sub-(infinity,1)-category

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# Contents
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## Idea

The analog of the notion of [[subcategory]] for [[(∞,1)-categories]].

## Definition

Say that an [[equivalence of quasi-categories|equivalence of (∞,1)-categories]] $D \stackrel{\simeq}{\to} C$ exhibits $D$ as a **0-subcategory** of $C$.

Then define recursively, for $n \in \mathbb{N}$:

an **$n$-subcategory** of an $(\infty,1)$-category $D$ for $n \geq 1$ is an [[(∞,1)-functor]]

$$
  F : D \to C
$$

such that for all objects $x,y \in D$ the component-$(\infty,1)$-functor on the [[hom-object in a quasi-category|hom-objects]]

$$
  F_{x,y} : Hom_D(x,y) \to Hom_C(F(x), F(y))
$$

exhibits an $(n-1)$-subcategory.

## Special cases

### 1-Subcategory / full subcategory

A **full subcategory** is a 1-subcategory, exhibited by a [[full and faithful (∞,1)-functor]].

#### For $sSet$-enriched and quasi-categories

Let $C$ and $D$ be incarnated specifically as [[model structure on sSet-categories|fibrant]] [[simplicially enriched categories]]. Then for $F : D \to C$ a full and faithful $(\infty,1)$-functor, choose in each preimage $F^{-1}(c)$ for each object $c \in C$ a representative, and let $C'$ be the full [[sSet]]-[[enriched category|enriched]] [[subcategory]] on these representatives.

Then the evident projection functor $D \stackrel{\simeq}{\to} D'$ is manifestly an [[equivalence of quasi-categories|equivalence]] and the original $F : D \to C$ factors as

$$
  F : D \stackrel{\simeq}{\to} D' \hookrightarrow C
  \,,
$$

where the second morphism is an ordinary inclusion of objects and hom-complexes.

#### Reflective sub-$(\infty,1)$-categories

If the $(\infty,1)$-functor $F : D \to C$ has a left [[adjoint (∞,1)-functor]] $L : C \to D$, then $F$ is full and faithful and hence exhibits a 1-subcategory precisely if the counit

$$
  L \circ F \stackrel{}{\to} Id_D
$$

is an [[equivalence in a quasi-category|equivalence]] of [[(∞,1)-functor]]s. (See also [[Higher Topos Theory|HTT, p. 308]]).

In this case $D$ is a [[reflective (∞,1)-subcategory]].

### 2-Subcategory

#### For $sSet$-enriched and quasi-categories

Let the $(\infty,1)$-categories $C$ and $D$ concretely be incarnated as [[model structure on sSet-categories|fibrant]] [[simplicially enriched categories]].

Write $h C := Ho(C)$ and $h D := Ho(D)$ for the corresponding [[homotopy category of an (∞,1)-category]] (hom-wise the connected components of the corresponding [[simplicially enriched category]]).

Let $h D \to h C$ be a [[faithful functor]]. Then if we have a [[pullback]] in [[sSet]]-[[Cat]]

$$
  \array{
    D &\to& C
    \\
    \downarrow && \downarrow
    \\
    h D &\to& h C
  }
$$

$D$ is a 2-sub-$(\infty,1)$-category of $C$.
This pullback manifestly produces the simplicially enriched category whose

* objects are those of $h D$;

* hom-complexes are precisely the unions of those connected components of the hom-complexes of $C$ whose equivalence class is present in $h D$.

Therefore the inclusion functor $D \to C$ is on each hom-complex a [[full and faithful (∞,1)-functor]]. Hence this identifies $D$ as a 2-subcategory of $C$.

If $h D \to h C$ is an inclusion on equivalence classes of objects then this is the definition of subcategory of an $(\infty,1)$-category that appears in [[Higher Topos Theory|HTT, section 1.2.11]].

#### As 2-subobjects {#2Subobjects}

Let $core(Set_*) \to core(Set)$ be the 2-subobject classifier in the [[(∞,1)-topos]] [[∞Grpd]]. Then
for $C \in \infty Grpd$ a 1-subobject is classified by an $\infty$-functor
$C \to Set$. This factors through the homotopy category of $C$ as $C \to h C \to Set$. Since $Set_* \to Set$ is the universal faithful functor, the pullback

$$
  \array{
    Q &\to& Set_*
    \\
    \downarrow && \downarrow
    \\
    h C &\to& Set
  }
$$

gives an ordinary subcategory of $h C$. This means that the total pullback $D \to C$

$$
  \array{
    D &\to& h D &\to& Set_*
    \\
    \downarrow && \downarrow && \downarrow
    \\
    C &\to& h C &\to& Set
  }
$$

gives a 2-sub-$(\infty,1)$-category $K$ of $X$ (where both happen to be $\infty$-groupoids) here.

## References

What we call a 2-subcategory of an $(\infty,1)$-category appears in  section 1.2.11 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_

[[!redirects sub-quasi-category]]
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[[!redirects sub-quasicategory]]
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[[!redirects sub-(infinity,1)-category]]
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sub-(infinity,1)-category - internal formulation

[[!redirects sub-(infinity,1)-catgeory - internal formulation]]
[[!redirects sub-(∞,1)-catgeory - internal formulation]]

> under construction

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## Idea

The following gives a characterization of full [[sub-(∞,1)-categories]] of an [[(∞,1)-topos]] entirely in terms of its [[internal logic of an (∞,1)-topos|internal logic]]/[[homotopy type theory]].

## Background and subtleties

(...)

## Statement
 {#Statement}

(...)

[[!redirects sub-(∞,1)-category - internal formulation]]


sub-bundle

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## Definition

A _sub-bundle_ is a [[subobject]] in a [[category]] of [[bundles]].

If $\left[ E \overset{fb}{\to} X\right]$ is a [[bundle]], then a sub-bundle is a [[monomorphism]]

$$
  S \hookrightarrow E
$$

in the given [[slice category]] over $X$, hence

$$
  \array{
    S && \hookrightarrow && E
    \\
    & {}_{\mathllap{\exists!}}\searrow && \swarrow_{\mathrlap{fb}}
    \\
    && X
  }
$$

## Examples

* Given a [[real vector bundle]] with [[orthogonal structure]], then the sub-bundle whose [[fibers]] are the [[unit spheres]] is called the [[unit sphere bundle]] of the vector bundle.

[[!redirects sub-bundles]]

[[!redirects subbundle]]
[[!redirects subbundles]]


subcanonical coverage

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## Definition

A [[coverage]] (resp. [[Grothendieck topology]], resp. [[Grothendieck pretopology]]) defining a [[site]] is called **subcanonical** if all [[representable functor|representable presheaves]] on this site are [[sheaf|sheaves]].  Of course, a **subcanonical site** is one whose coverage is subcanonical.

The term "subcanonical" comes about because the largest coverage for which the representables are sheaves is called the **[[canonical coverage]]**, and the subcanonical coverages are precisely the "sub-coverages" of the canonical one.

## Effective-epimorphic sieves ##

An alternate definition is that a Grothendieck coverage is subcanonical if and only if all of its covering [[sieve|sieves]] $R\hookrightarrow C(-,U)$ are **effective-epimorphic**, meaning that the morphisms $f:V\to U$ in $R$ form a colimit cone under the diagram consisting of _all_ morphisms between them over $U$.  To see this, first recall that if $R\hookrightarrow C(-,U)$ is a sieve, then a functor $X:C^{op}\to Set$ satisfies the [[sheaf]] axiom for $R$ if and only if

* for every family $(x_f)_{f\in R}$ which is compatible, in the sense that $X(g)(x_f) = x_{f g}$ whenever this makes sense, there exists a unique $x\in X(U)$ such that $x_f = X(f)(x)$.

Interpreting this when $X$ is a representable functor $C(-,Z)$, we obtain

* for every family of maps $(h_f:V\to Z)$, where $f:V\to U$ is in $R$, such that $h_f g = h_{f g}$ for any $g:V'\to V$, there exists a unique $k:U\to Z$ such that $h_f = k f$.

But this says precisely that $R$ is effective-epimorphic, as defined above.

In fact, since the covering sieves in a subcanonical coverage must also satisfy pullback-stability, they must be not only   effective-epimorphic but __universally effective-epimorphic__ (meaning that any pullback of them is effective-epimorphic).  It is then easy to see that the canonical coverage consists precisely of all the universally effective-epimorphic sieves.

Note also that if $f:V\to U$ is a single morphism having a [[kernel pair]] $p,q:V\times_U V \;\rightrightarrows\; V$, then the sieve generated by $f$ is effective-epimorphic iff $f$ is the [[coequalizer]] of its kernel pair, and thus iff $f$ is an [[effective epimorphism]].

## Examples {#Examples}

See at [[coverage#Examples|coverage]].

## Related concepts

* [[canonical topology]]

* [[trivial topology]]

[[!redirects subcanonical pretopology]]
[[!redirects subcanonical site]]
[[!redirects subcanonical topology]]

[[!redirects subcanonical Grothendieck topology]]
[[!redirects subcanonical Grothendieck topoloies]]


subcanonical pretopology > history

< [[subcanonical pretopology]]
[[!redirects subcanonical pretopology -- history]]


subcategory

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# Subcategories
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## Definition

Given a [[category]] $C$, a __subcategory__ $D$ consists of a subcollection of the collection of [[object|objects]] of $C$ and a subcollection of the collection of [[morphism|morphisms]] of $C$ such that:

* If the morphism $f : x \to y$ is in $D$, then so are $x$ and $y$.

* If $f : x \to y$ and $g : y \to z$ are in $D$, then so is the [[composition|composite]] $g f : x \to z$.

* If $x$ is in $D$ then so is the [[identity morphism]] $1_x$.

These conditions ensure that $D$ is a category in its own right and the inclusion $D\hookrightarrow C$ is a [[functor]].  Additionally, we say that $D$ is...

* A __[[full subcategory]]__ if for any $x$ and $y$ in $D$, every morphism $f : x \to y$ in $C$ is also in $D$ (that is, the inclusion [[functor]] $D\hookrightarrow C$ is [[full functor|full]]).

* A __[[replete subcategory]]__ if for any $x$ in $D$ and any isomorphism $f:x\cong y$ in $C$, both $y$ and $f$ are also in $D$.

* A __[[wide subcategory]]__ if every object of $C$ is also an object of $D$.

## Variants in accord with the principle of equivalence
 {#VariantsInAccordWithThePrincipleOfEquivalence}

Just as subsets of a [[set]] $X$ can be identified with isomorphism classes of [[monomorphism|monic]] functions into $X$, subcategories of a category $C$ can be identified with isomorphism classes of monic _functors_ into $C$.  A functor is easily verified to be monic iff it is [[faithful functor|faithful]] and injective on objects.  This can be generalized to monomorphisms in a [[strict 2-category]].

However, this notion violates the [[principle of equivalence]] since being injective-on-objects refers to equality of objects.  This raises the question: what is a good definition of **subobject** in a [[2-category]] in accord with the [[principle of equivalence]]?  It is the contention of the authors of this page that there are multiple such definitions.  Two evident ones are:

* A morphism $f: A\to B$ in a 2-category $K$ is **1-monic** if it is [[fully faithful|full and faithful]], i.e. $K(X,A) \to K(X,B)$ is full and faithful for all $X$.  A **1-subobject** of $B$ is an [[equivalence]] class of 1-monomorphisms into $B$, and a **1-subcategory** is a 1-subobject in $Cat$.
* Likewise, $f$ is **2-monic** if $K(X,A) \to K(X,B)$ is faithful for all $X$.  A **2-subobject** of $B$ is an equivalence class of 2-monomorphisms into $B$, and a **2-subcategory** is a 2-subobject in $Cat$.

The obvious generalizations (at least, obvious once you start thinking in terms of $k$-[[k-surjective functor|surjectivity]]) are that every morphism is 3-monic, while the 0-monic morphisms are the equivalences.  (Note that this numbering is offset by one from that used in [Baez and Shulman](http://arxiv.org/abs/math.CT/0608420).)  There is likewise an evident generalization to $k$-monomorphisms in any $n$-[[n-category|category]].

It is fairly undisputed that 1-subobjects, as defined above, are a good notion of subobject in a 2-category.  In particular, any [[full and faithful functor]] $C\to D$ in $Cat$ is [[equivalence of categories|equivalent]] to the inclusion of a [[full subcategory]] $C'\to D$ (here $C'$ is the full image of $C$).  Also, in a 1-category considered as a locally discrete 2-category, the 1-monomorphisms are precisely the usual sort of [[monomorphism]].

In fact, any faithful functor is likewise equivalent to the inclusion of a (non-full) subcategory, but in this case the codomain must be modified as well as the domain.  It is somewhat more disputable whether 2-subcategories all deserve to be called "subcategories;" for instance, is [[Grp]] a "subcategory" of [[Set]]?  Note also that any functor between [[discrete category|discrete categories]] is faithful, so that the [[terminal category]] has a proper class of inequivalent 2-subcategories, and similarly every morphism in a locally discrete 2-category is 2-monic.  However, kernels of morphisms between 2-groups are 2-subobjects, not 1-subobjects, and likewise for any subgroup of a [[group]] (considered as a 1-object category).  This motivates the term "2-subobject," to make it clear that there is some relationship with the sort of subobjects we are used to in 1-categories, but also some notable generalization.

Other types of morphism in a 2-category which have some claim to be considered "subobjects" include [[pseudomonic functor|pseudomonic]] morphisms and [[conservative functor|conservative]] morphisms.  Pseudomonic morphisms might merit a name such as **(2,1)-subcategory**, since a functor is pseudomonic iff it is faithful (a 2-subcategory) and its induced functor between [[underlying groupoid]]s is fully faithful (a 1-subcategory).  See also [[stuff, structure, property]].

### As Displayed Categories

A different way to describe subcategories is as [[displayed categories]] satisfying certain properties. This is analogous to defining a [[subset]] by its [[characteristic function]], which allows to define subsets without a notion of global membership.

A __subcategory__ of a category $C$ is a displayed category $D$ over $C$ such that all $D(c)$ and $D_f(A,B)$ are [[mere proposition|propositions]]. We can recover a subcategory in the original sense by taking the [[Grothendieck construction]] of $D$. The identity and composition in the displayed category correspond precisely to the inclusion of the subcategory being closed under identity and composition.

We can also capture the above flavors of subcategory by adding further conditions to $D$. We say $D$ is...

* A __full subcategory__ if all $D_f(A,B)$ are true.
* A __wide subcategory__ if all $D(c)$ are true.
* A __replete subcategory__ if $D$ is an [[isofibration]].

## Related concepts

[[!include notions of subcategory]]

* [[embedding of categories]]

[[!redirects subcategory]]
[[!redirects subcategories]]
[[!redirects inclusion functor]]


subdivision

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## Idea

**Subdivision** is an (often) functorial process which takes as input some combinatorial notion of space (for example, a [[simplicial complex]] or [[simplicial set]]) and produces as output a more finely meshed space. It is related to the notion of [[classical triangulation|classical subdivision]].

There is a more general use of the term in which one simplicial set or complex is a subdivision of another, although the relation may not be functorially generated.  We will briefly look at one such example later.

There are various standard forms of functorial subdivision, the most common of which is **barycentric subdivision**.

## Definition

### For simplicial complexes

Barycentric subdivision is easiest to define for [[simplicial complexes]].  We have a pair of [[functors]]

$$SimpComplex \stackrel{\overset{U}{\to}}{\underset{Flag}{\leftarrow}} Pos$$

where

* The functor $U$ sends a [[simplicial complex]] $(V, \Sigma)$ to $\Sigma$, regarded as a [[poset]] ordered by inclusion, and

* The functor $Flag$ sends a poset $P$ to the simplicial complex whose vertex set is $P$ and whose [[simplex|simplices]] are the underlying sets $\{x_0, \ldots, x_n\}$ of flags $x_0 \lt \ldots \lt x_n$.

The composite $Flag \circ U$ is called the **subdivision** $Sd$; it is an endofunctor of $SimpComplex$.  A vertex of $Sd(X)$ is a simplex of $X$.

Let $\alpha_X : X \to Sd(X)$ be a morphism of simplicial complexes that sends a vertex $v$ of $X$ to the vertex $\{v\}$ of $Sd(X)$. Then ${|f|}: {|X|}\xrightarrow{\cong} {|Sd(X)|}$ is an isomorphism, where $|-|$ is the usual [[geometric realization]] of simplicial complexes. In terms of categories, $\alpha$ is a natural transformation from the identity to the endofunctor $Sd$ whose geometric realization is a natural isomorphism.

This is what it looks like for $X$ the 2-simplex.

[[sdDelta2r.jpg:pic]]

The simplicial complex which is the 2-simplex has as its poset of simplices the partially ordered set of non-empty subsets of $\{0,1,2\}$.  The subdivided simplex therefore is the flags of that poset. The diagram shows that nerve of the corresponding category, (or rather its dual), as this is the simplicial set that one gets from this. The central vertex is the whole set $\{0,1,2\}$, the vertices on the long edges are the 2-element subsets and the three extremal vertices are the singletons.

### For simplicial sets

We can now define the barycentric subdivision of a [[simplicial set]] as follows.  [Recall](/nlab/show/simplicial+complex#vsSSet) that we can make any simplicial complex into a simplicial set.  If we do this for the standard $n$-simplex, which as a simplicial complex is the set $([n],P([n])$ for $[n] = \{0,1,\dots,n\}$, then we get the standard $n$-simplex simplicial set $\Delta^n$.  We can then define the subdivision of $\Delta^n$, as a simplicial set, to be the simplicial set corresponding to its simplicial-complex subdivision.  Finally, we can make this functorial on maps between standard simplices, and left [[Kan extension|Kan extend]] to a [[cocontinuous functor|cocontinuous]] endofunctor of $SSet$.

Explicitly:

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

Write $Sd \Delta[n]$ for the [[nerve]] of [[category of non-degenerate simplices]] in the standard $n$-[[simplex]].

For $X$ an arbitrary [[simplicial set]], its barycentric subdivision is

$$
  Sd X \simeq \underset{\to}{\lim}_{\Delta[n] \to X} Sd \Delta[n]
  \,.
$$

=--

A review is for instance around ([Fiore-Paoli def. 3.1](#FiorePaoli)).

## Properties

### Adjointness with the $Ex$-functor

By construction (or the [[adjoint functor theorem]]), the subdivision functor $Sd$ on simplicial sets has a [[right adjoint]], called $Ex$.  A countably infinite iteration of $Ex$, called $Ex^\infty$, can be used to construct [[Kan fibrant replacements]] in the [[model structure on simplicial sets]].

This functorial subdivision corresponds to the classical [[barycentric subdivision]].

##Other types of subdivision

Other [[classical triangulation|classical subdivision]]s that are frequently encountered include the middle edge subdivision.  This latter is closely related to the [[ordinal subdivision]] of simplicial sets.

## Relation to category of simplices
 {#RelationToCategoryOfSimplices}

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

If every non-degenerate simplex in $X$ is given by a [[monomorphism]] $\Delta^n \to X$, then the barycentric subdivision of def. \ref{ColimitFormulaForBarycentricSubdivision} is equivalently given by the [[nerve]] of the [[full subcategory]] of its [[category of simplices]] on the non-degenerate simplices.

=--

([Thomason, p. 311](#Thomason))

## References

* [[Rick Jardine]], _Simplicial approximation_, Theory and Applications of Categories, Vol. 12, 2004, No. 2, pp 34-72. ([web](https://www.emis.de/journals/TAC/volumes/12/2/12-02aabs.html))

A review is in section 3 of

* [[Thomas Fiore]], [[Simona Paoli]], _A Thomason model structure on the category of small $n$-fold categories_  ([arXiv:0808.4108](http://arxiv.org/abs/0808.4108))
 {#FiorePaoli}

The relation to the [[nerve]] of the [[category of simplices]] is discussed in

* R. W. Thomason, _Cat as a closed model category_, Cahiers Topologie Géom. Différentielle, 21(3):305–324, 1980.
 {#Thomason}

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subfield

## Definition

A [[subobject]] in the [[category of fields]], a [[subset]] of a [[field]] that inherits itself the structure of a field.

## Related concepts

* [[Galois theory]]

* [[field extension]]

[[!redirects subfield]]
[[!redirects subfields]]

[[!redirects sub-field]]
[[!redirects sub-fields]]


subformula property

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# Subformula property
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## Idea

The **subformula property** is a property of [[cut elimination|cut-free]] presentations of the [[sequent calculus]]. It says that in any derivation

$$\frac{\vdots}{\Gamma\vdash \Delta}$$

any type appearing above the line is a *subformula* of one of the types appearing in $\Gamma,\Delta$. This makes it easy to make arguments about provability: while it is not obvious in a cut-full system that the contradiction/empty sequent $\cdot\vdash\cdot$ is not derivable, it is a corollary of the subformula property that $\cdot \vdash \cdot$ is not derivable because all of the rules of the cut-free calculus involve a type below the line.

This property is a formal expression of the *modularity* of type theoretic connectives: each connective is given by rules which involve only its arguments.


subframe

A __subframe__ is a [[subobject]] in the category [[Frm]] of [[frames]].  More explicitly, if $L$ is a frame, then a __subframe__ of $L$ is a [[subset]] $M$ of the [[underlying set]] of $L$ such that $M$ is closed under arbitrary [[joins]] and finitary [[meets]] (including having the [[bottom]] and [[top]] elements).

In the correspondence between frames and [[locales]], a subframe corresponds to a kind of [[quotient object|quotient]] locale.  However, only [[regular epimorphism|regular]] quotients of locales behave as [[quotient space]]s for the purposes of [[topology]].  These correspond to [[regular subobject|regular]] subframes; not all subframes are regular.

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subfunctor

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## Definition

A **subfunctor** is a [[subobject]] in a [[functor category]].

A subfunctor of a [[functor]] $G:C\to D$ between [[categories]] $C$ and $D$ is a pair $(F,i)$ where $F:C\to D$ is a functor and $i:F\to G$ is a [[natural transformation]] such that its components $i_M:F(M)\to G(M)$ are [[monomorphism|monic]].

In fact one often by a subfunctor means just an [[equivalence class]] of such monic natural transformations; compare [[subobject]].

A subfunctor is also called a **sub[[presheaf]]** . A subfunctor of a [[representable functor]] $Hom(-,x)$ is precisely a [[sieve]] over the representing object $x$.

## Properties

In a [[concrete category]] with [[image]]s one can choose a representative of a subfunctor where the components of $i$ are genuine inclusions of the underlying sets; then a subfunctor is just a natural transformation whose components are inclusions.  The naturality in terms of concrete inclusions just says that for all $f:c\to d$, $F(f)=G(f)|_{F(c)}$. If the set-theoretic circumstances allow consideration of a [[functor category|category of functors]], then a subfunctor is a [[subobject]] in such a category.

A *subfunctor $(F,i)$ of the identity* $id_C:C\to C$ in a category with images is an often used case: it amounts to a natural assignment $c\mapsto F(c)\stackrel{i}\hookrightarrow c$ of a subobject to each object $c$ in $C$. For concrete categories with images then $F(f)=f|_{F(c)}$.

[[!redirects subfunctors]]

[[!redirects subpresheaf]]
[[!redirects subpresheaves]]


subgroup

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## Idea

A _subgroup_ of a [[group]] $G$ is a "smaller" group $K$ sitting inside $G$.

## Definition

A _subgroup_ is a [[subobject]] in the [[category]] [[Grp]] of [[group]]s: a [[monomorphism]] of groups

$$
  K \hookrightarrow G
  \,.
$$

Here $K$ is _a subgroup of $G$_.

## Special cases

* [[normal subgroup]]

* [[stabilizer subgroup]]

* [[maximal compact subgroup]]

* [[torsion subgroup]]

* [[Borel subgroup]]

* [[parabolic subgroup]]

* [[generalized Fitting subgroup]]

## Properties

### Of free groups

Every subgroup of a [[free group]] is itself free. This is the statement of the _[[Nielsen-Schreier theorem]]_.

### Of Lie groups

For $H \hookrightarrow G$ a sub-[[Lie group]] inclusion write $\mathbf{B}H  \to \mathbf{B}G$ for the induced map on [[delooping]] [[Lie groupoids]]. The [[homotopy fiber]] of this map (in [[Smooth∞Grpd]]) is the [[coset space]] $G/H$: there is a [[homotopy fiber sequence]]

$$
  G/H \to \mathbf{B}H \to \mathbf{B}G
  \,.
$$

Now let $H \hookrightarrow K \hookrightarrow G$ be a sequence of two subgroup inclusions. By the above this yields the [[diagram]]

$$
  \array{
    K/H &\to& G/H &\to& G/K
    \\
    \downarrow && \downarrow && \downarrow
    \\
    \mathbf{B}H &\to& \mathbf{B}H &\to& \mathbf{B}K
    \\
    \downarrow && \downarrow && \downarrow
    \\
    \mathbf{B}K &\to& \mathbf{B}G &\to& \mathbf{B}G
  }
$$

## Examples

* [[centralizer subgroup]]

* [[normalizer subgroup]]

* [[stabilizer subgroup]]

* [[conjugate subgroup]]

* [[congruence subgroup]]

* [[Young subgroup]]

## Related concepts

* [[lattice of subgroups]]

* [[maximal subgroup]]

* [[index of a subgroup]]

* [[Klein geometry]], [[Cartan geometry]]

* [[BN-pair]]

* [[subring]], [[submodule]]

## References

Discussion in [[univalent foundations of mathematics]] ([[homotopy type theory]] with the [[univalence axiom]], but for 1-groups):

* [[Martín Escardó]], *[Subgroups](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html#subgroups-sip)*, §33.12 in: *Introduction to Univalent Foundations of Mathematics with Agda* [[arXiv:1911.00580](https://arxiv.org/abs/1911.00580),  [webpage](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html)]

  > (in [[Agda]])

* [[Marc Bezem]], [[Ulrik Buchholtz]], [[Pierre Cagne]], [[Bjørn Ian Dundas]], [[Daniel R. Grayson]]: Chapter 5 of: *[[Symmetry]]* (2021) $[$[pdf](https://unimath.github.io/SymmetryBook/book.pdf)$]$

See also

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

[[!redirects subgroup]]
[[!redirects subgroups]]


subgroup series

Stub

* [wikipedia entry](http://groupprops.subwiki.org/wiki/Subgroup_series)


Subir Sachdev

* [webpage](http://sachdev.physics.harvard.edu/)

## Selected writings

Introducing the [[Sachdev-Ye-Kitaev model]]:

* {#SachdevYe93} [[Subir Sachdev]], Jinwu Ye, _Gapless Spin-Fluid Ground State in a Random Quantum Heisenberg Magnet_, Phys. Rev. Lett. 70:3339, 1993 ([arXiv:cond-mat/9212030](https://arxiv.org/abs/cond-mat/9212030))

On [[AdS/CFT in condensed matter physics]]:

* {#HartnollLucasSachdev16} [[Sean Hartnoll]], [[Andrew Lucas]], [[Subir Sachdev]], _Holographic quantum matter_, MIT Press 2018 ([arXiv:1612.07324](https://arxiv.org/abs/1612.07324), [publisher](https://mitpress.ublish.com/book/holographic-quantum-matter))

On the [[quantum dimer model]]:

* [[Subir Sachdev]], *[Quantum dimer model](http://qpt.physics.harvard.edu/qdimer.html)*

## Related $n$Lab entries

* [[AdS-CFT in condensed matter physics]]

* [[Sachdev-Ye-Kitaev model]]

category: people


Subir Sarkar

* [webpage](https://www2.physics.ox.ac.uk/contacts/people/sarkar)

## Selected writings

On [[inhomogeneous cosmology]] as a possible explanation for apparent [[dark energy]]:

* {#Sarkar08} [[Subir Sarkar]], _Is the evidence for dark energy secure?_, Gen. Rel. Grav.40:269-284, 2008 ([arXiv:0710.5307](https://arxiv.org/abs/0710.5307))

* {#SarkarEtAl18} J. Colin, [[Roya Mohayaee]], [[Mohamed Rameez]], [[Subir Sarkar]], _Evidence for anisotropy of cosmic acceleration_, Astronomy & Astrophysics Letters ([arXiv:1808.04597](https://arxiv.org/abs/1808.04597))

* {#Sarkar18} [[Subir Sarkar]], _Is the universe isotropic?_, talk at _[Current Themes in High Energy Physics and Cosmology 2018](https://indico.nbi.ku.dk/event/973/)_ ([pdf](https://indico.nbi.ku.dk/event/973/contributions/8344/attachments/2648/3858/Sarkar_Copenhagen18_cosmo.pdf))

* {#ColinMoymhayeeRamezSarkar19} Jacques Colin, [[Roya Mohayaee]], [[Mohamed Rameez]], [[Subir Sarkar]], _A response to Rubin & Heitlauf: "Is the expansion of the universe accelerating? All signs still point to yes"_ ([arXiv:1912.04257](https://arxiv.org/abs/1912.04257))

On the evidence for [[dark energy]] in [[cosmology]]:

* {#MRS21} [[Roya Mohayaee]], [[Mohamed Rameez]], [[Subir Sarkar]], *Do supernovae indicate an accelerating universe?* ([arXiv:2106.03119](https://arxiv.org/abs/2106.03119))

On evidence for [[inhomogeneous cosmology]] (large anisotropies) in observed radio galaxies and quasars:

* {#SHRMS22} Nathan Secrest, Sebastian von Hausegger, [[Mohamed Rameez]], [[Roya Mohayaee]], [[Subir Sarkar]], *A Challenge to the Standard Cosmological Model*, ApJ Letters (2022) [[arXiv2206.05624](https://arxiv.org/abs/2206.05624)]

## Related $n$Lab entries

* [[inhomogeneous cosmology]]

* [[dark energy]]

category: people


subjective idealism

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

## Related entries

* [[doctrine of ideas]]

* [[idealism]]

  * [[objective idealism]]

  * [[absolute idealism]]

## References

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


sublocale

# Sublocales
* table of contents
{: toc}

## Idea

A sublocale is a [[subspace]] of a [[locale]].

It is important to understand that, even for a [[topological locale]] $X$ (which can be identified with a [[sober space|sober]] [[topological space]]), most sublocales of $X$ are *not* topological.  Specifically, we have an [[inclusion function]] $Sub Top(X) \hookrightarrow Sub Loc(X)$ which, while [[injection|injective]], is usually far from [[surjection|surjective]].

Sublocales can also be defined for [[formal topologies]].

## Definitions

There are multiple equivalent ways to formally define this concept.

### In terms of regular monomorphisms of locales

Let $L$ be a [[locale]], which (as an [[object]]) is the same as a [[frame]].

A __sublocale__ of $L$ is a [[regular subobject]] of $L$ in [[Loc]], the [[category]] of locales.  Equivalently, it is a [[regular quotient]] of $L$ in [[Frm]], the category of [[frames]].

Equivalently, a sublocale of $L$ can be described as a [[morphism of frames]]
$L\to S$ whose underlying map of sets is surjective.

### In terms of subsets of opens

Given a [[regular monomorphism]] $i\colon S\to L$,
it induces an adjunction $f^*\dashv f_*$ between the underlying [[posets]].
The direct image map
$$f_*: S\to L$$
is an injection on underlying sets
such that if we equip its image $f_*(S)$ with the induced order,
it becomes a [[locale]] itself, and the inclusion $f_*(S)\to L$
is a localic map, i.e., it has a left adjoint that preserves finite meets.

Thus, sublocales of $L$ can be defined as subsets $S$ of $L$
such that $S$ is a [[locale]] once we equip it with the order induced from $L$,
and the left adjoint to the inclusion $S\to L$ exists and preserves finite meets.
(Proposition 2.2 in [Picado and Pultr](#PP).)

(The inclusion $S\to L$ need not preserve joins, so in particular,
the lattice structure on $S$ may be different from that of $L$,
only the ordering is the same.)

Equivalently, sublocales of $L$ can also be described
([2.1](#PP)) as subsets $S$ of $L$
that are closed under all meets and for any $s\in S$ and $x\in L$,
we have $(x\to s)\in S$.

### Equivalence of these two definitions

Given a surjective [[morphism of frames]] $h\colon L\to S$,
the subset $h_*(S)$ is a sublocale of $L$.

Given a subset $S$ of a locale $L$ that is a sublocale,
the left adjoint $j^*\colon L\to S$ to the inclusion $j_*\colon S\to L$
is a surjective [[morphism of frames]].

### In terms of nuclei

A sublocale is given precisely by a __[[nucleus]]__ on the underlying frame.  This is a [[function]] $j$ from the opens of $L$ to the opens of $L$ satisfying the following identities:

1.  $j(U \cap V) = j(U) \cap j(V)$,
2.  $U \subseteq j(U)$,
3.  $j(j(U)) = j(U)$.

In other words, a sublocale of $L$ is given by a [[meet]]-preserving [[monad]] on its frame of opens.

The precise reasons why nuclei correspond to quotient frames (and hence to sublocales) is given at [[nucleus]].  But the interpretation of the operation $j$ is this: we identify two opens if they 'agree on the sublocale'.  Given an open $U$, there will always be a largest open that is identified with $U$, so we can also describe a subspace of a locale as an operation that maps each open to its largest representative open in the sublocale.  This map is the nucleus $j$.

### Equivalence of nuclei and other definitions

Given a nucleus $j$ on a [[frame]] $L$,
we construct a [[congruence]] on $L$
given by the equivalence relation $E$
consisting of all pairs $(x,y)$ ($x,y\in L$)
such that $j(x)=j(y)$.
The quotient map $L\to L/E$ is a surjective [[morphism of frames]].

Vice versa, given a congruence $E$ on $L$,
define $j(x)=\bigvee_{y\colon (x,y)\in E}y$.
Then $j$ is the nucleus corresponding to the sublocale $L\to L/E$.

Given a nucleus $j$, we can restrict its codomain to its set-theoretical image,
obtaining a surjective [[morphism of frames]] $L\to j(L)$.

Vice versa, given a surjective [[morphism of frames]] $h^*\colon L\to S$,
the composition $h_* h^*\colon L\to L$ is a nucleus on $L$.

Given a nucleus $j\colon L\to L$, the subset $j(L)$ is a sublocale.

Vice versa, given a subset $S\subset L$ that is a sublocale,
the corresponding nucleus $j\colon L\to L$
is given by the formula $j(a)=\bigwedge\{s\in S\mid a\le s\}$.

## Special cases

Of course, every locale $L$ is a sublocale of itself.  The corresponding nucleus is given by
$$ j_L(U) \coloneqq U ,$$
so every open is preserved in this sublocale.  Conversely, every locale has an [[empty subspace|empty]] sublocale, given by
$$j_\empty(U) \coloneqq L .$$

Suppose that $U$ is an open in the locale $L$.
Then $U$ defines an [[open subspace]] of $L$, also denoted $U$, given by
$$ j_U(V) \coloneqq U \Rightarrow V ,$$
so $j_U(V)$ is the largest open which agrees with $V$ on $U$.
$U$ also defines a [[closed subspace]] of $L$, denoted $U'$ (or any other notation for a [[complement]]), given by
$$ j_{U'}(V) \coloneqq U \cup V ,$$
so $j_{U'}(V)$ is the largest open which agrees with $V$ except on $U$.
If $L$ is [[topological locale|topological]], then every open sublocale of $L$ is also topological; the same goes for closed sublocales, assuming [[excluded middle]].  Even in [[constructive mathematics]], however, open and closed sublocales are [[complements]] in the lattice of sublocales.

The [[double negation sublocale]] of $L$, denoted $L_{\neg\neg}$, is given by
$$ j_{\neg\neg}(U) \coloneqq \neg{\neg{U}} .$$
This is always a [[dense subspace]]; in fact, it is the *smallest* dense sublocale of $L$.  (As such, even when $L$ is topological, $L_{\neg\neg}$ is rarely topological; in fact, its only points are the [[isolated point]]s of $L$.)

## Relation to topological subspaces

The underlying [[locale]] of a [[topological space]]
will typically have many sublocales that are not [[spatial locale|spatial]],
i.e., do not come from any [[topological space]].

However, any [[subset]] $S$ of a [[topological space]] $X$ can be equipped
with the induced topology, which turns it into a subspace,
whose underlying [[locale]] is a sublocale of $\Omega(X)$.
This can be most easily seen in the language of surjective [[morphisms of frames]],
since open sets in the induced topology are by definition intersections
of $S$ and an open subset of $X$.

Thus, we have a canonical map from the set of (topological) subspaces of $X$
to the set of spatial sublocales of $\Omega X$.

This map is a bijection if $X$ satisfies the [[separation axiom]] $T_D$, e.g., is a [[Hausdorff space]] (VI.1.2 in [PP](#PP)).

This map is a surjection if $X$ is a [[sober space]] (Proposition VI.2.2.1 in [PP](#PP)).

## References

* _[[Stone Spaces]]_, II.2

* [[Jorge Picado]], [[Aleš Pultr]], _Frames and Locales_.
  {#PP}

* [[Steven Vickers]], Sublocales in formal topology. *The Journal of Symbolic Logic*, Volume 72, Issue 2, June 2007, pp. 463 - 482 ([doi:10.2178/jsl/1185803619](https://doi.org/10.2178/jsl/1185803619))

[[!redirects sublocale]]
[[!redirects sublocales]]
[[!redirects localic subspace]]
[[!redirects localic subspaces]]


submanifold

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

#Contents#
* table of contents
{:toc}

## Idea

A _submanifold_ is a [[manifold]] inside another manifold.

## Definition

For a homomorphism of [[differentiable manifolds]]

$$
  X \hookrightarrow Y
$$

to qualify as a submanifold inclusion it is usually required to be an [[embedding of differentiable manifolds]], hence

1. an [[embedding of topological spaces]];

1. an [[immersion of differentiable manifolds]].

## Related concepts

* [[embedding of differentiable manifolds]]

* [[hypersurface]]

## References

See also

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

[[!redirects submanifolds]]


submersion

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Étale morphisms
+--{: .hide}
[[!include etale morphisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Let $X$ and $Y$ be two [[smooth manifolds]] of finite [[dimension]] and let $f : X \to Y$ be a [[differentiable function]] between them

In components, the definition of submersion reads as follows.

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

The function $f : X \to Y$ is called a __submersion__ precisely if its [[differential]] $d f\colon T X \to T Y$ is for every point $x \in X$ a [[surjection]] $d f_x\colon T_x X \to T_{f(x)} Y$; hence if all points in its [[image]] are [[regular values]].

=--

More abstractly formulated, this means equivalently the following.

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

The function  $f : X \to Y$ is a submersion precisely if the canonical morphism

$$
  T X \to X \times_Y T Y \eqqcolon f^* T Y
$$

from the [[tangent bundle]] of $X$ to the [[pullback]] of the tangent bundle of $Y$ along $f$ is a [[surjection]].

=--

This morphism is the one induced by the [[universal property]] of the [[pullback]] from the [[commuting diagram]]

$$
  \array{
    T X &\stackrel{d f}{\to}& T Y
    \\
    \downarrow && \downarrow
    \\
    X &\stackrel{f}{\to}& Y
  }
  \,.
$$

In terms of coordinates, the map $f$ is a submersion at a point $p\colon X$ if and only if there exists a [[coordinate chart]] on $X$ near $p$ and a coordinate chart on $Y$ near $f(p)$ relative to which $f$ is the projection $f(x_1,\ldots,x_n) = (x_1,\ldots,x_m)$.  This definition applies to infinite-dimensional manifolds, to non-differentiable maps, even between non-differentiable manifolds.

## Properties

### Pullbacks

While the [[category]] [[Diff]] of (finite dimensional) [[smooth manifold]]s does not have all [[pullback]]s,  the [[pullback]] along a submersion always exists. This is because a submersion is [[transverse maps|transversal]] to every other smooth map into its codomain. Moreover, submersions are stable under pullback.

### Epimorphisms and coverings

The [[surjection|surjective]] submersions (that is the submersions that are also [[epimorphisms]] in [[Diff]]) are [[regular epimorphism]]s.

Surjective submersions form a singleton [[Grothendieck pretopology]] on [[Diff]], and so may be used in [[internal category|internal]] [[category theory]] when using $Diff$ as the ambient category. They appear notably in the definition of [[Lie groupoids]].

### Ehresmann's theorem

[[Ehresmann's theorem]] states that a [[proper morphism|proper]] submersion is a [[locally trivial bundle|locally trivial]] [[fibration]].

### Normal form

For $f : X \to Y$ a submersion, then around every point of $X$ there is an [[open neighbourhood]] on which $f$ restricts to a [[projection]].

### Characterization in infinitesimal cohesion

A [[smooth function]] $f : X \to Y$ between [[smooth manifolds]] is canonically regarded as a morphism in the [[cohesive (∞,1)-topos]] [[SynthDiff∞Grpd]]. With respect to the canonical [[infinitesimal cohesion|infinitesimal neighbourhood inclusion]] $i : $ [[Smooth∞Grpd]] $\hookrightarrow$ [[SynthDiff∞Grpd]] there is a notion of [[formally smooth morphism]] in $SynthDiff\infty Grpd$.

$f$ is a submersion precisely if it is formally smooth with respect to this infinitesimal cohesion.

See the discussion at [[SynthDiff∞Grpd]] for details.

## Variants

The [[algebraic geometry]] analogue of a submersion is a [[smooth morphism of schemes|smooth morphism]].

The analogue between arbitrary [[topological spaces]] (not manifolds) is simply an [[open map]]. There is also [[topological submersion]], of which there are two versions.

## Related concepts

* [[immersion]], **submersion**, [[local diffeomorphism]]

## References

For instance chapter XIV

* [[Serge Lang]], _Fundamentals of differential geometry_ Springer (1991)

Ehresmann's theorem is due to

* [[Charles Ehresmann]], _Les connexions infinitésimales dans un espace fibré différentiable_, Colloque de Topologie, Bruxelles (1950), 29-55.

[[!redirects submersion]]
[[!redirects submersions]]

[[!redirects surjective submersion]]
[[!redirects surjective submersions]]

[[!redirects submersion of differentiable manifolds]]
[[!redirects submersions of differentiable manifolds]]


submodule

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

#Contents#
* table of contents
{:toc}

## Definition

Thoughout let $R$ be some [[ring]]. Write $R$[[Mod]] for the [[category]] of [[module]] over $R$. Write $U R Mod \to$ [[Set]] for the [[forgetful functor]] that sends a module to its underlying [[set]].

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

For $N \in R Mod$ a module, a **submodule** of $N$ is a [[subset]] of $U(N)$ which

1. is a [[subgroup]] of the underlying group (closed under the addition in $N$);

1. is preserved by the $R$-[[action]].

=--

Equivalently this means:

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

A submodule of $N \in R Mod$ is a module [[homomorphism]] $i : K \to N$ whose underlying map $U(i)$ of sets is an [[injection]].

=--

And since the injections in $R$[[Mod]] are precisely the [[monomorphisms]], this means that equivalently

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

A submodule of $N \in R Mod$ is a [[monomorphism]] $i : K \hookrightarrow N$ in $R$[[Mod]]. Hence a submodule is a [[subobject]] in $R$[[Mod]].

=--

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

Given a submodule $K \hookrightarrow N$, the [[quotient module]] $\frac{N}{K}$ is the [[quotient group]] of the underlying abelian groups.

=--

## Examples

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

For $N = R$ regarded as a [[module]] over itself, a submodule is precisely an [[ideal]] of $R$.

=--

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

For $f : N_1 \to N_2$ a [[homomorphism]] of modules,

1. the [[kernel]] $ker(f) \hookrightarrow N_1$ is a submodule of $N_1$,

1. the [[image]] $im(f) \hookrightarrow N_2$ is a submodule of $N_2$.

=--

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

In example \ref{KernelAndImage} [[quotient module]] of $N_2$ by the image is the [[cokernel]] of $f$

$$
  coker(f) \simeq \frac{N_2}{im(f)}
  \,.
$$

=--

## Properties

### Of free modules

Let $R$ be a [[ring]].

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

Assuming the [[axiom of choice]], the following are equivalent

1. every submodule of a [[free module]] over $R$ is itself free;

1. every [[ideal]] in $R$ is a free $R$-module;

1. $R$ is a [[principal ideal domain]].

=--

A proof is in ([Rotman, pages 650-651](#Rotman)).

## Related concepts

* [[subgroup]], [[subring]]

## References

For instance

* Rotman _Advanced Modern Algebra_
 {#Rotman}

[[!redirects submodules]]


submodules of finitely generated free modules over a PID


submonoid

#Contents#
* table of contents
{:toc}

## Definition ##

A **submonoid** of a [[monoid]] $M$ with unit $1$ is a subset $N$ of $M$ containing $1$ which is also a monoid with respect to the inherited multiplication.

## Examples ##

* Given a group $G$, any subgroup is a submonoid.
* Given a category $C$ and a subcategory $D$, an object $X$ in both, the monoid $\Hom_D(X,X)$ is a submonoid of $\Hom_C(X,X)$.
* A [[commutative]] and [[cancellative]] monoid $M$ is a submonoid of its [[Grothendieck group]] $G(M)$.

## See also ##

* [[GCD ring]]


subnet

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

#Contents#
* table of contents
{:toc}

## Idea

The definition of the concept of sub-nets of a [[net]] requires some care. The point of the definition is to ensure that in [[classical mathematics]], [[compact spaces equivalently have converging subnets|compact spaces are equivalently those for which every net has a converging subnet]].

## Definition

There are several different definitions of '[[subnet]]' in the literature, all of which intend to generalise the concept of subsequences.  We state them now in order of increasing generality.  Note that it is Definition \ref{AA} which is correct in that it corresponds precisely to refinement of filters.  However, the other two definitions (def. \ref{Willard}, def. \ref{Kelley}) are sufficient (in a sense made precise by theorem \ref{EquivalenceOfDefinitionsOfSubnets} below) and may be easier to work with.

+-- {.num_defn #Willard}
###### Definition
(Willard, 1970).

Given a net $(x_{\alpha})$ with index set $A$, and a net $(y_{\beta})$ with an index set $B$, we say that $y$ is a __subnet__ of $x$ if:

We have a [[function]] $f\colon B \to A$ such that

*  $f$ maps $x$ to $y$ (that is, for every $\beta \in B$, $y_{\beta} = x_{f(\beta)}$);
*  $f$ is monotone (that is, for every $\beta_1 \geq \beta_2 \in B$, $f(\beta_1) \geq f(\beta_2)$);
*  $f$ is cofinal (that is, for every $\alpha \in A$ there is a $\beta \in B$ such that $f(\beta) \geq \alpha$).
=--

+-- {.num_defn #Kelley}
###### Definition
(Kelley, 1955).

Given a net $(x_{\alpha})$ with index set $A$, and a net $(y_{\beta})$ with an index set $B$, we say that $y$ is a __subnet__ of $x$ if:

We have a [[function]] $f\colon B \to A$ such that

*  $f$ maps $x$ to $y$ (that is, for every $\beta \in B$, $y_{\beta} = x_{f(\beta)}$);
*  $f$ is strongly cofinal (that is, for every $\alpha \in A$ there is a $\beta \in B$ such that, for every $\beta_1 \geq \beta \in B$, $f(\beta_1) \geq \alpha$).

=--

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

Notice that the function $f$ in definitions \ref{Willard} and \ref{Kelley} is _not_ required to be an [[injection]], and it need not be. As a result, a [[sequence]] regarded as a [[net]] in general has more sub-nets than it has sub-sequences.

=--

+-- {.num_defn #AA}
###### Definition
(Smiley, 1957; Årnes & Andenæs, 1972).

Given a net $(x_{\alpha})$ with index set $A$, and a net $(y_{\beta})$ with an index set $B$, we say that $y$ is a __subnet__ of $x$ if:

The [[eventuality filter]] of $y$  refines the eventuality filter of $x$.  (Explicitly, for every $\alpha \in A$ there is a $\beta \in B$ such that, for every $\beta_1 \geq \beta \in B$ there is an $\alpha_1 \geq \alpha \in A$ such that $y_{\beta_1} = x_{\alpha_1}$.)
=--

The equivalence between these definitions is as follows:

+-- {.num_theorem #EquivalenceOfDefinitionsOfSubnets}
###### Theorem
([Schechter, 1996](#Schechter96)).

1.  If $y$ is a (\ref{Willard})-subnet of $x$, then $y$ is also a (\ref{Kelley})-subnet of $x$, using the same function $f$.
2.  If $y$ is a (\ref{Kelley})-subnet of $x$, then $y$ is also a (\ref{AA})-subnet of $x$.
3.  If $y$ is a (\ref{AA})-subnet of $x$, then there is some net $z$ such that
    *  $z$ is equivalent to $y$ in the sense that $y$ and $z$ are (\ref{AA})-subnets of each other, and
    *  $z$ is a (\ref{Willard})-subnet of $x$, using some function.
=--

So from the perspective of definition (\ref{AA}), there are enough (\ref{Willard})-subnets and (\ref{Kelley})-subnets, up to equivalence.

## References

A textbook account is in

* {#Schechter96} [[Eric Schechter]], sections 7.14--7.21 of _[[Handbook of Analysis and its Foundations]]_, Academic Press (1996)

Lecture notes include

* {#Vermeeren10} [[Stijn Vermeeren]], _Sequences and nets in topology_, 2010 ([pdf](http://stijnvermeeren.be/download/mathematics/nets.pdf))

[[!redirects subnets]]

[[!redirects sub-net]]
[[!redirects sub-nets]]


subobject

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

# Subobjects
* table of contents
{: toc}

## Definition

### As classes of monomorphisms

A **subobject** of an [[object]] $c$ in a [[category]] $C$ is an [[isomorphism]] class of [[monomorphisms]]

$$
  i: a \hookrightarrow c

$$

into $c$. (Two morphisms $i: a \to c$, $j: b \to c$ are _isomorphic_ if there exists an isomorphism $k: a \to b$ such that $i = j k$.)

Monos into an object $c$ are [[preorder|preordered]] by a relation

$$
  (i: a \to c) \leq (j: b \to c)
$$

defined by the condition that there exists $k: a \to b$ such that $i = j k$. (There is at most one such $k$ since $j$ is monic, and such $k$ is monic since $i$ is monic.)

A subobject of $c$ may equivalently be defined as an element of the [[posetal reflection]] $Sub(c)$ of this preorder.

### In terms of over categories

Let $C_c$ be the [[full subcategory]] of the [[over category]] $C/c$ on [[monomorphism|monomorphisms]]. Then $C_c$ is the [[poset]] of subobjects of $c$ and the set of isomorphism classes of $C_c$ is the set of subobjects of $c$. However this "set" can be in fact a proper class in general, see [[well-powered category]].

### Generalizations

* More generally, in some contexts we may take "subobject" to mean an isomorphism class of morphisms $i: a\to c$ satisfying some suitable condition other than being a monomorphism (usually a stronger one).  Common choices are [[strong monomorphisms]], [[regular monomorphisms]], or the right class of some [[orthogonal factorization system]].  (The latter choice has the advantage that then [[images]] will automatically exist.)

  * For example, in [[Top]] a monomorphism is just a continuous injective function, whereas the strong and regular monomorphisms coincide and are the subspace embeddings.  In some contexts at least, one can argue that subspace embeddings are a more appropriate notion of "subobject" in $Top$ (for example, if one wants to exhibit it as a [[locally bounded category]]).  A similar thing happens in a [[quasitopos]].

* The [[partial order]] on the collection of subobjects internalizes into contexts more general than [[Set]]. For instance in every [[topos]] the [[subobject classifier]] $\Omega$ has the structure of an internal poset (see there).

## Properties

### The poset of subobjects.

* For $C$ an [[accessible category]], $c \in C$ any object, the poset $Sub(c)$ of subobjects of $c$ is a [[small category]].

* Suppose $E$ is a [[well-powered category]]. Denote by $Sub(X)$ the poset of subobjects of object $X$ in $E$. The correspondence $Sub:X\mapsto Sub(X)$ may be extended to a [[contravariant functor]] $E \to Pos$ (that is a functor $E^op \to Pos$), namely if $f: X\to Y$ is arbitrary and $m:S\hookrightarrow Y$ is an element of $Sub(Y)$, i.e. monic, then the pullback $f^*(m):f^*(S)\to X$ of $m$ along $f$ is automatically a monic; the correspondence $m\mapsto f^*(m)$ describes $Sub(f)$ at the level of representatives of subobjects.

### Limits and colimits of subobjects
 {#LimitsAndColimits}

Assume that the ambient category has all [[limit|limits]] and [[colimit|colimits]] considered in the following.

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

For $X \in C$ an [[object]], $Sub(X)$ the [[poset]] of subobjects and
$U_1, U_2 \hookrightarrow X$ two [[subobject|subobjects]],

*  their [[product]] in $Sub(X)$ is denoted $U_1 \cap U_2$ or $U_1 \wedge U_2$ and called the **[[intersection]]** or **[[meet]]** of the two subobjects;

* their [[coproduct]] in $Sub(X)$ is denoted $U_1 \cup U_2$ or $U_1 \vee U_2$ and called the **[[union]]** or **[[join]]** of the two subobjects.

Two subobjects $U_1, U_2 \in Sub_C(X)$ are called **disjoint** if $U_1 \cap U_2 = \emptyset$ is the [[initial object]].

=--

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

Let $C$ be a [[topos]].

1. The intersection of two subobjects in $Sub_C(X)$ is their [[fiber product]] in $C$: the [[diagram]]

   $$
     \array{
       U_1 \cap U_2 &\to& U_2
       \\
       \downarrow && \downarrow
       \\
       U_1 &\to& X
     }
   $$

   is a [[pullback]] diagram.

1. The union of two subobjects $U_1, U_2 \in Sub_C(X)$ is the [[pushout]]  $U_1 \coprod_{U_1 \cap U_2} U_2$ in $C$: the [[diagram]]

   $$
     \array{
       U_1 \cap U_2 &\to& U_2
       \\
       \downarrow && \downarrow
       \\
       U_1 &\to& U_1 \cup U_2
     }
   $$

   is a [[pushout]] diagram.

1. This last pushout diagram is also a [[pullback]] diagram.

=--

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

For the first point: Since [[monomorphism]] are (as discussed there) stable under [[pullback]] and composition, the fiber product is a subobject. Its [[universal property]] as a [[limit]] in $C$ then implies its universal property as a [[product]] in $Sub_C(X)$.

For the second point: by the same kind of argument, it is sufficient to show that the canonical morphism $U_1 \coprod_{U_1 \cap U_2} U_2 \to X$ exhibits the coproduct as a subobject.

Since [[monomorphism|monomorphisms]] (as discussed there) are characterized by the fact that the [[pullback]] along themselves is their domain, it is sufficient to show that

$$
  \array{
     U_1 \coprod_{U_1 \cap U_2} U_2 &\stackrel{Id}{\to}&  U_1 \coprod_{U_1 \cap U_2} U_2
     \\
     {}^{\mathllap{id}}\downarrow && \downarrow
     \\
     U_1 \coprod_{U_1 \cap U_2} U_2 &\stackrel{i}{\to}& X
  }
$$

is a [[pullback]] diagram. For showing this we use that in a [[topos]] we have [[universal colimits]], so that equivalently it is sufficient to show that

$$
  U_1 \coprod_{U_1 \cap U_2} U_2
  \simeq
  (i^* U_1) \coprod_{i^* (U_1 \cap U_2)} (i^* U_2)
  \,.
$$

To see this, again use [[universal colimits]] to get

$$
  \begin{aligned}
    i^* U_1 & \simeq U_1 \times_X (U_1 \coprod_{U_1 \cap U_2} U_2)
    \\
     & \simeq (U_1 \times_X U_1) \coprod_{U_1 \times_X (U_1 \cap U_2)} (U_1 \times_X U_2)
    \\
    & \simeq U_1 \coprod_{U_1 \cap U_2} (U_1 \times_X U_2)
    \\
    & \simeq U_1 \coprod_{U_1 \cap U_2} (U_1 \cap U_2)
    \\
    & \simeq U_1
  \end{aligned}
$$

and similarly

$$
  i^* U_2 \simeq U_2
$$

and

$$
  i^* (U_1 \cap U_2) \simeq (U_1 \cap U_2)
  \,.
$$

This proves the second point.

The third point is directly verified by checking the universal property.

=--

### Comparison with the notion of "subset"

The notion of subobject figures prominently in [[topos theory]] and in other approaches to [[set theory]] based on categories. It is not an exact translation of the usual notion of "subset" in traditional set theory; in ordinary set theory, the notion of subset is defined in terms of a global elementhood relation between sets, which one doesn't have in categorical set theory (and which one wouldn't necessarily want: it violates the [[principle of equivalence]] in the sense of not being invariant with respect to [[isomorphism]]).

Category-theoretically, the traditional notion of subset gives a way of picking out a canonical representative or "normal form" among all the monos in an isomorphism class. As we intimated, there is no intrinsic way of defining such representatives in the theory of toposes: such would have to be considered an extra structure on a topos. Mathematically, there is no particular gain in having such structure around; at best it enables a traditional mode of discourse in which subsets are concrete maps, and to this end it can function as a linguistic or psychological convenience.

On the other hand, there is no particular harm either in having such structure around, as long as one remembers that it is not an isomorphism invariant.  People will instinctively turn to canonical representatives whenever they can -- think of what we would tell a student who asks for help understanding how to multiply elements in $\mathbb{Z}_13$ -- and even category theorists do so when they are available.

## Examples
 {#Examples}

* A subobject in [[Set]] is a [[subset]].

* A subobject in [[Grp]] is a [[subgroup]].

* A subobject in [[Ring]] is a [[subring]].

* A subobject in $R$[[Mod]] is a [[submodule]].

* A subobject of a [[representation]] is a [[subrepresentation]].

* A subobject (a [[subfunctor]]) of a [[representable functor]]  is a [[sieve]].

* A subobject of a [[bundle]] (hence in a [[slice category]]) is a [[sub-bundle]].

## Related concepts

* **subobject**

* [[pure subobject]]

* [[improper subobject]]

* [[subobject in an (∞,1)-category]]

* [[property sup]]

* [[subtype]]

## References

Standard textbook references include section I.3 of

* [[Saunders MacLane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_
 {#MacLaneMoerdijk}

and

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_
 {#Johnstone}

[[!redirects subobjects]]

[[!redirects sub-object]]
[[!redirects sub-objects]]


subobject classifier

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

#Contents#
* table of contents
{:toc}

## Idea

Subsets $A$ of a set $X$ correspond precisely to maps from $X$ to the set of truth values of classical logic via their [[characteristic function]] $\chi_A:X\to \{0,1\}$. The concept of a **subobject classifier** generalizes this situation to [[toposes]] other than [[Set]]:

A _subobject classifier_ in a [[topos]] is a morphism
$true : * \to \Omega$ such that every [[monomorphism]] $A \hookrightarrow B$ in the topos (hence every [[subobject]]) is the [[pullback]] of this morphism along a unique morphism (the [[characteristic morphism]] of $A$) $B \to \Omega$.

In this sense $\Omega$ is the [[classifying space|classifying object]] for subobjects and $true : * \to \Omega$ the _generic_ subobject.

The existence of a subobject classifier in a category is a powerful property which induces much other structure that lies at the heart of [[topos theory]].

By restricting the class of monomorphism appropriately, the concept can be relativized to the concept of an _**M**-subobject classifier_[^weak]: e.g. demanding only classification of strong monomorphisms leads to [[quasitopos|quasitoposes]].

In [[type theory]], a [[type]] closely related to the subobject classifier is the [[type of propositions]], often denoted $Prop$ or (sometimes in [[homotopy type theory]]) $hProp$.

[^weak]: For **M** the class of strong monomorphisms, this is called a _weak subobject classifier_ in Johnstone (2002, p.120).

## Definition

\begin{definition}\label{DefinitionOfSubobjectClassifier}

In a [[category]] $C$ with [[finite limit]]s, a **subobject classifier** is a [[monomorphism]] $true \,\colon\, * \to \Omega$ out of the [[terminal object]], such that for every [[monomorphism]] $U \to X$ in $C$ there is a unique [[morphism]] $\chi_U \colon X \to \Omega$ such that the following [[commuting square|square commutes]] and is [[Cartesian square|Cartesian]] (a [[pullback]]):

$$
  \array{
    U &\longrightarrow& *
    \\
    \big\downarrow
    &{}^{{}_{(pb)}}&
    \big\downarrow{{}^\mathrlap{true}}
    \\
    X &\underset{\chi_U}{\longrightarrow}& \Omega
    \mathrlap{\,.}
  }
$$

\end{definition}

See for instance ([MacLane-Moerdijk, p. 32](#MacLaneMoerdijk)).

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

Some terminology:

If it exists, the object $\Omega$ is also called the **object of [[truth value]]s**, a [[global element]] $K \to \Omega$ is called a **[[truth value]]** and the element $true : * \hookrightarrow \Omega$ is the truth value **[[true]]**, where all these terms allude to the [[internal logic]] of the category $C$.

Note that the subobjects classified by the truth values are [[subterminal object]]s.

The morphism $\chi_U$ is also called the **[[characteristic map]]** or **[[classifying map]]** of the subobject $U \hookrightarrow X$.

=--

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

If $C$ has [[finite limit]]s and is in addition a [[locally small category]], then it has a subobject classifier precisely if the [[subobject]]-assigning [[presheaf]]

$$
  Sub : C^{op} \to Set
$$
$$
  X \mapsto \{U \hookrightarrow X\}/\sim
$$

is [[representable functor|representable]]. In this case the representing object is [[generalized the|the]] subobject classifier: there is a [[natural isomorphism]]

$$
  Sub(X) \simeq C(X, \Omega)
$$

in $X \in C$.

Moreover, in this case $C$ is [[well-powered category|well powered]].

=--

This appears for instance as ([MacLane-Moerdijk, prop. I.3.1](#MacLaneMoerdijk)).

In more detail: given a [[morphism]] $f: c \to d$ in $C$, the function

$$
  Sub(f): Sub(d) \to Sub(c)
$$

takes a [[subobject]] $i: t \hookrightarrow d$ to the subobject of $c$ obtained by [[pullback|pulling back]] $i$ along $f$. (Notice that [[monomorphism]]s, as discussed there, are stable under pullback.)

The representability of this functor means there is an object $\Omega$ together with a subobject $t: T \hookrightarrow \Omega$ which is *[[universal construction|universal]]*, meaning that given any subobject $i: s \hookrightarrow c$, there is a unique morphism $f: c \to \Omega$ such that $i$ is obtained as the pullback of $t$ along $f$.

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

To see that a subobject classifier induces such a natural isomorphism,
we need that the morphisms $Sub(f)$ for $f \in Mor(C)$ corresponds to the morphisms $C(f,\Omega)$. This is the [[pasting law]] for [[pullback]]s.

Conversely, to see that a subobjects-representing object $\Omega$ is a subobject classifier, use that by [[natural transformation|naturality]] we have for each morphism $ \phi : X \to \Omega$ a [[commuting diagram]]

$$
  \array{
    Sub(\Omega) &\stackrel{\simeq}{\to}& C(\Omega, \Omega)
    \\
    {}^{\mathllap{Sub(\phi)}}\downarrow && \downarrow^{\mathrlap{C(\phi,\Omega)}}
    \\
    Sub(X) &\stackrel{\simeq}{\to}& C(X, \Omega)
  }
$$

whose commutativity says that every element of $Sub(X)$ is the pullback along some $\phi : X \to \Omega$ of the subobject of $\Omega$ corresponding under the natural isomorphism to $Id : \Omega \to \Omega$.

By further playing around with this one finds that this latter subobject of $\Omega$ has to be a [[terminal object]].

=--

## Examples

### In $Set$

In the category [[Sets]] of [[sets]], the 2-element set $\mathbf{2} = \{f, t\}$ (the [[classical mathematics|classical]] *[[Boolean domain]]*) plays the role of $\Omega$; the morphism $t \colon  1 \to \mathbf{2}$ just names the element $t$. Given a [[subset]] $S \subseteq X$, the characteristic function $\chi_S: X \to \mathbf{2}$ is the function defined by $\chi_S(x) = t$ if $x \in S$, and $\chi_S(x) = f$ if $x \notin S$.

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

It is not usually true in [[toposes]] that $\Omega$ is the [[coproduct]] $\mathbf{2} = 1 + 1$; toposes where that occurs are called _[[Boolean topos|Boolean]]. Thus the category $Set$ of sets is a Boolean topos, as is the [[presheaf]] topos $Set^G$ when $G$ is a [[groupoid]].

=--

### In a presheaf topos

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

The subobject classifier in a [[presheaf topos]] $PSh(S)$ is the [[presheaf]] that sends each [[object]] $U \in S$ to the [[set]] $sieves(U)$ of [[sieves]] on it

$$
  \Omega
    \;\colon\;
  U \mapsto sieves(U)
$$

Here $sieves(U)$ is equivalently the set of [[subobjects]] of the [[representable functor|representable]] [[presheaf]] $Y(U)$.

The corresponding morphism $true : * \to \Omega$ of presheaves is the [[natural transformation]] that picks over each object the _maximal sieve_ $true_U = maximal_{sieves(U)} : * \to sieves(U)$.

=--

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

If one views a [[presheaf]] over a [[small category]] $S$ as a set varying (or evolving) over $S$, then the subobject classifier in $PSh(S)$ may be viewed as encapsulating the ways of an element to be in the set ranging from 'never' to 'always' being in the set.

In $Set = PSh(\ast)$ these two extremes are the only possibilities, but in the general case there are ways to become an element 'over time'.

It might be helpful to have a look at the simple example of a subobject classifier in the presheaf topos of directed graphs worked out at [[Quiv]] to get some intuition.

=--

#### In $G Set$

As a special case of presheaf toposes, for $G$ a [[discrete group]] and $G Set = [\mathbf{B} G, Set]$ the topos of [[G-sets]], there are precisely two [[sieves]] on the single object of the [[delooping]] [[groupoid]] $\mathbf{B}G$: the trivial one and the empty one. Hence the subobject classifier here is the 2-element set as in [[Set]], but now regarded as a $G$-set with trivial $G$-[[action]].

For toposes of (left) actions of general [[monoid|monoids]] $M$ the picture changes dramatically, since then there will usually exist non-trivial left ideals and, accordingly, the structure of $\Omega$ will become richer: Its underlying set has elements all left ideals, with $m\in M$ acting on a left ideal $L$ by mapping it to the left ideal $\{n\in M| n\cdot m\in L\}$. (Of course, the same description of $\Omega$ applies to the case of groups as well, it is just, that groups lack non-trivial left ideals!)

### In a sheaf topos
The following is adapted from p. 34 of [Caramello](#Caramello).

Let $(C, J)$ be a [[site|Grothendieck site]]. A [[sieve]] $S$ on an object $c \in C$ is $J$-closed when, for any arrow $f :d \to c$, if $f^*(S) \in J(d)$ then $f \in S$. It's called *closed* since this is a closure property: $S$ contains all arrows it covers. If $(C, J)$ is a [[locale]] with the open covers topology, then this condition means that $S$ is a collection of subopens of $c$ which is closed under unions. Clearly, every sieve on $c$ can be saturated to a closed one.

The subobject classifier of $\mathrm{Sh}(C,J)$ is defined on objects as
$$
  \Omega_J(c) = \{ S | \text{S is a J-closed sieve on c} \}
$$
and on morphisms as
$$
  \Omega_J(f : c \to d) = f^*
$$
where $f^*$ denotes, as above, pullback of [[sieve|sieves]].

The arrow $\mathsf{true} : 1 \to \Omega_J$ picks the maximal sieve on each object (the one generated by the identity).

Given a mono $m : B \to A$, the classifying arrow $\chi_m : A \to \Omega_J$ is given by
$$
  \chi_m(c)(x) = \{f : d \to c | A(f)(x) \in B(d)\},
$$
for any $c \in C$ and $x \in A(c)$.

### In a non-boolean topos

An example of a non-[[Boolean topos]] is the [[category of sheaves]] over a "typical" [[topological space]] $X$ such as the [[real line]] $\mathbb{R}$ in its usual [[topology]]. In this case, $\Omega$ is the sheaf where the set of sections over an [[open subset]] $U$ is the set of open subsets of $U$, with the obvious restriction maps; the [[sheaf and topos theory|sheaf topos]] in this case is guaranteed to be non-Boolean provided there are some non-regular open sets in $X$ (a open set is _regular_ if it is the interior of its closure). The "[[internal logic]]" of such a topos is [[intuitionistic logic|intuitionistic]].

### In a slice topos

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

Let $\mathcal{E}$ be a [[topos]] and $X \in \mathcal{E}$ any object. Write $\mathcal{E}/X$ for the corresponding [[over-topos]].

The subobject classifier of $\mathcal{E}/X$ is $p_2 : \Omega_{\mathcal{E}} \times X \to X$.

=--

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

This follows for instance from the statement that the [[inverse image]] of any [[base change geometric morphism]] is a [[logical functor]] and hence preserves subobject classifiers: Here we are looking at the base change along $p : X \to *$ and hence $p^* \Omega_{\mathcal{E}}\simeq \Omega_{\mathcal{E}} \times X$.

But the statement is also easily directly checked.

=--

### In a non-topos

The category $Set_\ast$ of [[pointed sets]] has a subobject classifier (specified up to unique isomorphism as the pointed set with two elements).

If one is willing to admit non-locally small categories, then the category of [[proper class|classes]] in [[ZFC]] is not a topos (it is not cartesian closed) but has a subobject classifier: any two-element set.

The opposite of the category of commutative von Neumann algebras has a subobject classifier given by $\mathbb{C}^2$ [according to Simon Henry on MathOverflow](https://mathoverflow.net/questions/384346/is-the-opposite-category-of-commutative-von-neumann-algebras-a-topos), but is not a topos because it is not cartesian closed (see Dmitri Pavlov's answer to the same MathOverflow question).

## Properties

Suppose a category $\mathbf{C}$ has a subobject classifier; this entails some striking structural consequences for $\mathbf{C}$. We list a few here:

+-- {: .num_prop #regular}
###### Proposition
Every [[monomorphism]] in $\mathbf{C}$ is a [[regular monomorphism]], i.e. is an [[equalizer]] of some pair of maps.
=--

+-- {: .proof}
###### Proof
For $\chi_i: X \to \Omega$ the characteristic map of a mono $i: A \to X$, we find that $i$ is the equalizer of a pair of maps $X \rightrightarrows \Omega$:

$$\array{
 & & 1 \\
 & \mathllap{!} \nearrow & \downarrow \mathrlap{t} \\
X & \underset{\chi_i}{\to} & \Omega.
}$$
=--

+-- {: .num_cor}
###### Corollary
$\mathbf{C}$ is [[balanced category|balanced]], i.e., a morphism in $\mathbf{C}$ is an isomorphism iff it is both [[monomorphism|monic]] and [[epimorphism|epic]].
=--

+-- {: .proof}
###### Proof
"Only if" is trivial. The "if" comes from the fact that an epic (epimorphic) equalizer must be an isomorphism, for if $i: A \to X$ is the equalizer of $f, g: X \rightrightarrows Y$ and $i$ is epic, then $f = g$, whence $1_X$ is their equalizer, so $i: A \to X$ must have been an isomorphism.
=--

+-- {: .num_prop}
###### Proposition
Any two epi-mono factorizations of a map in $\mathbf{C}$ are canonically isomorphic.
=--

+-- {: .proof}
###### Proof
Suppose $i p = j q$ where $p, q$ are epic and $i, j$ are monic. Since $j$ is regular, it is the equalizer of some parallel pair $f, g$ as in the diagram

$$\array{
A & \stackrel{p}{\to} & B & & \\
\mathllap{q} \downarrow & & \downarrow \mathrlap{i} & & \\
C & \underset{j}{\to} & D & \stackrel{\overset{f}{\to}}{\underset{g}{\to}} & E,
}$$

so that $f i p = f j q = g j q = g i p$, whence $f i = g i$ since $p$ is epic, whence $i$ factors through $j$ as $j$ is the equalizer: $i = j k$ for some $k: B \to C$. Then also $k p = q$ since $j k p = i p = j q$ and $j$ is monic. We have that $k$ is monic since $i$ is, and $k$ is epic since $q$ is. Thus $k$ is an isomorphism.
=--

Already these results impose some tight restrictions on $\mathbf{C}$. We get some more by exploiting the internal structure of $\Omega$.

The subobject classifier always comes with the structure of an internal [[partial order|poset]]; that is, a relation $\subseteq\, \hookrightarrow \Omega\times\Omega$ which is internally reflexive, antisymmetric, and transitive.  This can be constructed directly (see Proposition \ref{implication} below), or obtained via the [[Yoneda lemma]] since the collection of subobjects of any object is an external poset.

Similarly, since we assume that $\mathbf{C}$ is finitely complete, each subobject poset $Sub(X)$ has intersections (gotten as pullbacks or [[fiber products]] of pairs of monics $i: A \to X,j: B \to X$), and the intersection operation

$$\cap: Sub(X) \times Sub(X) \to Sub(X)$$

is natural in $X$. Hence we have a family of maps

$$\cap: \hom(X, \Omega \times \Omega) \to \hom(X, \Omega)$$

natural in $X$; by the [[Yoneda lemma]], we infer the presence of an internal intersection map

$$\wedge: \Omega \times \Omega \to \Omega$$

making $\Omega$ an internal [[meet-semilattice]].

More significantly, $\Omega$ is an internal [[Heyting algebra]]. More accurately, it's a Heyting algebra provided it has joins; without joins it is a [[cartesian closed category|cartesian closed]] poset:

+-- {: .num_prop #implication}
###### Proposition
There is an internal implication operator

$$\Rightarrow: \Omega \times \Omega \to \Omega$$

uniquely specified by the internal condition

$$w \wedge u \leq v \qquad iff \qquad w \leq u \Rightarrow v.$$
=--

+-- {: .proof}
###### Proof (sketch)
Construct $\subseteq \hookrightarrow \Omega \times \Omega$ as the equalizer of the pair of maps

$$\Omega \times \Omega \stackrel{\overset{\pi_1}{\longrightarrow}}{\underset{\wedge}{\longrightarrow}} \Omega$$

and then define $\Rightarrow: \Omega \times \Omega \to \Omega$ to be the characteristic map of $\subseteq \hookrightarrow \Omega \times \Omega$. Now if $\chi_u, \chi_v$ are two maps $X \to \Omega$, one calculates that $w \hookrightarrow X$ is contained in the subobject classified by $\chi_u \Rightarrow \chi_v$ iff $w \cap u = w \cap u \cap v$, which is just a way of saying $w \cap u \leq v$.
=--

+-- {: .num_cor}
###### Corollary
In every subobject poset $Sub(X)$, meets distribute over any joins that exist.
=--

+-- {: .proof}
###### Proof
Because $U \cap -$ is left adjoint to the external operator $U \Rightarrow -$ on $Sub(X)$, it preserves any joins that happen to exist in $Sub(X)$.
=--

Normally these results are proved in the context of [[toposes]], where we may say for example that $\Omega$ is an internal [[Boolean algebra]] if and only if the topos is [[Boolean topos|Boolean]]. But as the proofs above indicate, we need only exploit the definition of subobject classifier making reference only to finite limit structure.

In a topos, the subobject classifier $\Omega$ is always [[injective object|injective]], and, so is the _power object_ $\Omega^X$ for every object $X$ (See at [[injective object#Exponential_injectives|injective object]] for some of the details). In particular, every object $X$ embeds into an injective object by the singleton monomorphism $X\to\Omega^X$: 'A topos has enough injective objects!'. More generally, injective objects in a topos are precisely the ones that are retracts of some $\Omega^X$ (Cf. [Borceux 1994](#Borceux3), p.315; [Moerdijk-MacLane 1994](#MoerdijkMacLane), p.210).

### Johnstone's exercise

A curiosity from Johnstone's [Topos Theory](#J77), posed as an exercise, is that any monomorphism $\Omega \to \Omega$ is an isomorphism and even an [[involution]]. Thus $\Omega$ is a [[Hopfian object]].

An online proof may be found [here](https://ncatlab.org/toddtrimble/published/Monic+endomorphisms+on+the+subobject+classifier).

## Categories with a contractible subobject classifier

Suppose a [[topos]] $\mathcal{E}$ has a [[connected component|connected components functor]] $\Pi$ [[left adjoint]] to $\Delta\dashv \Gamma$ assigning to an object $X$ the set of it connected components. We call $X$ [[connected object|connected]] if $\Pi(X)$ is a [[singleton]] and [[contractible space|contractible]] if $\Pi(X^Y)$ is connected for all $Y\in\mathcal{E}$. It can be shown that provided $\Pi$ preserves [[finite products]], $\Omega$ is contractible iff $\Omega$ is connected which in turn implies the same for all other [[injective object|injective objects]] (see at [[sufficiently cohesive topos]]).

Now the subobject classifier $\Omega$ has always two disjoint points $\mathsf{true}$, $\mathsf{false}$ whence provided it is connected we can view it (together with its Heyting algebra structure) has a (highly nonlinear) generalized [[interval object]] and define a notion of [[homotopy]] relative to $\Omega$. The intuition here is that [[true|truth]] and [[false|falsity]] are continuously connected in such toposes and blend into each other, endowing (the logic in) $\mathcal{E}$ with a certain [[Hegel|Hegelian]] flavor. The contractability of $\Omega$ was taken as a key property of a [[gros topos]] of spaces by [[William Lawvere]]. Further information on this particular class of [[cohesive topos|cohesive toposes]] and discussion of properties of $\Omega$ relevant in this context is at [[sufficiently cohesive topos]].

## Categories without subobject classifiers

As the previous section indicates, having a subobject classifier is a very strong property of a category and "most" categories with finite limits don't have one.

For example, there is an easy condition ensuring a category[^1] with a terminal object can't have a subobject classifier: if there are no nonidentity morphisms out of the terminal object. This includes the following examples.

[^1]: We mean a nontrivial category, obviously, where "trivial" here means every object is terminal.

* Any top bounded partial order.

* In $Ring$, the [[category of rings]], there are no nonidentity morphisms out of the terminal object the [[zero ring]].

Here's another obstacle:

* If an [[abelian category]] had a subobject classifier, every subobject of every object would have to be the kernel of its classifying map. In particular, the subobject $0$ of every object $A$ would have to be the kernel of its classifying map, so that every object in this abelian category would embed into the subobject classifier $\Omega$ (including, say, all small products of $\Omega$ with itself) which in nontrivial cases would cause size issues.

But a real killer is the fact that all monos are regular, or its consequences of the category being balanced and uniqueness of epi-mono factorizations:

* The categories [[Pos]], [[Cat]], [[Top]] are not balanced (consider the map from a discrete structure on a set to an indiscrete structure on the same set, induced by the identity function). The category [[CMon]] is not balanced (consider the inclusion $\mathbb{N} \hookrightarrow \mathbb{Z}$ which is epic).

Even though all monos in [[Grp]] are regular, we can kill off $Grp$ by observing that if $t: 1 \to \Omega$ were a subobject classifier, the proof of Proposition \ref{regular} indicates that every mono $i: A \to X$ would have to be the [[kernel]] of $\chi_i$. But not all monos in $Grp$ are kernels.

Perhaps an even more decisive killer is the observation that meets distribute over (arbitrary) joins in subobject orders. This eliminates many categories from consideration:

* Lattices of subobjects in $Grp$ or $Ab$ are rarely distributive.

* For any nontrivial category with [[biproducts]], there are non-distributive subobject lattices. Take any object $A$, so that we have three subobjects $i_1: A \to A \oplus A$, $i_2: A \to A \oplus A$, and $\Delta: A \to A \oplus A$. Then $i_1 \vee i_2 = \top$, whereas $i_1 \wedge \Delta = \bot = i_2 \wedge \Delta$. Under distributivity we have
$$\Delta = \Delta \wedge \top = \Delta \wedge (i_1 \vee i_2) = (\Delta \wedge i_1) \vee (\Delta \wedge i_2) = \bot \vee \bot = \bot$$
but $\Delta = \bot$ forces $A = 0$. So the only such category that can have a subobject classifier is trivial.

Directly related to this distributivity property is the question when a [[variety]] of algebras has a subobject classifier (cf. Johnstone [1990](#JT90), pp.448f).

## Generalizations

### Weak subobject classifier

There are many subclasses of [[monomorphisms]], such as [[regular monomorphisms]], [[strong monomorphisms]], and [[extremal monomorphisms]]. As a result, there are weaker forms of subobject classifiers that only classify a subclass of monomorphisms instead of all monomorphisms.

Suppose there is a [[category]] $C$ with [[finite limit]]s. Let $S$ be a subclass of [[monomorphisms]] in $C$, whose elements we shall refer to as $S$-monomorphisms. An $S$-**subobject classifier** is an $S$-[[monomorphism]] $true : * \to \Omega$ out of the [[terminal object]], such that for every $S$-[[monomorphism]] $U \to X$ in $C$ there is a unique morphism $\chi_U : X \to \Omega$ such that there is a [[pullback]] [[diagram]] of the following form:

$$
  \array{
    U &\to& *
    \\
    \downarrow && \downarrow^{\mathrlap{true}}
    \\
    X &\stackrel{\chi_U}{\to}& \Omega
  }
  \,.
$$

* If $S$ is the class of all [[monomorphisms]] in $C$, then $\Omega$ is just a **subobject classifier**.
* If $S$ is the class of all [[extremal monomorphisms]] in $C$, then $\Omega$ is an **extremal subobject classifier**.
* If $S$ is the class of all [[strong monomorphisms]] in $C$, then $\Omega$ is a **strong subobject classifier**.
* If $S$ is the class of all [[regular monomorphisms]] in $C$, then $\Omega$ is a **regular subobject classifier**.
* If $S$ is the class of all [[effective monomorphisms]] in $C$, then $\Omega$ is an **effective subobject classifier**.
* If $S$ is the class of all [[strict monomorphisms]] in $C$, then $\Omega$ is a **strict subobject classifier**.
* If $S$ is the class of all $\kappa$-compact monomorphisms, for some [[regular cardinal]] $\kappa$, then $\Omega$ is a **$\kappa$-compact subobject classifer**.

Strong subobject classifiers in particular are important in the definition of a [[quasitopos]].

Let $\kappa$ be an [[inaccessible cardinal]]. Then $\kappa$-compact subobject classifers are used in the syntactic category of [[Martin-Löf type theories]] like [[book HoTT]].

### Object classifier

In higher topoi the subobject classifiers are the [[generalized universal bundle|universal fibrations]]:

In the [[(n,1)-topos|(n+1,1)-topos]] $n Grpd$ or $(n,0) Cat$ of [[n-groupoid|(n,0)-categories]] the subobject classifier is the [[stuff, structure, property|forgetful functor]]

$$
  (n,1) true : (n-1,0)Cat_* \to (n-1,0)Cat
$$

from the $(n,1)$-category of [[pointed object|pointed]] $(n-1,0)$-categories to that of $(n-1,0)$-categories, which forgets the point.

This is described in more detail at [[generalized universal bundle]]. See also the discussion at [[stuff, structure, property]].

In fact, using the notion of [[(-1)-category|(-1,0)-category]] the subobject classifier in [[Set]] does fit precisely into this pattern:

the set of truth values $\Omega$ may be regarded as the [[(0,1)-category]] of [[(-1)-category|(-1,0)-categories]] (of which there are two) and the one-element set $*$ is the [[(0,1)-category]] of [[pointed object|pointed]] [[(-1)-category|(-1,0)-categories]], of which there is one.

In the context of [[(∞,1)-topos]] [[Higher Topos Theory|theory]] subobject classifiers are discussed in section 6.1.6 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_ , Princeton UP 2009.

Whereas for 1-toposes the _subobject classifier_ is the key structural ingredient (besides the exactness properties), in [[higher topos theory]] this role is taken over by the [[object classifier]], as pointed out in Lurie (2009).

## Related concepts

* [[Lawvere interval]]

* [[boolean domain]]

* [[quasitopos]] where a weaker notion of subobject classifier only classifies [[strong monomorphism]]s.

* [[discrete object classifier]]

* [[(sub)object classifier in an (infinity,1)-topos]]

* [[type of propositions]], [[type of types]]

* [[classifying space]], [[classifying stack]], [[moduli space]], [[moduli stack]], [[derived moduli space]]

* [[classifying topos]]

* [[universal principal bundle]], [[universal principal ∞-bundle]]

* [[classifying morphism]]

* [[sufficiently cohesive topos]]

* [[localic topos]]

## References#

The concept was introduced in

* [[William Lawvere]], *Quantifiers and sheaves*, Actes Congrès intern. Math. **1** (1970) 329-334 [[pdf](https://raw.githubusercontent.com/mattearnshaw/lawvere/master/pdfs/1970-quantifiers-and-sheaves.pdf), [[Lawvere-QuantifiersAndSheaves.pdf:file]]]

Discussion of the concept can be found in the usual suspects

* [[Francis Borceux]], _[[Handbook of Categorical Algebra]] 3_ , Cambridge UP 1994. {#Borceux3}

* {#Goldblatt84} R. Goldblatt, *Topoi - The Categorical Analysis of Logic*, 2nd ed. North-Holland Amsterdam 1984. (Dover reprint New York 2006; [project euclid](http://projecteuclid.org/euclid.bia/1403013939))

* {#J77}[[Peter Johnstone]], _Topos Theory_ , Academic Press New York 1977.

* [[Peter Johnstone]], _[[Sketches of an Elephant]] I_ , Oxford UP 2002.

* {#MacLaneMoerdijk} [[Saunders MacLane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_ , Springer Heidelberg 1994. (section I.3-4)

See also

* [[Francis Borceux]], _When is $\Omega$ a cogenerator in a topos ?_ , Cah. Top. Géom. Diff. Cat. **XVI** no.1 (1975) pp.3-15. ([numdam](http://www.numdam.org/item?id=CTGDC_1975__16_1_3_0))

For sheaves

* {#Caramello}[[Olivia Caramello]], _Topos Theory: Lectures 7-14: Sheaves on a site_, [slides](https://www.oliviacaramello.com/Teaching/Lectures7_to_14.pdf)

The existence of subobject classifiers in categories of algebras is considered in

* {#JT90}[[Peter Johnstone]], _Collapsed Toposes and Cartesian Closed Varieties_ , JA **129** (1990) pp.446-480.

[[!redirects subobject classifiers]]


subobject in an (infinity,1)-category

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## Definition

Let $C$ be an [[(∞,1)-category]] and $X \in C$ an [[object]].

+-- {: .num_def}
###### Definition

A **subobject** of $X$ is a [[1-monomorphism]] $K \hookrightarrow X$ into $X$.

The [[(∞,1)-category]] $Sub(X)$ of **subobjects** of $X$ is the $(-1)$-[[truncated object in an (∞,1)-category|truncation]] of the [[slice-(∞,1)-category]] of $C$ over $X$

$$
  Sub(X) := \tau_{-1} C_{/X}
  \,.
$$

=--

This is the category whose objects are [[monomorphism in an (∞,1)-category|monomorphisms]] $U \hookrightarrow X$ in $C$ and whose morphisms are [[2-morphism]]s

$$
  \array{
     U_1 &&\to&& U_2
     \\
     & \searrow &\swArrow& \swarrow
     \\
     && X
  }
$$

in $C$.

## Properties

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

$Sub(X)$ is a [[(0,1)-category]] (a [[poset]]).

=--

This appears for instance in ([Lurie, section 6.2](#Lurie)).

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

If $C$ is a [[locally presentable (∞,1)-category]] then $Sub(X)$ is a [[small category]].

=--

## Related concepts

* [[subobject]]

* **subobject in an $(\infty,1)$-category**

* [[n-image]], [[n-truncated object in an (∞,1)-category]]

## References

* [[Jacob Lurie]], _[[Higher Topos Theory]]_
  {#Lurie}

[[!redirects subobject in an (∞,1)-category]]

[[!redirects subobjects in an (∞,1)-category]]
[[!redirects subobjects in an (infinity,1)-category]]


subquotient

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## Definition ##

A **subquotient** of an object $X$ in some category is

* a [[quotient object]] of a [[subobject]] of $X$,

or

* a subobject of a quotient object of $X$.

## Properties

* The two definitions are equivalent whenever quotients are [[pullback stability|stable under pullback]] and subobjects are stable under pushout, such as in a [[topos]].

* Just as with subobjects and quotient objects, we have that $X$ is a subquotient of itself, and subquotients of subquotients of $X$ are themselves subquotients of $X$ in a natural way.

* Just as subobjects of a set $X$ are in correspondence with [[predicate]]s on $X$ and quotients of $X$ are in correspondence with [[equivalence relation]]s on $X$, subquotients of $X$ are in correspondence with [[partial equivalence relation]]s on $X$.

## Examples

* In [[Grp]], all but six of the 26 [[sporadic finite simple groups]] are subquotients of the [[Monster group]], the largest such group.

[[!redirects subquotient object]]
[[!redirects subquotient set]]

[[!redirects subquotients]]

[[!redirects subquotient objects]]
[[!redirects subquotient sets]]


Subrahmanyan Chandrasekhar

_Subrahmanyan Chandrasekhar_ was an Indian American astrophysicist, a pioneer in study of gravitation, [[general relativity]] and in particular [[black hole]] physics; a Nobel laureate.

* [wikipedia](http://en.wikipedia.org/wiki/Subrahmanyan_Chandrasekhar)

* S. Chandrasekhar, The mathematical theory of black holes ([ZMATH entry](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0912.53053&format=complete)) (classics with lots of explicit details of analytic solutions of black hole spacetimes)

[[!redirects Chandrasekhar]]

category: people


subrepresentation

#Contents#
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{:toc}

## Definition

A _subrepresentation_ is a [[subobject]] in a [[category of representations]].

## Related concepts

* [[induced representation]]

* [[coinduced representation]]

[[!redirects subrepresentations]]

[[!redirects sub-representation]]
[[!redirects sub-representations]]


subring

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## Idea
 {#Idea}

In [[algebra]], a *subring* is a [[subobject]] of a [[ring]], i.e. a [[subobject]] of an [[object]] in the [[category]] [[Ring]] of rings with [[homomorphisms]] between them.

As usual, there is a little bit of variation of what exactly one takes to be the definition of "ring" (multiplicative [[unital magma|unitality]] is usually understood by default, while [[commutative ring|commutativity]] is usually not assumed by default) but in each case the general notion of [[subobject]] reduces to the appropriare notion of subring. For instance, a subring in the category of unital rings necessarily contains the [[unit]]-element of the ambient ring, etc.

## Related concepts

* [[subgroup]], [[submodule]]

## References

See also:

* Wikipedia, *[Subring](https://en.wikipedia.org/wiki/Subring)*

[[!redirects subrings]]

[[!redirects sub-ring]]
[[!redirects sub-rings]]


subscheme of an Abelian category

## Motivation

By the Gabriel-Rosenberg reconstruction theorem a scheme $X$ can be reconstructed from the category $Qcoh_X$ of quasicoherent sheaves on $X$. If $i : Y\hookrightarrow X$ is a subscheme of $X$ then we can identify the category quasicoherent sheaves on $Y$ with certain abelian subcategory of quasicoherent sheaves on $X$. One attempts to abstractly deefine the properties of such a subcategory which guarantee that it corresponds to a subcategory of that kind.

## Definition

([[Alexander Rosenberg]] in his 1996 book and MPI preprints with Lunts from 1996)

A **subscheme** of an abelian category $A$ is a [[coreflective subcategory|coreflective]] [[topologizing subcategory]] of $A$.

If a subscheme is also reflective then we call it Zariski closed.

## Properties

If an abelian category $A$ has the Gabriel's property (sup) then every

(a) The intersection of any set of subschemes of $A$ is a subscheme.

(b) The intersection of any set of Zariski closcd suhschemes of $A$ is a Zariski closed subscheme.

## Literature

* V. A. Lunts, A. L. Rosenberg, _Differential calculus in noncommutative algebraic geometry I. D-calculus on noncommutative rings_, MPI 1996-53 [pdf](http://www.mpim-bonn.mpg.de/preprints/send?bid=3894), II. D-Calculus in the braided case. The localization of quantized enveloping algebras, MPI 1996-76 [pdf](http://www.mpim-bonn.mpg.de/preprints/send?bid=3916)

[[!redirects subschemes of an abelian category]]
[[!redirects subscheme of an abelian category]]