more cleanly by disqualifying the empty space as connected (much as one disqualifies $1$ when one defines the notion of [[prime number]]).  See also the discussion at *[[empty space]]* and at *[[too simple to be simple]]*.

### Connected components

Every topological space $X$ admits an [[equivalence relation]] $\sim$ where $x \sim y$ means that $x$ and $y$ belong to some subspace which is connected. The [[equivalence class]] $Conn(x)$ of an element $x$ is thus the union of all connected subspaces containing $x$; it follows readily from Result \ref{2} that $Conn(x)$ is itself connected. It is called the **connected component** of $x$. It is closed, by Result \ref{5}. A space is connected if and only if it has exactly one connected component (or at most one, if you allow the empty space to be connected).

+-- {: .num_defn #ComponentsConnected}
###### Definition
**(connected components)**

For $(X,\tau)$ a [[topological space]], its _[[connected components]]_
are the [[equivalence classes]] under the [[equivalence relation]] on $X$ which
regards two points as equivalent if they both sit in some [[open subset]] which, as a [[topological subspace]] (example \ref{SubspaceTopology}), is [[connected topological space|connected]] (def. \ref{ConnectedTopologicalSpace}):

$$
  (x \sim y)
    \;\coloneqq\;
  \left(
    \underset{ U \subset X \; \text{\open} }{\exists}
    \left(
       \left( x,y \in U \right)
        \phantom{A}\text{and}\phantom{A}
       \left( U \, \text{is connected} \right)
    \right)
  \right)
  \,.
$$

=--

+-- {: .num_defn #QuasiComponentsConnected}
###### Remark
**(quasi-components)**

There is another [[equivalence relation]] $\sim_q$ modelling a notion of connected components,  where $x \sim_q y$ iff $f(x) = f(y)$ for every [[continuous map]] $f \colon X \to D$ to a _[[discrete space]]_ $D$.

The corresponding [[equivalence class]] of $x$ may alternatively be described as the [[intersection]] of all [[clopen set|clopen subsets]] that contain $x$. This is called the **quasi-component** of $x$, denoted here as $QConn(x)$.

It is easy to prove that this relates to the equivalence classes $Conn(x)$ from Def. \ref{ComponentsConnected} as

$$Conn(x) \subseteq QConn(x)$$

and that equality holds if $X$ is [[compact Hausdorff space|compact Hausdorff]] or is [[locally connected topological space|locally connected]], but also in other circumstances (such as the space of [[rational numbers]] as a [[topological subspace|subspace]] of the [[real line]]).

=--

+-- {: .num_example #comb}
###### Example
For an example where $Conn(x) \neq QConn(x)$, take $X$ to be the following [[subspace]] of $[0, 1] \times [0, 1]$:

$$X = \{(0, 0), (0, 1)\} \cup \bigcup_{n \geq 1} \{1/n\} \times [0, 1]$$

In this example, $Conn((0, 1)) = \{(0, 1)\}$, but $QConn((0, 1)) = \{(0, 0), (0, 1)\}$.
=--

+-- {: .num_remark}
###### Remark
**Warning**

It is not generally true that a [[topological space]] is the [[disjoint union space]] ([[coproduct]] in [[Top]]) of its [[connected components]], nor of its quasi-components.

The spaces such that all their open subspaces are the disjoint union of their connected components are the _[[locally connected topological spaces]]_.

=--

+-- {: .num_example}
###### Example

The connected components in [[Cantor space]] $2^{\mathbb{N}}$ (with its topology as a product of 2-point discrete spaces) are just the [[singletons]], but the coproduct of the singleton subspaces carries the [[discrete topology]], which differs from that of Cantor space.

Similarly for set of [[rational numbers]] with its absolute-value topology (the one induced as a [[topological subspace]] of the [[real line]]).

=--

## Examples

### Basic examples {#basic}

+-- {: .num_example #ContinuousImagesOfConnectedSpacesAreConnected}
###### Example
**([[continuous function|continuous]] [[images]] of connected spaces are connected)**

The [[regular monomorphism|regular]] [[image]] of a connected space $X$ under a continuous map $f: X \to Y$ (i.e., the set-theoretic image with the subspace topology inherited from $Y$) is connected. Or, what is essentially the same: if $X$ is connected and $f: X \to Y$ is [[epimorphism|epic]], then $Y$ is connected:

Let $X$ be a connected topological space, let $Y$ be any [[topological space]], and let

$$
  f \;\colon\; X \longrightarrow Y
$$

be a [[continuous function]]. This factors via continuous functions through the [[image]]

$$
  f \;\colon\; X \underoverset{surjective}{p}{\longrightarrow} f(X) \underoverset{injective}{i}{\longrightarrow} Y
$$

for $f(X)$ equipped either with the [[subspace topology]] relative to $Y$ or the [[quotient topology]] relative to $X$. In either case:

If $X$ is a connected topological space, then so is $f(X)$.

=--

+-- {: .proof}
###### Proof

Let $U_1,U_2 \subset f(X)$ be two [[open subsets]] such that $U_1 \cup U_2 = f(X)$ and $U_1 \cap U_2 = \emptyset$.
We need to show that precisely one of them is the [[empty set]].

Since $p$ is a [[continuous function]], also the [[pre-images]] $p^{-1}(U_1), p^{-1}(U_2) \subset X$
are open subsets and are still disjoint. Since preimages preserve unions it also follows that
$p^{-1}(U_1) \cup p^{-1}(U_2) = X$. Since $X$ is connected, it follows that one of these two pre-images $p^{-1}(U_i)$ is
the empty set. Since $p$ is [[surjective function|surjective]], this implies that $U_i$ is empty,
which means that $f(X)$ is connected.

=--

+-- {: .num_example #2}
###### Example
[[pushout|Wide pushouts]] of connected spaces are connected. (This would of course be false if the empty space were considered to be connected.) This follows from the hom-functor definition of connectedness, plus the fact that coproducts in $Set$ commute with [[wide pullbacks]]. More memorably: [[connected limit|connected colimits]] of connected spaces are connected.
=--

+-- {: .num_example #5}
###### Example
If $S \subseteq X$ is a connected subspace and $S \subseteq T \subseteq \overline{S}$ (i.e. if $T$ is between $S$ and its closure), then $T$ is connected. Or, what is essentially the same: if $T$ has a [[dense subspace|dense]] connected subspace $S$, then $T$ is connected.
=--

(see also prop. \ref{ClosureOfConnectedSubspaceIsConnected} below)

+-- {: .num_example #ProductSpaceOfConnectedSpacesIsConnected}
###### Example
**([[product space]] of connected spaces is connected)

Let $\{X_i\}_{i \in I}$ be a set of connected spaces. Then also their [[product topological space]]
$\underset{i \in I}{\prod}X_i$ (with the [[Tychonoff topology]]) is connected.

=--

+-- {: .proof}
###### Proof

This relies on some special features of [[Top]].
A general abstract proof is given at _[[connected object]]_ in [this theorem](connected+object#prod)
and [this remark](connected+object#prod2).

Here is an alternative elementary proof in [[point-set topology]]:

Let $U_1, U_2 \subset \underset{i \in I}{\prod}X_i$ be an [[open cover]]
of the product space by two disjoint open subsets. We need to show that precisely one of the two
is empty. Since each $X_i$ is connected and hence non-empty, the product space is not empty, and hence
it is sufficient to show that at lest one of the two is empty.

Assume on the contrary that both $U_1$ and $U_2$ are non-empty.

Observe first that if so, then we could find $x_1 \in U_1$ and $x_2 \in U_2$
whose coordinates differed only a a finite subset of $I$. This is since by the nature of the
[[Tychonoff topology]] $\pi_i(U_1) = X_i$ and $\pi_i(U_2) = X_i$ for all but a finite number of $i \in iI$.

Next observe that we then could even find $x'_1 \in U_1$ that differed only in a single coordinate from $x_2$:
Because pick one coordinate in which $x_1$ differs from $x_2$ and change it to the corresponding coordinate of $x_2$.
Since $U_1$ and $U_2$ are a cover, the resulting point is either in $U_1$ or in $U_2$. If it is in $U_2$, then
$x_1$ already differed in only one coordinate from $x_2$ and we may take $x'_1 \coloneqq x_1$.
If instead the new point is in $U_1$, then rename it to $x_1$ and repeat the argument. By [[induction]]
this finally yields an $x'_1$ as claimed.

Therefore it is now sufficient to see that it leads to a contradiction to assume
that there are points $x_1 \in U_1$ and $x_2 \in U_2$
that differ in only the $i_0$th coordinate, for some $i_0 \in I$ then $x_1 = x_2$.

Observe that the inclusion

$$
  \iota \colon X_{i_0} \longrightarrow \underset{i \in I}{\prod} X_i
$$

which is the identity on the $i_0$th component and is otherwise constant on the $i$th component of
$x_1$ or equivalently of $x_2$ is a continuous function, by the nature of the Tychonoff topology.

Therefore also the restrictions $\iota^{-1}(U_1)$ and $\iota^{-1}(U_2)$ are open subsets.
Moreover they are still disjoint and cover $X_i$. Hence by the connectedness of $X_i$,
precisely one of them is empty. This means that the $i_0$-component of both $x_1$ and $x_2$
must be in the other subset of $X_i$, and hence that $x_1$ and $x_2$ must both be in $U_1$
or both in $U_2$, contrary to the assumption.

=--

+-- {: .num_example #ConnectedSubspacesOfRealLineAreTheIntervals}
###### Example
**(connected subspaces of the real line are the intervals)

Regard the [[real line]] with its [[Euclidean space|Euclidean]] [[metric topology]]. Then a
[[subspace]] $S \subset \mathbb{R}$ is connected (def. \ref{ConnectedSpace})
precisely if it is an [[interval]], hence precisely if

$$
  \underset{x,y \in S \subset \mathbb{R}}{\forall}
  \underset{ r \in \mathbb{R}  }{\forall}
  \left(
   \left(
      x \lt r \lt y
   \right)
     \Rightarrow
   \left(
     r \in S
   \right)
  \right)
  \,.
$$

In particular for $\{ I_i \subset \mathbb{R} \}_{i \in I}$ a set of [[disjoint subset|disjoint]] intervals,
then $I$ is the set of connected components of the union $\underset{i \in I}{\cup} I_i$.

=--

+-- {: .proof}
###### Proof

Suppose on the contrary that we have $x \lt r \lt y$ but $r \notin S$. Then by the
nature of the [[subspace topology]] there would be a decomposition of $S$ as a [[disjoint union]]
of [[disjoint subset|disjoint]] [[open subsets]]:

$$
  S
    =
  \left(
    S \cap (r,\infty)
  \right)
    \sqcup
  \left(
    S \cap (-\infty,r)
  \right)
  \,.
$$

But since $x \lt r$ and $r \lt y$ both these open subsets were [[inhabited set|non-empty]], thus contradicting the
assumption that $S$ is connected. This yields a [[proof by contradiction]].

=--

### Exotic examples

The [basic results](basic) above give a plethora of ways to construct connected spaces. More exotic examples are sometimes useful, especially for constructing counterexamples.

+-- {: .num_example}
###### Example
The following, due to [Bing](#Bing), is a countable connected [[Hausdorff space]]. Let $Q = \{(x, y) \in \mathbb{Q} \times \mathbb{Q}: y \geq 0\}$, topologized by defining a basis of neighborhoods $N_{\epsilon, a, b}$ for each point $(a, b) \in Q$ and $\epsilon \gt 0$:

$$N_{a, b} \coloneqq \{(a, b)\} \cup \{(s, 0) \in Q: {|a + b/\theta - s|} \lt \epsilon\} \cup \{(s, 0) \in Q: {|a - b/\theta - s|} \lt \epsilon\}$$

where $\theta \lt 0$ is some chosen fixed irrational number. It is easy to see this space is Hausdorff (using the fact that $\theta$ is irrational). However, the _closure_ of $N_{\epsilon, a, b}$ consists of points $(x, y)$ of $Q \times Q$ with either $(x-a) - \epsilon \leq (y-b)/\theta \leq (x-a) + \epsilon$ or $(x-a) - \epsilon \leq -(y-b)/\theta \leq (x-a) + \epsilon$, in other words, the union of two infinitely long strips of width $2\epsilon$ and slopes $\theta$, $-\theta$. Clearly any two such closures intersect, and therefore the space is connected.
=--

+-- {: .num_example}
###### Example
This example is due to [Golomb](#Golomb). Topologize the set of natural numbers $\mathbb{N}$ by taking a basis to consist of sets $A_{a,b} \coloneqq \{a k + b | k = 1,2, \ldots\}$, where $a, b \in \mathbb{N}$ are relatively prime. The space is Hausdorff, but the intersection of the closures of two non-empty open sets is never empty, so this space is connected.
=--

## Properties

+-- {: .num_example #LocallyConstantFunctionsOnConnectedSpaces}
###### Example
**([[locally constant function]] on [[connected topological spaces]] are [[constant functions]])**

If $X$ is a connected topological space and $f \colon X \to Y$ is a [[locally constant function|locally constant]] [[continuous function]], then $f$ is in fact a [[constant function]].

=--

+-- {: .proof}
###### Proof

By definition of locally constant functions, every point $x \in X$ has an [[open neighborhood]] $U_x$ such that the restriction $f\vert_{U_x}$ is a constant function. The [[unions]] of these neighborhoods for a fixed constant value hence are disjoint open subsets that constitute a cover of $X$. By connectedness this cover must consist of a single non-empty element. But by construction this means that $f$ is constant.

=--

## Path-connectedness

An important variation on the theme of connectedness is path-connectedness.

+-- {: .num_defn #TopologicalSpacePath}
###### Definition
**(continuous path in a topological space)**

Let $(X,\tau)$ be a [[topological space]], Then a _continuous path_ (or just _path_, for short) in $(X,\tau)$ is a [[continuous function]] of the form

$$
  \gamma
   \;\colon\;
  [0,1]
    \longrightarrow
  (X,\tau)
$$

where the domain is the [[closed interval]] equipped with its [[Euclidean space|Euclidean]] [[subspace topology]].

One says that the path connects the point $\gamma(0) \in X$ with the point $\gamma(1) \in X$.

For $x \in X$ a fixed point, then the subset

$$
  PConn_x(X)
   \;\coloneqq\;
  \left\{
     y \in X
     \;\vert\;
    \underset{\text{path}\, \gamma}{\exists}
    \left(
      \left(
        \gamma(0) = x
      \right)
      \phantom{}
      \text{and}
      \phantom{A}
      \left(
        \gamma(1) = y
      \right)
    \right)
  \right\}
$$

is called the _path-connected component_ of $x$.

The set of path connected components of $X$ is denoted

$$
  \pi_0(X)
  \,.
$$

=--

The set $\pi_0(X)$ of path components (the 0th "[[homotopy group]]") is thus the [[coequalizer]] in

\[
  \label{ConnectedComponentsAsCoequalizer}
  \hom([0, 1], X)
    \underoverset
    {ev_1}
    {ev_0}
    {\rightrightarrows}
  \hom(1, X) \to \pi_0(X) .
\]

Observe that this is a [[reflexive coequalizer]], as witnessed by the mutual right inverse $\hom(!, X): \hom(1, X) \to \hom([0, 1], X)$.

(We can even topologize $\pi_0(X)$ by taking the coequalizer in $Top$ of

$$X^{[0, 1]} \stackrel{\overset{ev_0}{\to}}{\underset{ev_1}{\to}} X,$$

taking advantage of the fact that the locally compact Hausdorff space $[0, 1]$ is [exponentiable](http://ncatlab.org/nlab/show/exponential+law+for+spaces). The resulting quotient space will be [[discrete space|discrete]] if $X$ is [[locally path-connected space|locally path-connected]].)

We say $X$ is **path-connected** if it has exactly one path component.

It follows easily from the basic results [above](basic) that:

+-- {: .num_lemma #ConnectedPathConnectedSpace}
###### Lemma

A path connected space $X$ is connected.

=--

+-- {: .proof}
###### Proof

Assume it were not, then it would be covered by two disjoint inhabited open subsets $U_1, U_2 \subset X$. But by path connectedness there were a continuous path $\gamma \colon [0,1] \to X$ from a point in one of the open subsets to a point in the other. The continuity would imply that $\gamma^{-1}(U_1), \gamma^{-1}(U_2) \subset [0,1]$ were a disjoint open cover of the interval. This would be in contradiction to the fact that intervals are connected. Hence we have a [[proof by contradiction]].

=--

However, it need not be closed (and therefore need not be the connected component of $x$); see the following example. The path components and connected components do coincide if $X$ is [[locally path-connected space|locally path-connected]].

+-- {: .num_example #sine}
###### Example
The **topologist's sine curve**
$$ \{ (x, y) \in \mathbb{R}^2 \;:\; (0 \lt x \leq 1 \;\wedge\; y = sin(1/x)) \;\vee\; (0 = x \;\wedge\; -1 \leq y \leq 1) \} $$
provides a classic example where the path component of a point need not be closed. (Specifically, consider a point on the locus of $y = \sin(1/x)$.)
=--

The basic categorical Results \ref{ContinuousImagesOfConnectedSpacesAreConnected}, \ref{2}, and \ref{ProductSpaceOfConnectedSpacesIsConnected} above carry over upon replacing "connected" by "path-connected". (As of course does
example \ref{ConnectedSubspacesOfRealLineAreTheIntervals}, trivially.)

Finally, as a contrast to a path-connected space, a **totally path-disconnected** space is a space such that its set of path components is equal to the underlying set of the space. Equivalently, that there are no non-constant paths. This by far does not mean that the space is discrete!

### Arc-connectedness

A refinement of the notion of path-connected space is that of arc-connected (or arcwise-connected) space:

+-- {: .num_defn}
###### Definition
A space $X$ is **arc-connected** if for any two distinct $x, y \in X$ there exists an _injective_ continuous map $\alpha: I \to X$ such that $\alpha(0) = x$ and $\alpha(1) = y$.
=--

Arc-connected spaces are of course path-connected, but there are trivial examples (using an [[indiscrete topology]]) that the converse fails to hold. A rather nontrivial theorem is the following:

+-- {: .num_theorem #arc}
###### Theorem
A path-connected [[Hausdorff space]] $X$ is arc-connected.
=--

This immediately generalizes to the statement that in a Hausdorff space $X$, any two points that can be connected by a path $\alpha: I \to X$ can be connected by an arc: just apply the theorem to the image $\alpha(I)$.

For a proof of this theorem, see [Willard](#Willard), theorem 31.2. More precisely, that result states that a _Peano space_, i.e., a [[compact space|compact]], [[connected topological space|connected]], [[locally connected topological space|locally connected]], and [[metrizable space]], is arc-connected if it is path-connected. It then suffices to observe that the continuous image $\alpha(I) \subseteq X $ of a path is in fact a Peano space, so that the path $\alpha: I \to \alpha(I)$ can be replaced by an arc.

+-- {: .num_lemma}
###### Lemma
If $X$ is Hausdorff and there is a continuous surjection $f: I \to X$, then $X$ is a [[Peano space]].
=--

+-- {: .proof}
###### Proof
Obviously $X$ is compact (Hausdorff) and connected. $X$ is a [[quotient space]] of $I$, since $f$ is a closed surjection (using compactness of $I$ and Hausdorffness of $X$), and therefore $X$ is locally connected by [this lemma](https://ncatlab.org/nlab/show/locally+connected+topological+space#quot). Being compact Hausdorff, $X$ is [[separation axiom|regular]], so to show metrizability it suffices by the [[Urysohn metrization theorem]] to show $X$ is second-countable.

Let $\mathcal{B}$ be a countable base for $I$ and let $\mathcal{C}$ be the collection consisting of finite unions of elements of $\mathcal{B}$. We claim $\forall_f(\mathcal{C}) \coloneqq \{\forall_f(C) = \neg f(\neg C): C \in \mathcal{C}\}$ is an (evidently countable) base for $X$. Indeed, suppose $U \subseteq X$ is open and $p \in U$; then $f^{-1}(p)$ is compact, so there exist finitely many $B_1, \ldots, B_n \in \mathcal{B}$ with

$$f^{-1}(p) \subseteq B_1 \cup \ldots \cup B_n \subseteq f^{-1}(U).$$

Put $C = B_1 \cup \ldots \cup B_n$. The first inclusion is equivalent to $p \in \forall_f(C)$ by the adjunction $f^{-1} \dashv \forall_f$. The second inclusion implies $\forall_f(C) \subseteq \forall_f f^{-1}(U) = U$, where the equality $\forall_f f^{-1} = id$, equivalent to $\exists_f f^{-1} = id$, follows from surjectivity of $f$. Thus we have shown $\forall_f(\mathcal{C})$ is a base.
=--

The converse of this lemma is the celebrated [[Hahn-Mazurkiewicz theorem]]:

+-- {: .num_theorem}
###### Theorem
Let $X$ be a nonempty Hausdorff space. Then there exists a continuous surjection $\alpha: [0, 1] \to X$ if $X$ is a Peano space. In particular, a nonempty Peano space is path-connected.
=--

(The terminology "Peano space" is given in recognition of Peano's discovery of space-filling curves, as for example the unit square.)

### Path-components functor

As above, let $\pi_0 \colon Top \to Set$ be the functor which assigns to each space $X$ its set of path components $\pi_0(X)$.

+-- {: .un_prop}
###### Proposition
The functor $\pi_0 \colon Top \to Set$ preserves arbitrary products.
=--

+-- {: .proof}
###### Proof
Let $X_i$ be a family of spaces; we must show that the comparison map

$$\pi_0(\prod_i X_i) \to \prod_i \pi_0(X_i)$$

is invertible. Injectivity: suppose $(x_i), (y_i) \in \prod_i X_i$ are tuples that map to the same tuple of path-components $(c_i)$; we must show that $(x_i)$ and $(y_i)$ belong to the same path component. For each $i$, both $x_i$ and $y_i$ belong to $c_i$, so we may choose a path $\alpha_i: I \to X_i$ connecting $x_i$ to $y_i$. Then $\langle \alpha_i \rangle \colon I \to \prod_i X_i$ connects $(x_i)$ to $(y_i)$. (Note this uses the [[axiom of choice]].) Surjectivity: for any tuple $(c_i) \in \prod_i \pi_0(X_i)$, the component $c_i$ is nonempty for each $i$, so we may choose an element $x_i$ therein. Then $(x_i)$ maps to $(c_i)$. Again this uses the axiom of choice.
=--

An elegant proof of the previous proposition but for preservation of _finite_ products is as follows: both $\hom(I, -)$ and $\hom(1, -)$ preserve products, and a [[reflexive coequalizer]] of product-preserving functors $C \to Set$, being a [[sifted colimit]], is also product-preserving.

+-- {: .un_prop}
###### Proposition
The functor $\pi_0 \colon Top \to Set$ preserves arbitrary coproducts.
=--

+-- {: .proof}
###### Proof
The functor $\hom(I, -) \colon Top \to Set$ preserves coproducts since $I$ is connected, and similarly for $\hom(1, -)$. The coequalizer of a pair of natural transformations between coproduct-preserving functors is also a coproduct-preserving functor.
=--

### Pseudo-arcs

Point-set topology is filled with counterexamples. An unusual type of example is that of pseudo-arc:

+-- {: .num_defn}
###### Definition
A _pseudo-arc_ is a [[continuum|metric continuum]] with more than one point such that every subcontinuum (a subspace that is a continuum) cannot be expressed as a union of two proper subcontinua.
=--

A pseudo-arc $X$ is necessarily totally path-disconnected: two distinct points $x, y$ of $X$ cannot be connected by a path in $X$. Indeed, the image of such a# path would be a path-connected Hausdorff space, hence arc-connected by Theorem \ref{arc}. Letting $\alpha: [0, 1] \to X$ be an arc from $x$ to $y$, we have that the continuum $\alpha([0, 1])$ is a union of proper subcontinua $\alpha([0, 1/2])$ and $\alpha([1/2, 1])$, a contradiction. Thus, a pseudo-arc is an example of a compact connected metrizable space that is totally path-disconnected.

Remarkably, all pseudo-arcs are homeomorphic, and a pseudo-arc is a [[homogeneous space]]. Perhaps also remarkable is the fact that the collection of pseudo-arcs in the [[Hilbert cube]] $Q$ (or in any Euclidean space) is a dense $G_\delta$ set (see [[G-delta set]]) in the [[Polish space]] of all nonempty compact subsets of $Q$ under the [[Hausdorff metric]]; see [Bing2](#Bing2), theorem 2.

A typical way in which pseudo-arcs arise is through [[inverse limits]] of [[dynamical systems]]. One of the original constructions is due to Henderson:

+-- {: .num_theorem}
###### Theorem
There is a $C^\infty$ function $f: I \to I$ such that the limit of the diagram

$$\ldots \stackrel{f}{\to} I \stackrel{f}{\to} I \stackrel{f}{\to} I$$

is a pseudo-arc.
=--

Roughly speaking, Henderson's $f$ is a small "notched" perturbation of the squaring function $[0, 1] \to [0, 1]: x \mapsto x^2$, as illustrated on page 38 (of 58) [here](http://www.akasimikrasna.sk/uploads/1/4/1/0/14106449/spitalsky2014.pdf).

## Properties
 {#Properties}

+-- {: .num_prop}
###### Corollary
**([[intermediate value theorem]])**

Regard the [[real numbers]] $\mathbb{R}$ with their [[Euclidean space|Euclidean]] [[metric topology]],
and consider a [[closed interval]] $[a,b] \subset \mathbb{R}$ equipped with its [[subspace topology]].

Then a [[continuous function]]

$$
  f \colon [a,b] \longrightarrow \mathbb{R}
$$

takes every value in between $f(a)$ and f(b).

=--

+-- {: .proof}
###### Proof

By example \ref{ConnectedSubspacesOfRealLineAreTheIntervals} the interval $[a,b]$ is connected.
By example \ref{ContinuousImagesOfConnectedSpacesAreConnected} also its image $f([a,b]) \subset \mathbb{R}$
is connected. By example \ref{ConnectedSubspacesOfRealLineAreTheIntervals} that image is hence
itself an interval. This implies the claim.

=--

+-- {: .num_prop #ClosureOfConnectedSubspaceIsConnected}
###### Proposition
**([[topological closure]] of connected subspace is connected)**

Let $(X,\tau)$ be a [[topological space]] and let $S \subset X$ be a [[subset]] which, as a [[subspace]], is connected. Then also the [[topological closure]] $Cl(S) \subset X$ is connected.

=--

+-- {: .proof}
###### Proof

Suppose that $Cl(S) = A \sqcup B$ with $A,B \subset X$ [[disjoint subsets|disjoint]] [[open subsets]]. We need to show that one of the two is empty.

But also the intersections $A \cap S\,,B \cap S \subset S$ are disjoint subsets, open as subsets of the subspace $S$ with $S = (A \cap S) \sqcup (B \cap S)$. Hence by the connectedness of $S$, one of $A \cap S$ or $B \cap S$ is empty. Assume $B \cap S$ is empty, otherwise rename. Hence $A \cap S = S$, or equivalently: $S \subset A$. By disjointness of $A$ and $B$ this means that  $S \subset Cl(S) \setminus B$. But since $B$ is open, $Cl(S) \setminus B$ is still closed, so that

$$
  (S \subset Cl(S) \setminus B)
  \Rightarrow
  (Cl(S) \subset Cl(S) \setminus B)
  \,.
$$

This means that $B = \emptyset$, and hence that $Cl(S)$ is connected.

=--

+-- {: .num_prop #ConnectedComponentsAreClosed}
###### Proposition
**(connected components are closed)**

Let $(X,\tau)$ be a [[topological space]]. Then its connected components (def. \ref{ComponentsConnected}) are [[closed subsets]].

=--

+-- {: .proof}
###### Proof

By definition, the connected components are [[maximal elements]] in the set of connected subspaces [[preorder|pre-ordered]] by inclusion. By prop. \ref{ClosureOfConnectedSubspaceIsConnected} this means that they must contain their closures, hence they must equal their closures.

=--

+-- {: .num_remark}
###### Remark

Prop. \ref{ConnectedComponentsAreClosed} implies that when a space has a [[finite set]] of connected components, then they are not just closed but also open, hence [[clopen subsets]] (because then each is the [[complement]] of a finite union of closed subspaces).

For a non-finite set of connected components this remains true if the space is [[locally connected topological space|locally connected]]. See [this prop.](locally+connected+topological+space#AlternativeCharacterizationsOfLocalConnectivity)

=--

## Related concepts

* [[locally connected topological space]]

* [[connected topos]], [[locally connected topos]]

* [[cohesive topos]]

* [[contractible topological space]]

* [[almost connected topological group]]

* A space in which the only connected subspaces are the singletons and the empty set is called [[totally disconnected space]].

* The connected subsets of a space form a [[connectology]].

\linebreak

* [[simply connected topological space]]

* [[simple topological space]]

* [[nilpotent topological space]]

* [[compact connected space]]

## References

Examples of countable connected Hausdorff spaces were give in

* {#Bing} R.H. Bing, A connected countable Hausdorff space, Proc. Amer. Math. Soc. 4 (1953), 474.

* {#Golomb} Solomon W. Golomb, _A Connected Topology for the Integers_, Amer. Math. Monthly, Vol. 66 No. 8 (Oct. 1959), 663-665.

Discussion of connectedness in [[constructive mathematics]] is in

* {#Taylor10} [[Paul Taylor]], Def. 13.2 of: *A lambda calculus for real analysis*, In: Journal of Logic & Analysis **2** 5 (2010) 1–115 $[$[L&A:63/25](http://logicandanalysis.org/index.php/jla/article/view/63/25), [pdf](http://logicandanalysis.org/index.php/jla/article/download/63/25), [webpage](https://www.paultaylor.eu/ASD/lamcra/)$]$

Related discussion in [[cohesive homotopy type theory]]:

* {#Shulman18} [[Mike Shulman]], Thm. 11.1, 11.3 *Brouwer’s fixed-point theorem in real-cohesive homotopy type theory*, Mathematical Structures in Computer Science **28** 6 (2018) 856-941 $[$[arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147)$]$

Material on arc-connected spaces and the Hahn-Mazurkiewicz theorem can be found in Chapter 31 of

* {#Willard} Stephen Willard, _General Topology_, Addison-Wesley 1970. ([online](http://www.scribd.com/doc/132023793/44283132-General-Topology-Stephen-Willard#scribd))

Material on pseudo-arcs can be found in

* {#Bing2} R.H. Bing, _Concerning hereditarily indecomposable continua_, Pacific J. Math. Volume 1, Number 1 (1951), 43-51. ([Project Euclid](http://projecteuclid.org/euclid.pjm/1102613150))

[[!redirects compact connected]]
[[!redirects compact connectedness]]
[[!redirects compact connected space]]
[[!redirects compact connected spaces]]

[[!redirects connected space]]
[[!redirects connected spaces]]
[[!redirects connected topological space]]
[[!redirects connected topological spaces]]

[[!redirects path-connected space]]
[[!redirects path-connected space]]
[[!redirects path-connected spaces]]

[[!redirects path connected space]]
[[!redirects path connected topological space]]
[[!redirects path connected spaces]]

[[!redirects pathwise connected space]]
[[!redirects pathwise connected spaces]]
[[!redirects pathwise connected topological space]]
[[!redirects pathwise connected topological spaces]]
[[!redirects path-connected topological space]]
[[!redirects path-connected topological spaces]]
[[!redirects pathwise connected]]

[[!redirects connected component]]
[[!redirects connected components]]

[[!redirects path component]]
[[!redirects path components]]

[[!redirects path-component]]
[[!redirects path-components]]


connected sum

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Manifolds and cobordisms
+-- {: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[differential topology]] the _connected sum_ of two [[manifolds]] equipped with a common [[submanifold]] is the result of gluing them along their [[boundaries]] of the [[tubular neighbourhoods]] of this submanifold.

Often this concept is considered by default for the case that the given submanifold is the [[point space]]. In this case the correspinding neighbourhoods are [[balls]] with [[boundary]] [[n-spheres]] along which the two manifolds are being glued.

## Properties

+-- {: .num_prop #EulerCharacteristicOfConnectedSums}
###### Proposition
**([[Euler characteristic]] of [[connected sums]])**

Let $X$ and $Y$ be [[closed manifolds]] ([[topological manifold|topological]] or [[differentiable manifold|differentiable]])  of [[even number|even]] [[dimension]]. Then the [[Euler characteristic]] of their [[connected sum]] is the [[sum]] of their separate [[Euler characteristics]], minus 2:

$$
  \chi
  \big(
    X \sharp Y
  \big)
  \;=\;
  \chi
  \big(X\big)
  +
  \chi
  \big(Y\big)
  -2
$$

=--

(e.g. [here](http://math.ucr.edu/~res/miscpapers/csums+echars.pdf))

## Related entries

* [[exotic smooth structure]]

* [[twisted connected sum]]

* [[twisted connected sum G2-manifolds]]

## References

See also

* Wikipedia, _[Connected sum](https://en.wikipedia.org/wiki/Connected_sum)_

[[!redirects connected sums]]