subsequential space

# Subsequential spaces
* table of contents
{: toc}

## Idea

A **subsequential space** is a [[set]] equipped with a notion of _[[convergence of a sequence|sequential convergence]]_, giving it a "[[topological space|topology]]" in an informal sense.

Any [[topological space]] (or more generally, any [[pseudotopological space]]) becomes a subsequential space with its standard notion of [[convergence]], but only for a [[sequential space]] can the topology be recovered from sequential convergence.  In the other direction, not every subsequential space is induced by a topological one.  Despite these apparent drawbacks, subsequential spaces have a number of advantages; see below.

## Definition

A **subsequential space** is a set $X$ equipped with a [[relation]] between sequences and points, called "converges to", with the following properties:

1. For every $x\in X$, the constant sequence $(x)$ converges to $x$.

2. If a sequence $(x_n)$ converges to $x$, then so does any subsequence of $x$.

3. If, for some sequence $(x_n)$ and some point $x$, every subsequence of $(x_n)$ contains a further subsequence converging to $x$, then $(x_n)$ itself converges to $x$.

The final property can be stated less [[constructive mathematics|constructively]] as "if $(x_n)$ does not converge to $x$, then there is a subsequence $(x_{n_k})$ of $(x_n)$ such that no subsequence of $(x_{n_k})$ converges to $x$."

Note that this definition matches the definition of [[pseudotopological space]] except for the restriction to sequences instead of general [[net]]s.  Accordingly, one may call a subsequential space a __sequential pseudotopological space__.

Subsequential spaces are also known as Kuratowski _limit spaces_, or _L-spaces_; see [Menni & Simpson (2002)](#MenniSimpson02).

A subsequential space is said to be **sequentially [[Hausdorff space|Hausdorff]]** if each sequence converges to at most one limit.

## Properties

The definition of a subsequential space is arguably easier and more intuitive than that of a [[topological space]].  Continuity of functions between subsequential spaces is likewise easy to define by preservation of convergent sequences.

As mentioned above, the [[category]] $SeqTop$ of [[sequential space|sequential (topological) spaces]] is a full [[reflective subcategory]] of the category $SeqPsTop$ of subsequential spaces. Thus, subsequential spaces include many spaces of interest to topologists, including all [[metric space|metrizable]] spaces and all [[CW complex|CW complexes]], and so they can be regarded as a sort of [[nice topological space]].

Not every subsequential space is a sequential (topological) space, but somewhat surprisingly, every sequentially Hausdorff subsequential space _is_ necessarily a sequential space.  Note, though, that while any Hausdorff space is sequentially Hausdorff, the converse is not true even for sequential spaces (though it is true for first-countable spaces).  Also of note is that $SeqTop$ is coreflective in $Top$.

Furthermore, $SeqPsTop$ is _also_ a [[nice category of spaces]]: it is [[locally cartesian closed category|locally cartesian closed]] and in fact a [[quasitopos]].  Since it is a "Grothendieck quasitopos" (the category of presheaves on a category which are [[sheaf|sheaves]] for one [[Grothendieck topology]] and separated for another one), it is also [[locally presentable category|locally presentable]].  In particular, it is [[complete category|complete]] and cocomplete, and has a small [[generator|generating set]].

Of course, the embedding of $SeqTop$ in $SeqPsTop$ [[preserved limit|preserves]] all [[limits]], [[right adjoints preserve limits|since]] it has a [[left adjoint]], but somewhat surprisingly it also preserves many [[colimits]].  In particular, it preserves all the colimits used in the construction of a CW complex; thus it makes no difference whether you carry out the construction of a CW complex in [[Top]] and then regard the result as a subsequential space, or carry out the construction in $SeqPsTop$ to begin with.

It follows that the [[geometric realization]] functor from [[simplicial sets]] can equally well be regarded as landing in $Top$, $SeqTop$, or $SeqPsTop$.  Of course, it has a [[singular simplicial complex]] functor as a right adjoint in any of these three cases (by [[nerve and realization]]).  In the cases of $SeqTop$ and $SeqPsTop$, geometric realization actually preserves all [[finite limits]]; in fact it and the singular complex functor form a [[geometric morphism]] between $SimpSet$ and a Grothendieck topos that contains $SeqPsTop$ as a reflective subcategory (the "[[topological topos]]" of Johnstone's paper).  Recall that geometric realization landing in $Top$ doesn't even preserve finite products, unless we replace $Top$ by (for instance) compactly generated spaces.

These properties of subsequential spaces should be compared with analogous ones for [[convergence space]]s and their relatives, such as [[pseudotopological space]]s.  The category $Conv$ of convergence spaces is also a complete and cocomplete quasitopos (hence, in particular, locally cartesian closed) and includes *all* of $Top$ as a reflective subcategory.  However, $Conv$ is not locally presentable and has no generator, and while the embedding of $Top$ into $Conv$ also preserves all limits (since it has a left adjoint), it actually preserves _fewer_ colimits than the embedding of $SeqTop$ into $SeqPsTop$.  In particular, it does _not_ preserve the colimits used in the construction of CW complexes: if you carry out the construction of a CW complex in $Conv$, in general the result won't even be a topological space.

## See also

* [[sequential preconvergence space]]

## References

* [[Peter Johnstone]], _On a topological topos_.  Proc. London Math. Soc. (3) 38 (1979) 237--271 doi:[10.1112/plms/s3-38.2.237](https://doi.org/10.1112/plms/s3-38.2.237)

* {#MenniSimpson02} [[Matias Menni]], [[Alex Simpson]]: _Topological and Limit-space Subcategories of Countably-based Equilogical Spaces_, Math. Struct. in Comp. Science **12** (2002) 739-770 [[pdf](http://homepages.inf.ed.ac.uk/als/Research/Sources/subcats.pdf), [doi:10.1017/S0960129502003699](https://doi.org/10.1017/S0960129502003699)]

* Sean Moss, [Blog post](https://golem.ph.utexas.edu/category/2014/04/on_a_topological_topos.html) at the $n$-Category café

[[!redirects subsequential space]]
[[!redirects subsequential spaces]]

[[!redirects sequential pseudotopological space]]
[[!redirects sequential pseudotopological spaces]]
[[!redirects sequential convergence space]]
[[!redirects sequential convergence spaces]]

[[!redirects Kuratowski limit space]]
[[!redirects Kuratowski limit spaces]]
[[!redirects limit space]]
[[!redirects limit spaces]]
[[!redirects L-space]]
[[!redirects L-spaces]]


subset

# Subsets
* table of contents
{: toc}

## Idea

A subset of a given [[set]] $A$ is a set $B$ that may be viewed as contained within $A$.

## Definitions

### In material set theory

In [[material set theory]], a __subset__ of a [[set]] $A$ is a set $B$ with the *property* that
$$ x \in B \;\Rightarrow\; x \in A $$
for any object $x$ whatsoever.  One writes $B \subseteq A$ or $B \subset A$ (depending on the author) if $B$ has this property.  Set theory\'s [[axiom of extensionality]] says that $A = B$ if (and only if) $A \subseteq B$ and $B \subseteq A$ (although this is only strong enough for [[well-founded set | well-founded sets]]).

### In structural set theory

In [[structural set theory]], this definition doesn\'t make sense, because there is no global membership predicate $\in$ (and there may not be a global [[equality predicate]] either).  Accordingly, we define a __subset__ of $A$ to be a set $B$ with the *structure* of an [[injection]]
$$ i\colon B \hookrightarrow A .$$

We can still define a *local* membership predicate $\in_A$ as follows:  Given an element $x$ of $A$ and a subset $B$ (technically, $(B,i)$) of $A$,
\[\label{indef} x \in_A B \;\Leftrightarrow\; \exists(y\colon B),\; x = i(y) .\]
Similarly, we can define a local containment predicate $\subseteq_A$ as follows:  Given subsets $B$ and $C$ of $A$,
$$ B \subseteq_A C \;\Leftrightarrow\; \forall(x\colon A),\; x \in B \;\Rightarrow\; x \in C .$$
We can also define a local equality predicate $=_A$ on subsets of $A$:
$$ B =_A C \;\Leftrightarrow\; B \subseteq C \;\wedge\; C \subseteq B .$$

In foundations that already have a global equality predicate on sets (and functions between equal sets), this local equality predicate must be interpreted as an [[equivalence relation]]; then a __subset__ of $A$ is technically an [[equivalence class]] of injections to $A$ rather than simply an injection to $A$.

In any case, if $A$ is a subset of $B$, then $B$ is a __superset__ of $A$.

### Internal subsets

One could define the subset relation as a [[set]] using the [[internal logic]] of a [[set theory]]. The inclusion [[relation]] between two sets $A$ and $B$ is defined as the [[support set|support]] of the [[injection set]] between $A$ and $B$:

$$A \subseteq B \coloneqq \left[\mathrm{Inj}(A, B)\right]$$

A **internal subset** of a [[set]] $B$ is a set $A$ with an element $p \in A \subseteq B$. A **internal superset** of a [[set]] $A$ is a set $B$ with an element $p \in A \subseteq B$.

## Remarks

As you can see from the right-hand sides of the above sequence of definitions, one usually suppresses the subscript $A$ from the notation.  Even the right-hand side of (eq:indef) may use a local equality relation on elements of $A$.  It may be necessary to distinguish $x\colon E$ (the _[[type|typing declaration]]_ introducing a variable $x$ for an element of a given set $E$) from $x \in_A E$ (the _[[proposition]]_ that a given element $x$ of a given set $A$ is a member of a given subset $E$ of $A$).  Some authors may use $x \in A$ for either or both of these, trusting on context (particularly whether $x$ has been introduced before) to distinguish them.  Another notational convenience is to suppress the injection $i$, writing $y$ instead of $i(y)$.

The concept of subset as it appears here generalises to [[subobject]] in category theory.  To be precise, a subset of $A$ is exactly a subobject of $A$ when $A$ is thought of as an object of the category [[Set]].  The concept of superset then generalises to a notion of [[extension]] analogous to that of [[field extension]].

When the abstract set $A$ is fixed, a subset $B$ of $A$ may be thought of as a __[[concrete structure|concrete]] set__.

## Related concepts

* [[subtype]]

* [[cofinite subset]]

* [[saturated subset]]

* [[subspace]]

* [[causally closed subset]]

* [[space of finite subsets]]

[[!redirects subset]]
[[!redirects subsets]]

[[!redirects superset]]
[[!redirects supersets]]

[[!redirects internal subset]]
[[!redirects internal subsets]]

[[!redirects internal superset]]
[[!redirects internal supersets]]

[[!redirects concrete set]]
[[!redirects concrete sets]]


subset collection

# Subset collection
* table of contents
{: toc}

## Idea

In [[set theory]] and the [[foundations]] of mathematics, the axiom of __fullness__ is an axiom intermediate in strength between the existence of [[power sets]] and [[function sets]].  Of course, these can only be distinguished in [[constructive mathematics]], since classically power sets and function sets are equivalent (at least if you have the set $\mathbf{2} = \{\top, \bot\}$).

The axiom of __subset collection__ combines fullness with the [[axiom of collection]] to apply in the situations where fullness is used.  In the presence (I think) of either collection or [[full separation]], subset collection is equivalent to fullness.

## Statements

In its guise as _fullness_, the axiom states that, given any [[sets]] $X$ and $Y$, there is a set $F_{X,Y}$ of _enough [[entire relations]]_ from $X$ to $Y$ in this sense:

*  Every element of $F_{X,Y}$ is an entire relation from $X$ to $Y$;
*  If $R$ is an entire relation from $X$ to $Y$, then for some $S \in F_{X,Y}$, $S \subseteq R$.

We have written this in a [[material set theory|material]] fashion; [[structural set theory|structurally]], the axiom says that $F_{X,Y}$ *parametrises* enough entire relations in this sense:

*  There is a [[subset]] $E_{X,Y}$ of $F_{X,Y} \times X \times Y$ such that:

   *  for every element $r$ of $F_{X,Y}$ and every element $a$ of $X$, there is an element $b$ of $Y$ such that $(r,a,b) \in E_{X,Y}$;

   (we think of $E_{X,Y}$ as giving the evaluation, as a [[truth value]], of $r$, as an entire relation, on the elements $a$ and $b$).

*  If $R$ is an entire relation from $X$ to $Y$, then for some $s$ in $F_{X,Y}$:

   *  for every element $a$ of $X$ and $b$ of $Y$, if $(s,a,b) \in E_{X,Y}$, then $a$ is $R$-related to $b$;

   (we think of $s$ as an entire relation via $E_{X,Y}$ and say that it is contained in $R$).

In the context of material set theory, the structural axiom is weaker than the material axiom, but they become equivalent in the presence of [[axiom of replacement|restricted replacement]], which is quite weak.

You can also write down an [[internalization|internal]] version of fullness by adding arbitrary additional parameters to the structural version above (analogously to the generalisation from [[power set]] to [[power object]], although more complicated since $F_{X,Y}$ is not given by a [[universal property]]).

Given [[function sets]], fullness is equivalent to the set theory having sets of [[inhabited]] [[subsets]], since an entire relation between $A$ and $B$ is a [[function]] from $A$ to the set of inhabited subsets of $B$. (Power sets on the other hand are sets of *all* subsets).

In its guise as _subset collection_, the axiom states ...
+-- {: .query}
I need to look this up again, including the conditions for its equivalence with fullness.  In the meantime, see [the Stanford Encyclopedia's list of axioms](http://plato.stanford.edu/entries/set-theory-constructive/axioms-CZF-IZF.html).
=--

### In dependent type theory

In [[dependent type theory]], given a [[Russell universe]] $U$, the type of all [[entire relations]] between types $A:U$ and $B:U$ is given by the type

$$\sum_{R:A \times B \to U} \prod_{x:A} \left(\prod_{y:A} \mathrm{isProp}(R(x, y))\right) \times \exists y:A.R(x, y)$$

In general, this type is not [[small type|$U$-small]]. The **axiom of fullness** for $U$ states that the above type is $U$-small, and is given by the following [[inference rules]]:

$$\frac{\Gamma \vdash A:U \quad \Gamma \vdash B:U}{\Gamma \vdash \mathrm{EntRel}(A, B):U}$$

$$\frac{\Gamma \vdash A:U \quad \Gamma \vdash B:U}{\Gamma \vdash \mathrm{EntRel}(A, B) \equiv \sum_{R:A \times B \to U} \prod_{x:A} \left(\prod_{y:A} \mathrm{isProp}(R(x, y))\right) \times \exists y:A.R(x, y) \; \mathrm{type}}$$

Similarly, for [[Tarski universes]] $(U, T)$, the type of all entire relations between $U$-small types $A:U$ and $B:U$ is given by the type

$$\sum_{R:T(A) \times T(B) \to U} \prod_{x:T(A)} \left(\prod_{y:T(A)} \mathrm{isProp}(T(R(x, y)))\right) \times \exists y:T(A).T(R(x, y))$$

There are similar inference rules for [[strict Tarski universes]]:

$$\frac{\Gamma \vdash A:U \quad \Gamma \vdash B:U}{\Gamma \vdash \mathrm{EntRel}(A, B):U}$$

$$\frac{\Gamma \vdash A:U \quad \Gamma \vdash B:U}{\Gamma \vdash T(\mathrm{EntRel}(A, B)) \equiv \sum_{R:T(A) \times T(B) \to U} \prod_{x:T(A)} \left(\prod_{y:T(A)} \mathrm{isProp}(T(R(x, y)))\right) \times \exists y:T(A).T(R(x, y)) \; \mathrm{type}}$$

For [[weak Tarski universes]], one uses an [[equivalence of types]] instead of [[judgmental equality]]

$$\frac{\Gamma \vdash A:U \quad \Gamma \vdash B:U}{\Gamma \vdash \mathrm{fullness}(A, B):T(\mathrm{EntRel}(A, B)) \simeq \sum_{R:T(A) \times T(B) \to U} \prod_{x:T(A)} \left(\prod_{y:T(A)} \mathrm{isProp}(T(R(x, y)))\right) \times \exists y:T(A).T(R(x, y))}$$

and the two inference rules could be reduced down to the element:

$$\mathrm{fullness}:\sum_{F:U \times U \to U} \prod_{A:U} \prod_{B:U} T(F(A, B)) \simeq \sum_{R:T(A) \times T(B) \to U} \prod_{x:T(A)} \left(\prod_{y:T(A)} \mathrm{isProp}(T(R(x, y)))\right) \times \exists y:T(A).T(R(x, y))$$

If the dependent type theory does not have any type universes at all, one could still express the axiom of fullness by explicitly postulating the [[type of all entire relations]] using [[inference rules]].

## Relation to other axioms

In the presence of the usual basic axioms of set theory (in particular, the axioms of [[bounded separation]] and [[axiom of pairing|pairing]] in a material set theory and the existence of [[equalisers]] and [[products]] in a structural set theory), we have:

> power sets → fullness → function sets

On the one hand, $F_{X,Y}$ can be constructed as the [[subset]] of the [[power set]] $\mathcal{P}(X \times Y)$ consisting of *all* entire relations from $X$ to $Y$.  On the other hand, the [[function set]] $Y^X$ may be constructed as a subset of $F_{X,Y}$ consisting of those elements $S$ such that $S \subseteq R$ for $R$ (the graph of) some function from $X$ to $Y$; (the key here is that $S = R$ must follow when $S$ is entire and $R$ is a function).  Conversely, if we have function sets, then the [[axiom of choice]] states precisely that the function set $Y^X$ always qualifies as $F_{X,Y}$ (although function sets and [[excluded middle]] are already sufficient for power sets and thus fullness).

Fullness also follows from function sets in the presence of [[COSHEP]], a fairly strong but constructive (in that it implies no general [[principle of omniscience]]) form of the [[axiom of choice]].  If $P$ is a [[projective set]] with a [[surjection]] $p\colon P \to X$ (which this axiom asserts must exist), then we may take $Y^P$ as $F_{X,Y}$, with $(r,a,b) \in E_{X,Y}$ iff, for some $c$ in $P$, $p(c) = a$ and $r(c) = b$.

Note that $F_{X,Y}$ is not categorical (in the logicians\' sense); that is, there may be many sets $F, F', \ldots$ that all satisfy the axiom without being (in an appropriate sense) equal.  This is the downside to adopting the fullness axiom, compared to power sets or function sets; this is also why the internal version is complicated.

As fullness is weaker than power sets in a constructive framework, we may consider it [[predicative mathematics|predicative]] in the sense of the constructive school; this is one reason for adopting it while rejecting power sets.  One reason for adopting it instead of (only) function sets is this: it is strong enough to construct the set $\mathbf{R}$ of (located Dedekind) [[real numbers]] from the set $\mathbf{N}$ of [[natural numbers]] *without* assuming [[excluded middle]] or [[countable choice]]; see below.

## Relation to type theory

Subset collection is justified by predicative versions of constructive [[type theory]] in the sense of Martin--Löf or Thierry Coquand.  To see this, note that these theories justify [[COSHEP]]; their types define (not sets in general but) _[[presets]]_ (following Bishop) or _[[completely presented sets]]_, which (as sets) are projective.  Analogously, an element of $F_{X,Y}$ may be considered an _operation_ or _prefunction_ (following 'preset') from $X$ to $Y$.

But note that the type of prefunctions from $X$ to $Y$, like the preset underlying $X$, is a categorical (unique) construction in type theory, while $F_{X,Y}$, like COSHEP, is not.  Thus type theory (or rather preset theory) gives us constructions that are not available in set theory (even with fullness or even COSHEP), such as the predicate that states whether two functions have equal underlying prefunctions.

## Definition of $\mathbf{R}$

We consider this application in some detail; see also [[real number object]].

Using [[function sets]], one can easily construct the set $\mathbf{R}_C$ of [[Cauchy reals]] as a [[subquotient]] of the function set $\mathbf{N}^{\mathbf{N}}$.  Both classical and constructive mathematicians usually adopt axioms in which one can prove that $\mathbf{R} = \mathbf{R}_C$; this is equivalent to a [[weak countable choice]] principle (identified by [[Fred Richman]] et al) that follows from either [[excluded middle]] (classically accepted) or [[countable choice]] (usually constructively accepted).  But there are (for example) some [[Grothendieck toposes]] in which $\mathbf{R} = \mathbf{R}_C$ (as a statement in the [[Mitchell–Bénabou language]]) fails, so any sort of constructive mathematics that wants to apply to such toposes cannot rely on this method to define the [[real numbers object]] $\mathbf{R}$.

Of course, in a topos, there are other ways to define $\mathbf{R}$, but what if you want your theory to *also* be predicative?  If you adopt the fullness axiom, then you can modify the definition of $\mathbf{R}_C$ to produce a definition of $\mathbf{R}$ *without* assuming any choice principle.  Instead of using $\mathbf{N}^{\mathbf{N}}$, you use $F_{\mathbf{N},\mathbf{N}}$ instead; $\mathbf{R}$ can be constructed as a subquotient of that set.

Let an element of $F_{\mathbf{N},\mathbf{N}}$ be a _pre[[sequence]]_ of natural numbers, and fix a [[bijection]] $\phi\colon \mathbf{N} \to \mathbf{Q}$ between $\mathbf{N}$ and the set $\mathbf{Q}$ of [[rational numbers]].  We call such a presequence $r$ _regular_ if, whenever $i,j,x,y$ are natural numbers such that $r(i,x)$ and $r(j,y)$, we have
$$ {|\phi(x) - \phi(y)|} \leq 1/i + 1/j .$$
(It is common, but not really necessary in the presence of function sets, to use regular sequences instead of the more general Cauchy sequences when defining $\mathbf{R}_C$ in constructive mathematics, and we follow that here.)  We call two regular presequences $r,s$ _equivalent_ if, whenever $i,j,x,y$ are natural numbers such that $r(i,x)$ and $s(j,y)$, we have
$$ {|\phi(x) - \phi(y)|} \leq 1/i + 1/j .$$
Then this defines an [[equivalence relation]] on the set of regular presequences, and the [[quotient set]] is $\mathbf{R}$.  (That is, it may be given, in the usual way, the structure of a [[linear order|linearly ordered]] [[Heyting field]] which is [[Dedekind-complete]] relative to located pairs of upper and lower sets.)

category: foundational axiom

[[!redirects fullness]]
[[!redirects axiom of fullness]]
[[!redirects fullness axiom]]

[[!redirects axiom of subset collection]]
[[!redirects subset collection]]


subsets are closed in a closed subspace precisely if they are closed in the ambient space

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

#Contents#
* table of contents
{:toc}

## Statement

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

Let $(X,\tau)$ be a [[topological space]], and let $C \subset X$ be a [[closed subset]], regarded as a [[topological subspace]] $(C,\tau_{sub})$. Then a subset $S \subset C$ is a [[closed subset]] of $(C,\tau_{sub})$ precisely if it is closed as a subset of $(X,\tau)$.

=--

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

If $S \subset C$ is closed in $(C,\tau_{sub})$ this means equivalently that there is an open open subset $V \subset C$ in $(C, \tau_{sub})$ such that

$$
  S = C \backslash V
  \,.
$$

But by the definition of the subspace topology, this means equivalently that there is a subset $U \subset X$ which is open in $(X,\tau)$ such that $V = U \cap C$. Hence the above is equivalent to the existence of an open subset $U \subset X$ such that

$$
  \begin{aligned}
    S & = C \backslash V
    \\
    & = C \backslash (U \cap C)
    \\
    & = C \backslash U
  \end{aligned}
  \,.
$$

But now the condition that $C$ itself is a closed subset of $(X,\tau)$ means equivalently that there is an open subset $W \subset X$ with $C = X \backslash W$. Hence the above is equivalent to the existence of two open subsets $W,U \subset X$ such that

$$
  S = (X \backslash W) \backslash U
   =
   X \backslash (W \cup U)
  \,.
$$

Since the union $W \cup U$ is again open, this implies that $S$ is closed in $(X,\tau)$.

Conversely, that $S \subset X$ is closed in $(X,\tau)$ means that there exists an open $T \subset X$ with  $S = X \backslash T \subset X$. This means that  $S = S \cap C = (X \backslash T) \cap C = C\backslash T = C \backslash (T \cap C)$, and since $T \cap C$ is open in $(C,\tau_{sub})$ by definition of the subspace topology, this means that $S \subset C$ is closed in $(C, \tau_{sub})$.

=--

## Related statements

* [[closed subsets of compact spaces are compact]]

* [[compact subspaces of Hausdorff spaces are closed]]

* [[closed subspaces of compact Hausdorff spaces are equivalently compact subspaces]]


subsets of lctvs

# Lexicography of Subsets of Topological Vector Spaces

## Idea

There are many different types of special subset of topological vector space.  The purpose here is to list many of them, to note easy relationships between them, and to remark on their use.

## Lexicography

Some of these only make sense if the vector space is the dual of some other space.

* Absorbing
* Bounded
* Bornivorous
* Closed
* Circled
* Compact
* Convex
* Equicontinuous
* Precompact
* Radial
* Weakly Compact
* Weakly Bounded
* Weakly Closed

[[!redirects Absorbing subset of a tvs]]
[[!redirects absorbing subset of a tvs]]
[[!redirects Absorbing subset of a lctvs]]
[[!redirects absorbing subset of a lctvs]]
[[!redirects Bounded subset of a tvs]]
[[!redirects bounded subset of a tvs]]
[[!redirects Bounded subset of a lctvs]]
[[!redirects bounded subset of a lctvs]]
[[!redirects Bornivorous subset of a tvs]]
[[!redirects bornivorous subset of a tvs]]
[[!redirects Bornivorous subset of a lctvs]]
[[!redirects bornivorous subset of a lctvs]]
[[!redirects Closed subset of a tvs]]
[[!redirects closed subset of a tvs]]
[[!redirects Closed subset of a lctvs]]
[[!redirects closed subset of a lctvs]]
[[!redirects Circled subset of a tvs]]
[[!redirects circled subset of a tvs]]
[[!redirects Circled subset of a lctvs]]
[[!redirects circled subset of a lctvs]]
[[!redirects Compact subset of a tvs]]
[[!redirects compact subset of a tvs]]
[[!redirects Compact subset of a lctvs]]
[[!redirects compact subset of a lctvs]]
[[!redirects Convex subset of a tvs]]
[[!redirects convex subset of a tvs]]
[[!redirects Convex subset of a lctvs]]
[[!redirects convex subset of a lctvs]]
[[!redirects equicontinuous]]
[[!redirects Equicontinuous subset of a tvs]]
[[!redirects equicontinuous subset of a tvs]]
[[!redirects Equicontinuous subset of a lctvs]]
[[!redirects equicontinuous subset of a lctvs]]
[[!redirects Precompact subset of a tvs]]
[[!redirects precompact subset of a tvs]]
[[!redirects Precompact subset of a lctvs]]
[[!redirects precompact subset of a lctvs]]
[[!redirects Radial subset of a tvs]]
[[!redirects radial subset of a tvs]]
[[!redirects Radial subset of a lctvs]]
[[!redirects radial subset of a lctvs]]
[[!redirects Weakly Compact subset of a tvs]]
[[!redirects weakly compact subset of a tvs]]
[[!redirects Weakly Compact subset of a lctvs]]
[[!redirects weakly compact subset of a lctvs]]
[[!redirects Weakly Bounded subset of a tvs]]
[[!redirects weakly bounded subset of a tvs]]
[[!redirects Weakly Bounded subset of a lctvs]]
[[!redirects weakly bounded subset of a lctvs]]


subsingleton

A **subsingleton** generally refers to a [[subset]] (of some ambient set $A$) having at most one element.  That is, it is a subset $B$ of $A$ such that any two elements of $B$ are equal.

Of course, classically any subsingleton is either [[empty set|empty]] or a [[singleton]], but [[constructive mathematics|constructively]] this need not hold.  In a [[topos]], the "object of subsingletons in $A$" is the [[partial map classifier]] for $A$, often denoted $A_\bot$.

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

Sometimes a slightly different convention is used: There what we call subsingletons are called **[[subterminal object|subterminals]]**, and a subset $B$ of $A$ is a *subsingleton* if and only if there exists an element $a \in A$ such that every element of $B$ is equal to $a$. With this nomenclature, any subsingleton is a subterminal, but the converse doesn't hold in general. (See _[[flabby sheaf]]_ for a class of examples where the converse does hold.)

=--

## See also

* [[h-proposition]]
* [[support]]
* [[decidable subsingleton]]

[[!include types and logic - table]]

[[!redirects subsingleton]]
[[!redirects subsingletons]]
[[!redirects subterminal]]
[[!redirects subterminals]]


subspace

# Subspaces
* table of contents
{: toc}

## Idea

Various more or less geometrical concepts are called [[spaces]], to name a few [[vector spaces]], [[topological spaces]], [[algebraic spaces]], ....  If such objects form a [[category]], it is natural to look for the [[subobjects]] and to call them _subspaces_.  However, often the natural subspaces in the field are the [[regular subobjects]]; conversely, it is also often the case that variants which are not subobjects in the categorical sense are allowed, such as an [[immersed submanifold]]. (These may have self-intersections, and then the immersion map is not monic, nor can this map be replaced by the inclusion of the image, since this image is usually not a manifold.)

## Definitions and examples

### Vector subspaces

These are very well behaved; as a [[vector space]] $X$ is simply a [[module]] over a [[field]], so a subspace of $X$ is simply a [[submodule]].  More generally, this is a special case of a [[subalgebra]].

Vector subspaces are precisely the [[subobjects]] in [[Vect]].

### Topological subspaces

Given a [[topological space]] $X$ (in the sense of [[Bourbaki]], that is: a set $X$ and a topology $\tau_X$) and a [[subset]] $Y$ of $X$, a topology $\tau_Y$ on a set $Y$ is said to be the topology **[[induced topology|induced]]** by the [[inclusion function|set inclusion]] $Y\subset X$ if $\tau_Y = \tau_X \cap_{pw} \{Y\} = \{ U \cap Y | U\in\tau_X \}$. The pair $(Y,\tau_Y)$ is then said to be a (topological) **subspace** of $(X,\tau_X)$.

If a [[continuous map]] $f:Z\to X$ is a [[homeomorphism]] onto its [[image]] $f(Z)$ in the induced topology on $f(Z)$, this *inclusion* map is sometimes called an [[embedding]]; $Z$ is thus isomorphic in [[Top]] to a subspace of $X$.

See at _[[topological subspace]]_.

### Topological vector subspaces

A 'subspace' of a [[topological vector space]] usually means simply a [[linear subspace]], that is a subspace of the underlying discrete vector space.

However, the subspaces that we really want in categories such as [[Ban]] are the *closed* linear subspaces.  (Essentially, this is because we want our subspaces to be [[complete space|complete]] whenever our objects are complete.)

### Sublocales

Given a [[locale]] $L$, which can also be thought of as a [[frame]], a [[sublocale]] of $L$ is given by a [[nucleus]] on the frame $L$.  Even if $L$ is [[topological locale|topological]], so that $L$ can be identified with a [[sober space|sober]] [[topological space]], still there are generally many more sublocales of $L$ than the topological ones.

### Submanifolds

... [[submanifold]] ...

## Subsites

For [[Grothendieck topology | Grothendieck topologies]], one instead of a subspace has a concept of a [[subsite]].

## Related concepts

* [[subobject]]

* [[linear subspace]]

[[!redirects subspace]]
[[!redirects subspaces]]


subspace topology

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

# Contents
* table of contents
{: toc}

## Definition

+-- {: .num_defn #SubspaceTopology}
###### Definition
**([[topological subspace|subspace topology]])**

Let $(X, \tau_X)$ be a [[topological space]], and let $Y \subset X$ be a [[subset]] of its underlying [[set]]. Then the corresponding _[[topological subspace]]_ has $Y$ as its underlying set, and its [[open subsets]] are those subsets of $Y$ which arise as restrictions of open subsets of $X$ (i.e. [[intersections]] of open subsets of $X$ with $Y$):

$$
  \left(
    U_Y \subset Y\,\,\text{open}
  \right)
    \,\Leftrightarrow\,
  \left(
     \underset{U_X \in \tau_X}{\exists} \left( U_Y = U_X \cap Y \right)
  \right)
  \,.
$$

In other words, $\tau_Y$ is the [[finer topology|smallest topology]] on $Y$ such that the inclusion $Y \hookrightarrow X$ is [[continuous map|continuous]] (the [[initial topology]] on that map).

The picture on the right shows two open subsets inside the [[square]], regarded as a [[topological subspace]] of the [[plane]] $\mathbb{R}^2$:

> graphics grabbed from [Munkres 75](#Munkres75)

=--

The pair $(Y,\tau_Y)$ is then said to be a *topological [[subspace]]* of $(X,\tau_X)$. The induced topology is for that reason sometimes called the **subspace topology** on $Y$.

A continuous function that factors as a [[homeomorphism]] onto its [[image]] equipped with the subspace topology is called an _[[embedding of topological spaces]]_.  Such a map is referred to as a _subspace inclusion_.

A property of topological spaces is said to be **hereditary** if its satisfaction for a topological space $X$ implies its satisfaction for all topological subspaces of $X$.

## Examples

* The [[open subsets]] of a [[closed interval]] $[0,1] \subset \mathbb{R}$ regarded as a topological subspace of the [[real line]] equipped with its [[Euclidean space|Euclidean]] [[metric topology]] is generated from the [[sub-base of a topology|sub-base]] $\beta = \{ [0, a) ,(a,1]\}_{a \in [0,1]}$.

The image on the right shows open subsets in the closed square $[0,1]^2$, regarded as a topological subspace of the Euclidean plane

* A subset of a [[metric space]] equipped with its [[induced metric]] is a subspace with respect to the corresponding [[metric topologies]].

## Properties

### Universal property
 {#UniversalProperty}

+-- {: .num_prop #UniversalPropertyOfSupspaceInclusion}
###### Proposition
**([[universal property]] of [[subspace topology]])**

Let $U \overset{i}{\longrightarrow} X$ be an [[injective function|injective]] [[continuous function]] between [[topological spaces]]. Then this is a [[topological subspace|subspace inclusion]] (Def. \ref{SubspaceTopology}) precisely if it satisfies the following [[universal property]]:

* For $Z$ any topological space, a [[function]] $Z \overset{f}{\longrightarrow} U$ (of underlying [[sets]]) is [[continuous function|continuous]] precisely if the [[composition]] $i \circ f$ is continuous as a function to $X$:

  $$
    \array{
       Z &\overset{f}{\longrightarrow}& U
       \\
       &{}_{\mathllap{i \circ f}}\searrow& \Big\downarrow{}^{\mathrlap{i}}
       \\
       && X
    }
  $$

=--

The elementary **proof** is spelled out, for instance, in [Terilla 14, theorem 1](#Terilla14). Of course this is just another way to speak of the [[initial topology]].

The universal characterization of Prop. \ref{UniversalPropertyOfSupspaceInclusion} lends itself to formalization via axioms for [[cohesion]]:

+-- {: .num_defn #SharpModalityOnTopologicalSpaces}
###### Definition
**([[sharp modality]] on [[topological spaces]])**

Let

$$
  Set
    \underoverset
      {\underset{coDisc}{\longrightarrow}}
      {\overset{\Gamma}{\longleftarrow}}
      {\bot}
  Top
$$

be the pair of [[adjoint functors]] given by sending a [[topological space]] $X$ to its underlying [[set]] $\Gamma(X)$, and by equipping a [[set]] $S$ with the [[codiscrete topology]] making it a [[codiscrete space]] $coDisc(X)$.

Write

$$
  \sharp \;\coloneqq\; coDisc \circ \Gamma
  \;\colon\;
  Top
    \longrightarrow
  Top
$$

for the induced [[modal operator]] on [[Top]] ([[sharp modality]]). We write

$$
  id \overset{\eta^{\sharp}}{\longrightarrow} \sharp
$$

for the [[unit of an adjunction|unit]] morphism of this [[adjunction]].

Notice that this means that for any [[topological space]] $Z$, _every_ function of underlying sets

$$
  Z \longrightarrow \sharp X
$$

is [[continuous functions]], hence that continuous functions into $\sharp X$ are in [[natural bijection]] to underlying functions of [[sets]]. This is the statement of the adjunction hom-isomorphism:

$$
  Hom_{Top}( Z, \sharp X )
  \;\simeq\;
  Hom_{Set}(\Gamma(Z), \Gamma(X))
  \,.
$$

=--

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

Let $U \overset{i}{\longrightarrow} X$ be an [[injective function|injective]] [[continuous function]] between [[topological spaces]]. Then this is a [[topological subspace|subspace inclusion]] (Def. \ref{SubspaceTopology}) precisely if its [[natural transformation|naturality square]] of the $\sharp$-[[unit of an adjunction|unit]] (Def. \ref{SharpModalityOnTopologicalSpaces})

$$
  \array{
    U &\overset{ \eta^\sharp_U }{\longrightarrow}& \sharp U
    \\
    {}^{\mathllap{i}}\Big\downarrow && \Big\downarrow{}^{\mathrlap{\sharp i}}
    \\
    X &\underset{\eta^\sharp_X}{\longrightarrow}& \sharp X
  }
$$

is a [[pullback]] square.

=--

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

By the [[universal property]] of a [[pullback]]/[[fiber product]] and the nature of $\sharp$, we have $U \simeq X \times_{\sharp X} \sharp U$ precisely if continuous functions out of some topological space $Z$ into $U$ are in [[natural bijection]] with continuous functions  $Z \to X$ whose underlying function $Z \to X \to \sharp X$ factors through the underlying function of $i$. This implies the statement by Prop. \ref{UniversalPropertyOfSupspaceInclusion}.

=--

+-- {: .num_remark #FormulationInCohesiveHoTT}
###### Remark
**(formulation in [[cohesive homotopy type theory]])**

The pullback square of the $\sharp$-unit in Prop. \ref{UniversalPropertyViaSharpModality} should correspond (after generalizing from [[topological spaces]] to suitable [[topological ∞-groupoids]]) to the [[categorical semantics]] of what in [[cohesive homotopy type theory]] is the statement that the [[characteristic function]]

$$
  \chi_U \;\colon\; X \to Prop
$$

to the [[universe of propositions]] factors through the universe of [[sharp modality|sharp]]-[[modal types]]. In this form topological subspace inclusions are characterized in [Shulman 15, Remark 3.14](#Shulman15).

=--

### General

A subspace $i: Y \hookrightarrow X$ is **closed** if $Y$ is [[closed subset|closed]] as a subset of $X$ (or if $i$ is a [[closed map]]), and is **open** if $Y$ is open as a subset of $X$ (or if $i$ is an [[open map]]).

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

Topological subspace inclusions ([[topological embedding]]) are precisely the [[regular monomorphisms]] in the [[category]] [[Top]] of all [[topological spaces]].

=--

For example, the [[equalizer]] of two maps $f, g \colon X \stackrel{\to}{\to} Y$ in [[Top]] is computed as the equalizer at the underlying-set level, equipped with the subspace topology.

+-- {: .num_lemma #pushout}
###### Lemma
The [[pushout]] in [[Top]] of any (closed/open) subspace $i \colon A \hookrightarrow B$ along any [[continuous function]] $f \colon A \to C$,

$$\array{
A & \stackrel{i}{\hookrightarrow} & B \\
\mathllap{f} \downarrow & po & \downarrow \mathrlap{g} \\
C & \underset{j}{\hookrightarrow} & D,
}$$

is a (closed/open) subspace $j: C \hookrightarrow D$.
=--

+-- {: .proof}
###### Proof
Since $U = \hom(1, -): Top \to Set$ is [[faithful functor|faithful]], we have that monos are reflected by $U$; also monos and pushouts are preserved by $U$ since $U$ has both a [[left adjoint]] and a [[right adjoint]]. In $Set$, the pushout of a mono along any map is a mono, so we conclude $j$ is monic in $Top$. Furthermore, such a pushout diagram in $Set$ is also a pullback, so that we have the [[Beck-Chevalley condition|Beck-Chevalley equality]] $\exists_i \circ f^\ast = g^\ast \exists_j \colon P(C) \to P(B)$ (where $\exists_i \colon P(A) \to P(B)$ is the [[direct image]] map between [[power sets]], and $f^\ast: P(C) \to P(A)$ is the [[inverse image]] map).

To prove that $j$ is a subspace, let $U \subseteq C$ be any open set. Then there exists open $V \subseteq B$ such that $i^\ast(V) = f^\ast(U)$ because $i$ is a subspace inclusion. If $\chi_U \colon C \to \mathbf{2}$ and $\chi_V \colon B \to \mathbf{2}$ are the maps to [[Sierpinski space]] that classify these open sets, then by the universal property of the pushout, there exists a unique continuous map $\chi_W \colon D \to \mathbf{2}$ which extends the pair of maps $\chi_U, \chi_V$. It follows that $j^{-1}(W) = U$, so that $j$ is a subspace inclusion.

If moreover $i$ is an open inclusion, then for any open $U \subseteq C$ we have that $j^\ast(\exists_j(U)) = U$ (since $j$ is monic) and (by Beck-Chevalley) $g^\ast(\exists_j(U)) = \exists_i(f^\ast(U))$ is open in $B$. By the definition of the topology on $D$, it follows that $\exists_j(U)$ is open, so that $j$ is an open inclusion. The same proof, replacing the word "open" with the word "closed" throughout, shows that the pushout of a closed inclusion $i$ is a closed inclusion $j$.
=--

A similar (but even simpler) line of argument establishes the following result.

+-- {: .num_lemma #transfinite}
###### Lemma
Let $\kappa$ be an [[ordinal]], viewed as a [[preorder]] category, and let $F: \kappa \to Top$ be a functor that preserves [[directed colimits]]. Then if $F(i \leq j)$ is a (closed/open) subspace inclusion for each morphism $i \leq j$ of $\kappa$, then the canonical map $F(0) \to colim_{i \in \kappa} F(i)$ is also a (closed/open) inclusion.
=--

## Variations

There is also a notion of a [[Grothendieck topology]] induced along a [[functor]] from a Grothendieck topology on another category (actually the input can be a somewhat more general [[coverage]], then the topology induced along the [[identity functor]] will serve as a sort of a completion). (this will be explained later).

A topology may be induced by more than a [[function]] other than a subset inclusion, or indeed by a family of functions out of $Y$ (not necessarily all with the same [[target]]). However, the term 'induced topology' is often (usually?) restricted to subspaces; the general concept is called a [[weak topology]].  (This construction can be done in any [[topological concrete category]]; in this generality it is often called an [[initial structure]] for a [[sink|source]].)  The dual construction (involving functions to $Y$) is a [[strong topology]] (or [[final structure]] for a [[sink]]); an example is the [[quotient topology]] on a [[quotient space]].

## Related concepts

* [[embedding]]

* [[embedding of topological spaces]]

* [[embedding of smooth manifolds]]

[[!include universal constructions of topological spaces -- table]]

## References

* {#Bourbaki71} [[Nicolas Bourbaki]], _Elements of Mathematics -- General topology_, 1971, 1990

* {#Munkres75} [[James Munkres]], _Topology_, Prentice Hall (1975, 2000)

* {#Terilla14} [[John Terilla]],  _Notes on categories, the subspace topology and the product topology_ 2014 ([pdf](https://math.mit.edu/~jhirsh/terilla_subspace_quotient.pdf))

* {#Shulman15} [[Mike Shulman]], _Brouwer's fixed-point theorem in real-cohesive homotopy type theory_, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))

[[!redirects subspace topology]]
[[!redirects subspace topologies]]

[[!redirects topological subspace]]
[[!redirects topological subspaces]]

[[!redirects subspace of a topological space]]
[[!redirects subspaces of a topological space]]


substance

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

## In philosophy

In [[philosophy]] a concept called _substance_ plays the central role in [[Spinoza's system]] and in [[Descartes]]'s thoughts ([LectHistPhil-Descartes](Lectures+on+the+History+of+Philosophy#DescartesSubstance)).

In [[Kant]]'s _critique of pure reason_ this appears as one of the _[[category (philosophy)|categories of pure thought]]_. Here "substance" is to go along with _substantial_, as opposede to _accidental_.

In [[Hegel]]'s _[[Science of Logic]]_ this is claimed to be recovered from a more  fundamental "process" as "the final unity of essence and being" ([§1235](Science%20of%20Logic#1235)).

## References

* Stanford Encyclopedia of Philosophy, _[Substance](http://plato.stanford.edu/entries/substance/)_


substitution

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

# Contents
* table of contents
{: toc}

## Idea

Syntactic _substitution_ of/for [[variables]] is one of the basic operations in formal mathematics, such as in [[formal logic]] and [[type theory|type theories]], where it is part of the *[[structural rules]]*.

## Syntactic substitution

Substitution is usually defined as an *operation* on expressions (such [[strings (computer science)|strings]] of letters from an [[alphabet]] and representing [[terms]], [[formulas]], [[propositions]], [[dependent types]], etc.) containing [[variables]].

Suppose that $P$ is an expression in the [[context]] of a [[variable]] $x \,\colon\, X$ of [[type]], and that $t \colon X$ is an expression which has the same [[type]] as $x$.  Then one denotes by

$$
  P[t/x]
$$

the result of **substituting** $t$ for all occurrences of $x$ in $P$.

For example, if $P$ is $x^2 + 2x y + 3$ and $t$ is $(y+z)$, then $P[t/x]$ is $(y+z)^2 + 2(y+z)y + 3$.

\begin{remark}
**(substitution is a meta-operation on syntax)**
\label{AsOperationOnSyntax}
In this approach, substitution is an *operation on syntax*, not an element of [[syntax]] itself. In particular, the bracket notation $[t/x]$ is part of "meta-syntax", not the syntax in question.

That is, the literal string of symbols "$P[t/x]$" is not itself an expression in the language under consideration, but *denotes* such an expression, in the same way that "$2+2$" is not literally an [[integer]] but *denotes* the [[numeral]] "$4$".

Compare also Rem. \ref{TypesettingByActualSubstitution} below.
\end{remark}

### Simultaneous substitution

Substitution for multiple variables does not, in general, commute.  That is, the expressions

$$
  P[t/x][s/y]
  \qquad\text{and}\qquad
  P[s/y][t/x]
$$

are not in general the same: The former substitutes $s$ for occurrences of $y$ in $t$, but not $t$ for occurrences of $x$ in $s$, while the latter has the opposite behavior.  We also write

$$ P[t,s/x,y] $$

to denote the *simultaneous substitution* of $t$ for $x$ and $s$ for $y$, in which *neither* occurrences of $x$ in $s$ nor occurrences of $y$ in $t$ are substituted for; this is generally not the same as either iterated substitution.

### Avoiding variable capture

If the language in question contains variable [[bound variable|binders]], then there is a subtlety to substitution: if $t$ contains free variables that are bound in $P$, then we cannot simply textually substitute $t$ for $x$ and obtain an expression with the desired meaning.

For instance, if $P$ is $\exists y (x + y = 1)$, and $t$ is the free variable $y$, then a literal interpretation of $P[t/x]$ would produce $\exists y (y + y = 1)$.  But $P$ is true (universally in its free variables) if the variables have type $\mathbb{Z}$, while $\exists y (y + y = 1)$ is not.  The free variable $y$ in $t$ has been "captured" by the binder $\exists y$ in $P$.

We say that $t$ is **substitutable** for $x$ in $P$ if performing a literal textual substitution as above would not result in undesired variable capture.  If $t$ is not substitutable for $x$ in $P$, then we can always replace $P$ by an [[alpha-equivalent]] expression in which $t$ *is* substitutable for $x$.  Since we often consider formulas only up to $\alpha$-equivalence anyway, one usually defines the notation "$P[t/x]$" to include an $\alpha$-conversion of $P$, if necessary, to make $t$ substitutable for $x$.

In computer implementations of type theories, however, the issue of variable binding and capture is one of the trickiest things to get right.  Performing $\alpha$-conversions is difficult and tedious, and other solutions exist, such as using [[de Bruijn indices]] to represent bound variables.

### As an admissible type inference rule

A general property of [[type theories]] (and other formal mathematics) is that *substitution is an admissible rule*.

Roughly, this means that if $P$ is an [[term|expression]] of some [[type]], then so is the result $P[t/x]$ of substitution (as long as $t$ and $x$ have the same type).  This is generally not a rule "put into" the theory, but rather a property one *proves* about the theory; type theorists say that substitution is an [[admissible rule]] rather than a [[derivable rule]].

For instance, in the language of [[dependent type theory]] the following *substitution rule* (one of the [[structural rules]]) is an [[admissible rule]]:

\[
  \label{SubstitutionRuleInDependentTypeTheory}
  Sub
  \frac{
    \Gamma,\; (x:A) \vdash P \;type
    \qquad
    \Gamma \vdash (t:A)
  }{
   \Gamma \vdash P[t/x] \;type
  }
  \,.
\]

Here "admissibility" means that if there exist derivations of $\Gamma \vdash (t:A)$ and $(x:A) \vdash P \;type$, then there also exists a derivation of $\Gamma \vdash P[t/x] \;type$.  By contrast, saying that this is a derivable rule would mean that it can occur itself as *part* of a derivation, rather than being a meta-statement *about* derivations.

The substition rule is closely related to the [[cut rule]], and admissibility of such rules is generally proven by [[cut elimination]].

\begin{remark}
\label{TypesettingByActualSubstitution}
**(alternative typesetting of substitution rule)**
\linebreak
In view of Remark \ref{AsOperationOnSyntax} we may re-express the substitution rule (eq:SubstitutionRuleInDependentTypeTheory) by using actual syntactic substitutions instead of the meta-instruction "[t/x]" to perform these, if only we make explicit the [[variable]] [[dependent type|dependency]] of all [[terms]], say by a subscript:

$$
  Sub
  \frac{
      \gamma \colon \Gamma
      ,\;\;
      a_\gamma \colon A_\gamma
      \;\;\;
      \vdash
      \;\;\;
      P_{a_\gamma} \;type
      \;\;\;\;\;\;\;\;\;\;\;
      \gamma \colon \Gamma
      \;\;\;
      \vdash
      \;\;\;
      t_\gamma \colon A
  }{
    \gamma \colon \Gamma
    \;\;\;
      \vdash
    \;\;\;
    P_{t_\gamma} \;type
  }
$$

\end{remark}

## Explicit substitution

An alternative approach to substitution is to make substitution part of the object language rather than the metalanguage.  That is, the notation

$$P[t/x]$$

is now actually itself a string of the language under consideration.  One then needs reduction or equality rules describing the relationship of this string $P[t/x]$ to the result of actually substituting $t$ for $x$ as in the usual approach.  See [[explicit substitution]] for more details.

## Categorical semantics
 {#CategoricalSemantics}

### Definition

In the [[categorical semantics]] of [[type theory]]:

* Recalling that [[terms]] are interpreted by [[morphisms]], substitution of a term into another term is interpreted by [[composition]] of the relevant morphisms.

* Recalling that [[propositions]] are interpreted by [[subobjects]], substitution of a term $t$ into a proposition $P$ is interpreted by [[pullback]] or [[inverse image]] of the subobject interpreting $P$ along the morphism interpreting $t$.

* Recalling that [[dependent types]] are interpreted by [[display maps]], substitution of a term $t$ into a dependent type $B$ is interpreted by [[pullback]] of the display map interpreting $B$ along the morphism interpreting $t$.

* Or else, since [[dependent types]] are also given by [[classifying morphisms]] into a [[type of types]], substitution corresponds to [[composition]] of these classifying morphisms with the given morphism.

In the third case, there is a [[coherence]] issue: syntactic substitution in the usual approach is strictly [[associativity|associative]], whereas pullback in a category is not.  One way to deal with this is by using explicit substitution as described above.  Another way is to strictify the category before modeling type theory; see _[[categorical model of dependent types]]_. For literature see ([Curien-Garner-Hofmann](#CurienGarnerHofmann), [Lumsdaine-Warren 13](#LumsdaineWarren13)).

### Examples

Let $\mathcal{C}$ be a suitable ambient [[category]] in which we are [[categorical semantics|interpreting]] [[logic]]/[[type theory]].

Suppose $X$ and $Y$ are [[types]], hence interpreted as [[objects]] of $\mathcal{C}$. Then a [[term]] of [[function type]] $f : X \to Y$ is interpreted by a [[morphism]], going by the same symbols.

Now a [[proposition]] about [[terms]] of [[type]] $Y$

$$
  y : Y \vdash P(y)
$$

is interpreted as an [[object]] of the [[slice category]] $\mathcal{C}_{/Y}$, specifically as a [[truncated object of an (infinity,1)-category|(-1)-truncated]] object if it is to be a [[proposition]], hence by a [[monomorphism]]

$$
  \array{
    P
    \\
    \downarrow
    \\
    Y
  }
  \,.
$$

For instance if $\mathcal{C} = $ [[Set]] then this is the inclusion of the [[subset]] of elements of $Y$ on which $P$ is true. And generally we may write

$$
  P = \{y : Y | isInhab(P(y)) \}
  \,.
$$

Now finally the **substitution** of $f(x)$ for $y$ in $P$, hence the proposition

$$
  \array{
    P(f(-))
    \\
    \downarrow
    \\
    X
  }
$$

is interpreted as the [[pullback]]

$$
  \array{
    P(f(-)) \coloneqq & f^* P &\to& P
    \\
    & \downarrow && \downarrow
    \\
    & X &\stackrel{f}{\to}& Y
  }
  \,.
$$

Notice that [[monomorphisms]] are preserved by pullback, so that this is indeed again the correct interpretation of a proposition.

Specifically, if $X$ is the [[unit]] type it is interpreted as a [[terminal object]] of $\mathcal{C}$, and then the [[function]] $f$ is identified simply with a [[term]] $y_0 \coloneqq f(*) $. In this case the substitution is _evaluation_ of the proposition at $y_0$, the resulting monomorphism

$$
  \array{
    P(y_0)
    &\longrightarrow&
    P
    \\
    \big\downarrow
      &&
    \big\downarrow
    \\
    \ast
    &
    \overset
      {y_0}
      {\longrightarrow}
    &
    Y
  }
$$

over the terminal object is a [[truth value]]: the truth value of $P$ at $y_0$.

## Related concepts

[[!include substitution natural deduction - table]]

* [[structural rules]]

  * [[contraction rule]]

  * [[weakening rule]]

  * [[exchange rule]]

  * [[variable rule]]

  * [[substitution rule]]

* [[explicit substitution]]

* [[context extension]]

* [[telescope]]

* [[dependent product]], [[universal quantifier]]

* [[dependent sum]], [[existential quantifier]]

## References
 {#References}

Discussion of the substitution rule in [[intuitionistic type theory|intuitionistic]] [[dependent type theory]]:

* {#Jacobs98} [[Bart Jacobs]], p. 123 in: *Categorical Logic and Type Theory*, Studies in Logic and the Foundations of Mathematics **141**, Elsevier (1998)  [[ISBN:978-0-444-50170-7](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/141), [pdf](https://people.mpi-sws.org/~dreyer/courses/catlogic/jacobs.pdf)]

  > (emphasis on the [[categorical model of dependent types]])

* {#UFP13} [[Univalent Foundations Project]], p. 423 of: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

  > (in the context of [[homotopy type theory]])

* {#Rijke18} [[Egbert Rijke]], p. 4 of: *Dependent type theory* [[pdf](https://www.andrew.cmu.edu/user/erijke/hott/dtt.pdf)], Lecture 1 in: *[[Introduction to Homotopy Type Theory]]*, lecture notes, CMU (2018) [[pdf](http://www.andrew.cmu.edu/user/erijke/hott/hott_intro.pdf), [[Rijke-IntroductionHoTT-2018.pdf:file]], [webpage](https://www.andrew.cmu.edu/user/erijke/hott/)]

  > (in the context of [[homotopy type theory]])

The observation that substitution forms an [[adjoint pair]]/[[adjoint triple]] with [[quantifiers]] is due to

* [[Bill Lawvere]], _Adjointness in Foundations_, ([TAC](http://www.emis.de/journals/TAC/reprints/articles/16/tr16abs.html)), Dialectica 23 (1969), 281-296

and further developed in

* [[Bill Lawvere]], _Equality in hyperdoctrines and
comprehension schema as an adjoint functor_, Proceedings of the AMS Symposium on Pure Mathematics XVII (1970), 1-14.

Exposition of the interpretation of substitution as pullback:

* [[Andrej Bauer]], _[Substitution is pullback](http://math.andrej.com/2012/09/28/substitution-is-pullback/)_, 2012

The [[coherence]] issue involved in making this precise is discussed in

* {#CurienGarnerHofmann} [[Pierre-Louis Curien]], [[Richard Garner]], [[Martin Hofmann]], _Revisiting the categorical interpretation of dependent type theory_ ([[CurienGarnerHofmann.pdf:file]])

* {#LumsdaineWarren13} [[Peter LeFanu Lumsdaine]], [[Michael Warren]], _An overlooked coherence construction for dependent type theory_, CT2013  ([[LumsdaineWarren2013.pdf:file]])

[[!redirects substitution]]
[[!redirects substitutions]]
[[!redirects variable substitution]]
[[!redirects variable substitutions]]

[[!redirects variable capture]]
[[!redirects variable captures]]

[[!redirects substitution rule]]
[[!redirects substitution rules]]


substitution natural deduction - table

| | [[type theory]] | [[category theory]] |
|--|--|--|
|  | [[syntax]] | [[semantics]] |
|  | [[natural deduction]] |  [[universal construction]] |
|  | **[[substitution]]**......................... | **[[pullback]] of [[display maps]]** |
|  | $\frac{ x_2 \colon X_2\; \vdash\; A(x_2) \colon Type \;\;\;\; x_1 \colon X_1\; \vdash \; f(x_1)\colon X_2}{ x_1 \colon X_1 \;\vdash  A(f(x_1)) \colon Type}$  |$\,$  $\array{ f^* A &\to& A \\ \downarrow && \downarrow \\ X_1 &\stackrel{f}{\to}& X_2 }$ |


substructural logic

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

#Contents#
* table of contents
{:toc}

## Idea

**Substructural logic** is a general term for [[logics]] in which the *[[structural inference rules]]*:

* the [[contraction rule]]

* the [[weakening rule]]

* the [[exchange rule]]

are not necessarily assumes, or only with restrictions.

Some particular substructural logics include:

* [[linear logic]], perhaps the best-known to category theorists, omits the contraction and weakening rules.  "Noncommutative linear logic" omits also the exchange rule.

* [[affine logic]] omits only the contraction rule.  One might call it "coaffine logic" if we omit only the weakening rule.

* Some forms of [[relevant logic]] and [[paraconsistent logic]] can be regarded as substructural logics (often of the coaffine variety).

## Related concepts

* [[bunched logic]]

* [[structural rules]]

## References

* [[Kosta Došen]], Peter Schroder-Heister, *Substructural Logics*, Clarendon Press (1993) [[ISBN:9780198537779](https://global.oup.com/academic/product/substructural-logics-9780198537779?cc=de&lang=en&)]

[[!redirects substructural logics]]


substructure complete theory

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

#Contents#
* table of contents
{:toc}

## Idea
One wants an equivalent and purely [[semantics|semantic]] characterization of [[quantifier elimination]].

A [[first-order theory|theory]] $T$ is substructure complete if after fixing a _substructure_, the theory of the [[elementary class]] of all $T$-models into which that substructure embeds is _complete_.

## Definition

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

A [[first-order theory]] $T$ in a language $\mathcal{L}$ is **substructure complete** if for any $M \in \mathbf{Mod}(T)$ and any $\mathcal{L}$-[[first-order structure|substructure]] $A \hookrightarrow M,$ the theory $T_{\mathsf{Diag}(A)}$ (expansion by $A$'s [[quantifier-free diagram]])---whose models are precisely those models $N \in \mathbf{Mod}(T)$ such that there exists an embedding of $\mathcal{L}$-structures $A \hookrightarrow N$---is complete.

=--

## Equivalence with quantifier elimination

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

$T$ [[quantifier elimination|eliminates quantifiers]] if and only if $T$ is substructure-complete.

=--

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

$(\Rightarrow)$ Suppose $T$ eliminates quantifiers. Fix a substructure $A$. Since quantifier-free sentences are automatically transferred by embeddings (whereas without the additional property of quantifier-freeness, we would require the additional property that an embedding be [[elementary embedding|elementary]]), then for any two models $M, N \in \mathbf{Mod}(T)$

$$
M \leftarrow A \rightarrow N
$$

containing $A$, we have that for every sentence $\varphi$ satisfied by $N$, $T \models \varphi \leftrightarrow \psi$ where $\psi$ is quantifier-free. Therefore, $\psi$ transfers to $A$ and $M$, so $M \models \varphi$ as well, and vice-versa.

$(\Leftarrow)$ Now suppose $T$ is substructure complete. Fix $\varphi(x)$ an $\mathcal{L}$-formula. Consider the [[type (in model theory)|type]] $\Phi$ of its quantifier-free consequences,i .e.

$$
\Phi(x) \overset{\operatorname{df}}{=} \{\varphi^*(x) \text{ quantifier-free } \operatorname{|} T \models \varphi \implies \varphi^*\}.
$$

(This thing is like the quantifier-free principal ultrafilter on $\varphi$ in the Boolean algebra of definable sets.)

Let $d_1, \dots, d_n$ be distinct new constant symbols. Write $d = \vec{d} = (d_1, \dots, d_n)$.

Claim: $T \cup \Phi(d) \models \varphi(d).$

(This claim says, roughly, that knowing the principal ultrafilter of quantifier-free definable sets containing $\varphi(x)$ suffices to pin down $\varphi(x)$. After establishing the claim, it will be easy to improve it to full quantifier elimination.)

Proof of claim: suppose towards a contradiction that there exists an $A \models T \cup \Phi(d) \cup \{\not \varphi(d)\}.$ (This is a structure in the language $\mathcal{L}$ of $T$, expanded by $d$.) Let $d^A$ be the interpretation of the new constants $d$ in $A$. Consider the finitely-generated $\mathcal{L}$-substructure of $A$ around $d^A$:

$$
C \overset{\operatorname{df}}{=} \langle d^A \rangle_{\mathcal{L}}.
$$

Subclaim: $T \cup \{\varphi(d^A)\} \cup \mathsf{Diag}(C)$ _is_ satisfiable.

(This subclaim amounts to saying that while $A$ models $\neg \varphi(d^A)$, we can find another model $B \models T$, embedding $C$, such that $B$ _does_ models $\varphi(d^A)$.)

Proof of subclaim: this will be a standard [[compactness theorem|compactness argument]]. Suppose not; by compactness there exist finitely many sentences $\theta_1, \dots, \theta_m \in \mathsf{Diag}(C)$ such that

$$
T \cup \{\varphi(d^A)\} \cup \{\theta_i\}_{1 \leq i \leq m}
$$

is not satisfiable. By the definition of [[quantifier-free diagram]],  we can write $\theta_i = \theta'_i(d^A)$, where $\theta'_i$ is a quantifier-free $\mathcal{L}$-formula.

Therefore,

$$
T \cup \{\varphi(d^A), \theta'_0(d^A), \theta'_1(d^A), \dots, \theta_m'(d^A)\}
$$

is not satisfiable, where (remembering that $d^A$ is now just a tuple of constant symbols) $\theta'_0(x)$ specifies that all the entries of $x$ are distinct.

Therefore,

$$
T \models \varphi(d^A) \rightarrow \bigvee_{i = 0}^m \neg \theta'_i(d^A).
$$

Since there are no constraints on the constant symbols $d^A$ and $\theta'_0$ specifies that they are all distinct, we may generalize them, so that

$$
T \models \forall x \left( \varphi(x) \to \bigvee_{i = 0}^m \neg \theta'_i(x) \right)
$$

hence

$$
\bigvee_{i = 0}^m \neg \theta'_i(x) \in \Phi(x).
$$

Now, by assumption, $A \models \Phi(d)$. This means in particular that $A \models \bigvee_{i = 0}^m \neg \theta'_i(d^A).$

Since that disjunction is quantifier-free, it transfers down to $C$. Therefore, there is some $0 \leq j \leq m$ such that $C \models \neg \theta'_j(d^A).$ Since the $d^A$ were distinct when we obtained $A$ (from which we obtained $C$), we can rule out $j \neq 0$. But for each $j = 1, \dots, m$, $\theta_j = \theta'_j(d^A) \in \mathsf{Diag}(C)$. This is a contradiction.

This proves the subclaim.

Now we proceed with proving the claim.

Let $\mathbf{B} \models T \cup \{\varphi(d^A)\} \cup \mathsf{Diag}(C).$

Then:

$$
A \models \neg \varphi(d^A), \text{ and } B \models \varphi(d^A),
$$

which contradicts substructure completeness over $C$.

This proves the claim.

Now we proceed with proving the theorem.

With the claim in hand, the [[compactness theorem]] tells us that the entailment $T \cup \Phi(x) \models \varphi(d)$ is finitely supported, so that there are finitely many $\varphi^*_1, \dots, \varphi^*_m \in \Phi(x)$ such that

$$
T \cup \{\varphi_i^*(d)\}_{i \leq m} \models \varphi(d).
$$

Write $\varphi^* \overset{\operatorname{df}}{=} \bigvee_{i \leq m} \varphi_i^*.$ Again, we generalize the constants $d$, obtaining

$$
T \models \forall x \left( \varphi \leftrightarrows \varphi^* \right).
$$

Since the $\varphi^*_i$ are all quantifier-free, and $\varphi$ was arbitrary, we have proved that $T$ eliminates quantifiers.
=--

## Examples

* By the above theorem, any theory which [[quantifier elimination|eliminates quantifiers]] is substructure complete.

* So e.g. the [[countable random graph]], the theory [[ACF]] of algebraically closed fields, the theory [[DLO]] of dense linear orders...

## Remarks
* The analogous (and weaker) property where one asks that the theory of the elementary class of $T$-models embedding a fixed $T$-model is complete is called [[model complete theory | model completeness]].

* Of course, substructure-completeness implies model-completeness.

## Related concepts
* [[quantifier elimination]]

* [[model complete theory]]

* [[existentially closed model]]

* [[diagram of a first-order structure]]

## References

* {#Marker}Dave Marker, _Model theory: an introduction_, theorem 3.14

[[!redirects substructure-complete]]

[[!redirects substructure completeness]]

[[!redirects substructure-completeness]]

[[!redirects substructure complete theories]]

[[!redirects substructure-complete theories]]

[[!redirects substructure complete]]


subsystem

#Contents#
* table of contents
{:toc}

## Idea

For _subsystem_ of a [[physical system]] in [[quantum mechanics]], see:

* [subsystem](quantum%20mechanics#Subsystems)

  * [independent subsystem](quantum%20mechanics#Subsystems), [[entanglement]]

## Related concepts

* [[local quantum field theory]], [[local prequantum field theory]]

* [[causal locality]], [[Einstein locality]]

* [[entanglement entropy]]

* [[parameterized quantum system]]

[[!redirects subsystems]]

[[!redirects independent subsystem]]
[[!redirects independent subsystems]]


subterminal object

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

#Contents#
* table of contents
{:toc}

## Definition

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

An [[object]] $U$ in a [[category]] $C$ is **subterminal** or **preterminal** if any two [[morphism]]s with [[target]] $U$ and the same source are equal.  In other words, $U$ is subterminal if for any object $X$, there is at most one morphism $X\to U$.

=--

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

An __[[umbrella category]]__ is a nonempty category $C$ such that for every object $X$ in $C$, there is at least one subterminal object $T$ such that $C(X,T)$ is nonempty (hence being a singleton).

=--

## Properties

If $C$ has a [[terminal object]] $1$, then $U$ is subterminal precisely if the unique morphism $U \to 1$ is [[monomorphism|monic]], so that $U$ represents a [[subobject]] of $1$; hence the name "sub-terminal."

This is equivalent to the hypothesis that the [[cone]] given by identity morphisms $U \leftarrow U \rightarrow U$ is a [[product]] cone, or that *some* [[product]] $U \times U$ exists and the [[diagonal]] $U \to U \times U$ is an [[isomorphism]].

Therefore for a [[sheaf topos]] over a [[topological space]] the subterminal objects of the topos are the [[open subsets]] of the topological space. Accordingly, the subterminal objects in any [[topos]] are also called _open objects_ (e.g. [Johnstone 77, p. 94](#Johnstone77))

The [[classifying topos]] for subterminal objects (hence open objects) in [[toposes]] is the [[Sierpinski topos]] (see e.g. [Johnstone 77, p. 117](#Johnstone77)).

## Examples

The subterminal objects in a [[topos]] can be viewed as its "external [[truth value]]s."  For example, in the topos $Sh(X)$ of [[sheaf|sheaves]] on a [[topological space]] $X$, the subterminal objects are precisely the open sets in $X$.

The *support* of an object $X$ in a topos is the [[image]] $U \hookrightarrow 1$ of the unique map $X \to 1$. Any map $U \to X$ is necessarily a [[section]] of $X \to U$.

## Related concepts

* [[subsingleton]]

* [[support object]]

[[!include types and logic - table]]

## References

* {#Johnstone77} [[Peter Johnstone]], _Topos theory_, London Math. Soc. Monographs __10__, Acad. Press 1977

* [[Dieter Pumplün]], _Initial morphisms and monomorphisms_, Manuscripta mathematica 32 (1980): 309-333.

[[!redirects subterminal]]
[[!redirects subterminals]]
[[!redirects subterminal objects]]

[[!redirects preterminal]]
[[!redirects preterminal object]]
[[!redirects preterminal objects]]

[[!redirects open object]]
[[!redirects open objects]]


subtopos

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

# Contents
* table of contents
{:toc}

##Idea
A **subtopos** of a topos is a generalization of the concept of a subspace of a [[topological space]].

##Definition
For $\mathcal{E}$ a [[topos]], a **subtopos** is another topos $\mathcal{F}$ equipped with a  [[geometric embedding]] $\mathcal{F} \hookrightarrow \mathcal{E}$.

If this is an [[open geometric morphism]] (or an [[essential geometric morphism]]) one speaks of an **[[open subtopos]]** (an **[[essential subtopos]]**, respectively, also called a _[[level of a topos]]_).

##Properties

### Sheaves, localization, closure and reflection

If $\mathcal{E}$ is an [[elementary topos]] then subtoposes correspond to [[Lawvere-Tierney topologies]] $j$ on $\mathcal{E}$,  to [[localization|localizations]] of $\mathcal{E}$ as well as to [[closure operator|universal closure operators]] on $\mathcal{E}$.

### For classifying toposes

Every Grothendieck topos $\mathcal{E}$ over $Set$ is (equivalent to) the [[classifying topos]] of some [[geometric theory]] $T$ and it can be shown that subtoposes of $\mathcal{E}$ correspond precisely to **deductively closed quotient theories** of $T$ ([Caramello (2009)](#Caramello09); thm. 3.6) i.e. passage to a subtopos corresponds to adding further geometric [[axioms]] to $T$ - localizing geometrically amounts to theory refinement logically.

### The lattice of subtoposes
 {#LatticeOfSubtoposes}

The inclusions induce an ordering on the subtoposes of $\mathcal{E}$ that makes them a [[lattice]] with a [[co-Heyting algebra]] structure i.e. the [[join]] operator has a [[left adjoint]] _[[subtraction]]_ operator. This corresponds to a dual [[Heyting algebra]] structure on the [[Lawvere-Tierney topologies]].

The lattice structure is further analyzed in ([Johnstone (2002)](#Johnstone02), [Caramello (2009)](#Caramello09)). We contend us to report

+-- {: .num_prop #subtopos_meet}
###### Proposition
Let $Sh_j(\mathcal{E})$, $Sh_k(\mathcal{E})$ be two subtoposes of a topos  $\mathcal{E}$ and $j,k$ the corresponding topologies with $j\vee k$ their join in the lattice of topologies. Then the following holds: $Sh_j(\mathcal{E})\wedge Sh_k(\mathcal{E})=Sh_{j\vee k}(\mathcal{E})=Sh_j(\mathcal{E})\cap Sh_k(\mathcal{E})$.
=--

(cf. [Johnstone (2002)](#Johnstone02), p.217). In other words, the meet of two subtoposes is just their intersection (and is equivalently given by the subtopos corresponding to the join of their topologies).

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

The _[[atoms]] in the lattice of subtoposes of $\mathcal{E}$ are precisely the two-valued [[Boolean topos|Boolean]]  subtoposes_ of $\mathcal{E}$.

=--

([Caramello (2009)](#Caramello09); prop. 10.1). This follows from the fact that two-valued and Boolean toposes are opposite extremes when it comes to dense subtoposes: in a two-valued topos every non-trivial subtopos is dense, whereas a Boolean topos has no non-trivial dense subtopos (cf. at [[dense subtopos]] for further details).

## Related pages

* [[open subtopos]]

* [[closed subtopos]]

* [[locally closed subtopos]]

* [[dense subtopos]]

* [[level]]

* [[co-Heyting boundary]]

* [[Artin gluing]]

##References

* {#SGA4}[[M. Artin]], [[A. Grothendieck]], [[J. L. Verdier]], _Théorie des Topos et Cohomologie Etale des Schémas ([[SGA4]])_, LNM **269** Springer Heidelberg 1972. (exposé IV, section 9, pp.431ff)

* {#BK91}[[Francis Borceux|F. Borceux]], M. Korostenski, _Open Localizations_ , JPAA **74** (1991) pp.229-238.

* {#Caramello09} [[Olivia Caramello|O. Caramello]], pp.15,58 of _Lattices of theories_,  (2009). ([arXiv:0905.0299](http://arxiv.org/abs/0905.0299))

* H. Forssell, _Subgroupoids and quotient theories_ , TAC **28** no.18 (2013) pp.541-551. ([pdf](http://www.emis.de/journals/TAC/volumes/28/18/28-18.pdf))

* {#Johnstone02}[[Peter Johnstone]], _Sketches of an [[Elephant]] I_, Oxford UP 2002. (pp.195-223)

* {#Johnstone77}[[Peter Johnstone]], _Topos Theory_ , Academic Press New York 1977. (Dover reprint 2014, section 3.5)

* {#KL89}[[G. M. Kelly]], [[William Lawvere|F. W. Lawvere]], _On the Complete Lattice of Essential Localizations_ , Bull. Soc. Math. de Belgique **XLI** (1989) 261-299 [[[Kelly-Lawvere_EssentialLocalizations.pdf:file]]]

[[!redirects Subtopos]]
[[!redirects subtoposes]]
[[!redirects Subtoposes]]

[[!redirects lattice of subtoposes]]
[[!redirects lattices of subtoposes]]

[[!redirects subtopos lattice]]
[[!redirects subtopos lattices]]


subtraction

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

# Subtraction
* table of contents
{: toc}

## Idea

The basic example of subtraction is, of course, the partial operation in the monoid of natural numbers or in the integers. It is often the first illustration of a non-associative operation met in abstract algebra. We think of subtraction as an operation $s:\mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}$, where, of course, $s(m,n)=m-n$.

There are numerous related abstractions of this, relating to different aspects of the basic operation.

## Abstract

...

## Definition

1. (Universal Algebra) In the sense of Ursini  in the context of a varietal theory,  a _subtraction term_, $s$, is a binary term $s$ satisfying $s(x, x) = 0$ and $s(x, 0) = x$.
(see [[subtractive variety]]

1. In a [[co-Heyting algebra]], which are models of [[subtractive logic]], subtraction is the operation [[left adjoint]] to the [[join]] operator:
$$ (- \setminus y) \dashv (y \vee -)$$

## Properties

...

## Examples

* Subtraction in the [[commutative monoid]] of the [[natural numbers]] $\mathbb{N}$, is only partially defined. It is given by the [[monus]]/truncated subtraction operator:

$$\dot - \; : \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}.$$

* Subtraction in the [[abelian group]] of the [[integers]] $\mathbb{Z}$, is well known to be an [[entire relation]] on the integers,

$$- \; :\mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}.$$

* Subtraction in a [[boolean ring]] is the [[symmetric difference]], and is the same as [[addition]] due to a boolean ring having a [[characteristic]] of 2.

## References

...

## Related concepts

* [[difference]]

[[!redirects subtraction]]
[[!redirects subtract]]
[[!redirects subtracts]]


subtractive category

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

\tableofcontents

## Definition

A [[category]] is __subtractive__
if it is [[pointed]], admits [[finite limits]],
and left punctual [[reflexive relations]] are right punctual.

Here a [[relation]] $r\colon R\to X\times Y$ is left (respectively right)
__punctual__ if $(id_X,0)\colon X\to X\times Y$ (respectively $(0,id_Y)\colon Y\to X\times Y$) factors through $r$.

## Example

A [[variety of algebras]] is a subtractive category
if and only it is a __subtractive variety__ in the sense of Ursini,
meaning its theory contains a binary operation $s$ and constant $0$
such that $s(x,x)=0$ and $s(x,0)=x$.

## Relation to Malcev categories

A [[category]] $C$ with [[finite limits]] is a [[Malcev category]]
if and only if for any object $B$ of $C$
the category $Pt(B)$ is subtractive.

Here $Pt(B)$ is the [[pointed category]]
whose objects are triples $(A\in C,\alpha\colon A\to B,\beta\colon B\to A)$
such that $\alpha\circ\beta=id_B$.
Morphisms $(A,\alpha,\beta)\to(A',\alpha',\beta')$
are morphisms $f\colon A\to A'$ such that $\alpha'\circ f=\alpha$ and $f\circ\beta=\beta'$.

## References

* {#Janelidze2005} [[Zurab Janelidze]], _Subtractive categories_, Appl. Categ. Struct 13, No. 4, 343-350 (2005) ([doi:10.1007/s10485-005-0934-8](https://doi.org/10.1007/s10485-005-0934-8))

* [[Zurab Janelidze]], _Closedness properties of internal relations_.  A series of papers.

  * I. A unified approach to Malʹtsev, unital and subtractive categories.

  * II. Bourn localization.

  * III. Pointed protomodular categories.

  * IV. Expressing additivity of a category via subtractivity.

  * V. Linear Malʹtsev conditions.

  * VI. Approximate operations.


subtractive logic

\tableofcontents

(
This is a stub)

This logic is a conservative extension of propositional intuitionistic logic with a new
connective (subtraction) dual to implication.

## Related concepts

* [[continuation semantics]]

## References:

* [[Tristan Crolard]], [_Subtractive logic_](http://www.sciencedirect.com/science/article/pii/S0304397599001243),  Theoretical Computer Science, 254, Issues 1–2, 6 March 2001, Pages 151-185


subtractive variety

(Stub)

A *subtractive variety* in the sense of Ursini  is a [[variety]] $V$ (in the sense of universal algebra)  such that the theory of $V$ contains a constant 0 and a _[[subtraction]] term_ $s$, i.e. a binary term $s$ satisfying $s(x, x) = 0$ and $s(x, 0) = x$.

##References

* Ursini, A.: _On subtractive varieties_, I. Algebra Universalis 31, 204–222 (1994)


subtype

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

\tableofcontents

## Idea
 {#Idea}

In [[type theory]], the notion of *subtypes* is the analog of *[[subsets]]* in [[set theory]].

In the presence of a [[type universe]] [[Prop|$Prop$]] of [[propositions]] (alternatively denoted "$\Omega$"), the subtypes of a given $A \colon Type$ may be simply be identified with the [[function type]] from $A$ to $Prop$:

$$
  Sub(A)
  \;\;
  =
  \;\;
  Prop_A
  \;\;
  \equiv
  \;\;
  A \to Prop
  \;\;
  \equiv
  \;\;
  A \to \Omega
  \,.
$$

In [[topos theory|topos]]-[[categorical semantics of dependent type theory|semantics]], $Prop$ is interpreted as the *[[subobject classifier]]* and subtypes as *[[subobjects]]*.

Notice that with $Prop$ itself being a subtype of a [[type universe]] $Type$, the subtypes of $A$ are thus a special case of the $A$-[[dependent types]]

$$
  Type_A
  \;\;
  \equiv
  \;\;
  A \to Type
  \,,
$$

namely those whose [[fiber type]] over $a \colon A$ is the proposition asserting that "$a$ is contained in the subtype".

This is equivalent to other definitions of subtypes, which one can make sense of also in the absence of a Prop-universe.

Notice here that in [[set theory]], there are in fact two notions of [[subsets]]:

* In [[material set theory]], a subset of a set $A$ is a set $B$ with the property that for all $x$, if $x \in B$, then $x \in A$, which by the definition of [[power set]] in [[material set theory]], is the same thing as an element $B \in \mathcal{P}(A)$.

* In [[structural set theory]], a subset of a set $A$ is a set $B$ with an [[injection]] $i \colon B \hookrightarrow A$.

In [[dependent type theory]], a similar distinction can be made between the two notions of subtypes, resulting in *material subtypes* and *structural subtypes*.

## Definition ##

### Material subtypes

We work with a [[Russell type of all propositions]] $\Omega$.

\begin{definition}
The **[[power set]]** of a type $A$ is defined as the function type from $A$ to the Russell type of all propositions $\Omega$, $\mathcal{P}(A) \coloneqq A \to \Omega$.
\end{definition}

The power set of a type $A$ is always a set because its [[codomain]], the Russell type of all propositions $\Omega$, is a set due to the [[univalence axiom]] or [[propositional extensionality]].

\begin{definition}
A **material subtype** on $A$ is an element of the power set $B:\mathcal{P}(A)$.
\end{definition}

\begin{definition}
The **local membership relation** $x \in_A B$ between elements $x:A$ and material subtypes $B:\mathcal{P}(A)$ is defined as
$$x \in_A B \coloneqq B(x)$$
\end{definition}

\begin{theorem}
Given a type $A$ and a [[subtype]] $P:A \to \mathrm{Prop}$, one can construct a subtype $[P]_0:[A] \to \mathrm{Prop}$ of the [[set truncation]] of $A$ such that for all $x:A$, $P(x) = [P]_0([x]_0)$.
\end{theorem}

\begin{proof}
By [[propositional extensionality]], the [[type of propositions]] $\mathrm{Prop}$ is a set, and by the [[universal property of set truncations]], given any type $A$, set $B$, and function $f:A \to B$, one can construct a [[uniqueness quantifier|unique]] function $u:[A]_0 \to B$ such that for all $x:A$, $f(x) = u([x]_0)$. This is just the special case of the universal property of set truncations where the [[codomain]] is the set of propositions.
\end{proof}

### Structural subtypes

\begin{definition}
A **structural subtype** of $A$ is a type $B$ with an embedding $i:B \hookrightarrow A$. $A$ is a **supertype** of $B$.
\end{definition}

### Comparison between material subtypes, structural subtypes, and predicates

| material subtype | predicate | structural subtype |
|-|-|-|
| $B:\mathcal{P}(A)$ | $x:A \vdash x \in_A B \coloneqq B(x)$ | $\sum_{x:A} x \in_A B$ |
| $\lambda x:A.B(x):\mathcal{P}(A)$ | $x:A \vdash B(x) \qquad x:A \vdash p(x):\mathrm{isProp}(B(x))$ | $\sum_{x:A} B(x)$ |
| $\left(\lambda x:A.\sum_{y:B} i(y) =_A x\right):\mathcal{P}(A)$ | $x:A \vdash \sum_{y:B} i(y) =_A x$ | $B \qquad i:B \hookrightarrow A$ |

## Operations on material and structural subtypes

### Material subtypes

\begin{definition}
Given a type $A$, the **[[improper subset|improper]] material subtype** on $A$ is the material subtype $\Im_A:\mathcal{P}(A)$ such that for all elements $x:A$, $x \in_A \Im_A$ is [[contractible type|contractible]].
\end{definition}

\begin{definition}
Given a type $A$, there is a binary operation $(-)\cap(-):\mathcal{P}(A) \times \mathcal{P}(A) \to \mathcal{P}(A)$ on the power set of $A$ called the **binary [[intersection]]**, such that for all material subtypes $B:\mathcal{P}(A)$ and $C:\mathcal{P}(A)$, $B \cap C$ is defined as
$$(B \cap C)(x) \coloneqq (x \in_A B) \wedge (x \in_A C)$$
for all $x:A$, where $A \wedge B \coloneqq [A \times B]$ is the [[disjunction]] of two [[types]] and $[T]$ is the propositional truncation of $T$.
\end{definition}

\begin{definition}
Given a type $A$ and a type $I$, there is an function $\bigcap:(I \to \mathcal{P}(A)) \to \mathcal{P}(A)$ called the **$I$-indexed [[intersection]]**, such that for all families of material subtypes $B:I \to \mathcal{P}(A)$, $\bigcap_{i:I} B(i)$ is defined as
$$\left(\bigcap_{i:I} B(i)\right)(x) \coloneqq \forall i:I.x \in_A B(i)$$
for all $x:A$, where
$$\forall x:A.B(x) \coloneqq \left[\prod_{x:A} B(x)\right]$$
is the [[universal quantification]] of a [[type family]] and $[T]$ is the propositional truncation of $T$.
\end{definition}

\begin{definition}
Given a type $A$, the **empty material subtype** on $A$ is the material subtype $\emptyset_A:\mathcal{P}(A)$ such that for all elements $x:A$, $x \in_A \emptyset_A$ [[equivalence of types|is equivalent to]] the [[empty type]].
\end{definition}

\begin{definition}
Given a type $A$, there is a binary operation $(-)\cup(-):\mathcal{P}(A) \times \mathcal{P}(A) \to \mathcal{P}(A)$ on the power set of $A$ called the **binary [[union]]**, such that for all material subtypes $B:\mathcal{P}(A)$ and $C:\mathcal{P}(A)$, $B \cup C$ is defined as
$$(B \cup C)(x) \coloneqq (x \in_A B) \vee (x \in_A C)$$
for all $x:A$, where $A \vee B \coloneqq [A + B]$ is the [[disjunction]] of two [[types]] and $[T]$ is the propositional truncation of $T$.
\end{definition}

\begin{definition}
Given a type $A$ and a type $I$, there is an function $\bigcup:(I \to \mathcal{P}(A)) \to \mathcal{P}(A)$ called the **$I$-indexed [[union]]**, such that for all families of material subtypes $B:I \to \mathcal{P}(A)$, $\bigcup_{i:I} B(i)$ is defined as
$$\left(\bigcup_{i:I} B(i)\right)(x) \coloneqq \exists i:I.x \in_A B(i)$$
for all $x:A$, where
$$\exists x:A.B(x) \coloneqq \left[\sum_{x:A} B(x)\right]$$
is the [[existential quantification]] of a [[type family]] and $[T]$ is the propositional truncation of $T$.
\end{definition}

\begin{definition}
Given a type $A$, there is a binary relation $B:\mathcal{P}(A), C:\mathcal{P}(A) \vdash B \subseteq_A C$ on the power set of $A$ called the **subtype inclusion relation**, such that for all material subtypes $B:\mathcal{P}(A)$ and $C:\mathcal{P}(A)$, $B \subseteq_A C$ is defined as
$$B \subseteq_A C \coloneqq \prod_{x:A} (x \in_A B) \to (x \in_A C)$$
\end{definition}

\begin{definition}
Given a type $A$, there is a binary relation $B:\mathcal{P}(A), C:\mathcal{P}(A) \vdash \mathrm{Eq}_A(B, C)$ on the power set of $A$ called **observational equality**, such that for all material subtypes $B:\mathcal{P}(A)$ and $C:\mathcal{P}(A)$, $\mathrm{Eq}_{\mathcal{P}(A)}(B, C)$ is defined as
$$\mathrm{Eq}_{\mathcal{P}(A)}(B, C) \coloneqq \prod_{x:A} (x \in_A B) \simeq (x \in_A C)$$
\end{definition}

\begin{proposition}
**[[Axiom of extensionality]]**: Given a type $A$, for all subtypes $B:\mathcal{P}(A)$ and $C:\mathcal{P}(A)$, there is an equivalence of types between $B =_{\mathcal{P}(A)} C$ and $\mathrm{Eq}_{\mathcal{P}(A)}(B, C)$.
\end{proposition}

\begin{proof}
For all $x:A$ and $B:\mathcal{P}(A)$, since $x \in_A B$ is propositions, by the introduction rules of the [[Russell type of all propositions]], $x \in_A B:\Omega$. The [[univalence axiom]] for $\Omega$ states that $(x \in_A B) \simeq (x \in_A C)$ is equivalent to $(x \in_A B) =_\Omega (x \in_A C)$. Thus, $\mathrm{Eq}_{\mathcal{P}(A)}(B, C)$ is equivalent to $\prod_{x:A} (x \in_A B) =_\Omega (x \in_A C)$. But by [[function extensionality]], there is an equivalence of types between $B =_{\mathcal{P}(A)} C$ and $\prod_{x:A} (x \in_A B) =_\Omega (x \in_A C)$; hence there is an equivalence of types between $B =_{\mathcal{P}(A)} C$ and $\mathrm{Eq}_{\mathcal{P}(A)}(B, C)$.
\end{proof}

\begin{definition}
Given a type $A$, a **singleton subtype** on $A$ is a material subtype $S:\mathcal{P}(A)$ such that [[uniqueness quantifier|there exists a unique]] $x:A$ such that $x \in S$:
$$\mathrm{isSingleton}(S) \coloneqq \exists!x:A.x \in S$$
\end{definition}

\begin{definition}
Given a type $T$, there is a binary function $(-) \times (-):\mathcal{P}(T) \times \mathcal{P}(T) \to \mathcal{P}(T \times T)$ called the **[[cartesian product]] of subtypes**, defined by
$$(A \times B)(a, b) \coloneqq (a \in_T A) \wedge (b \in_T B)$$
for all elements $a:T$, $b:T$ and subtypes $A:\mathcal{P}(T)$ and $B:\mathcal{P}(T)$.
\end{definition}

### Structural subtypes

\begin{definition}
Given a type $T$, $T$ is the **improper structural subtype**, with embedding given by the identity function on $T$.
\end{definition}

\begin{definition}
Given a type $T$ and structural subtypes $R$ and $S$ with embeddings $i_R:R \hookrightarrow T$ and $i_S:S \hookrightarrow T$, the **binary intersection** of $R$ and $S$ is the [[pullback]] of the two embeddings into $T$, or equivalently the dependent sum type
$$R \cap S \coloneqq \sum_{r:R} \sum_{s:S} i_R(r) =_T i_S(s)$$
\end{definition}

\begin{definition}
Given a type $T$, an index type $I$, and a family of structural subtypes $(R(i))_{i:I}$ with a family of embeddings $i_R:\prod_{i:I} R(i) \hookrightarrow T$, the **$I$-indexed intersection** of $(R(i))_{i:I}$ is the [[dependent pullback]] of the embeddings into $T$, or equivalently the dependent sum type
$$\bigcap_{i:I} R(i) \coloneqq \sum_{t:T} \prod_{i:I} \sum_{r:R(i)} i_R(i, r) =_T t$$
\end{definition}

By definition of [[dependent pullback]], the intersection of a family of structural subsets is a [[wide span]], with dependent function

$$\lambda j:I.\lambda p:\bigcap_{i:I} R(i).\pi_1(\pi_2(p)(j)):\prod_{j:I} \left(\bigcap_{i:I} R(i)\right) \to R(j)$$

\begin{definition}
Given a type $T$, the [[empty type]] is the **empty structural subtype**, with embedding given by any function from the empty type to $T$.
\end{definition}

\begin{definition}
Given a type $T$ and structural subtypes $R$ and $S$ with embeddings $i_R:R \hookrightarrow T$ and $i_S:S \hookrightarrow T$, the **binary [[union]]** of $R$ and $S$ is the [[pushout type]]
$$R \cup S \coloneqq R \sqcup^{R \cap S}_{\pi_1 ; \pi_1' \circ \pi_2} S$$
where $\pi_1$ and $\pi_2$ are the first and second projection functions for the first dependent sum type and $\pi_1'$ is the first projection function for the second dependent sum type for the intersection as defined above ($\sum_{r:R} \sum_{s:S} i_R(r) =_T i_S(s)$)
\end{definition}

\begin{definition}
Given a type $T$, an index type $I$, and a family of structural subtypes $(R(i))_{i:I}$ with a family of embeddings $i_R:\prod_{i:I} R(i) \hookrightarrow T$, the **$I$-indexed union** of $(R(i))_{i:I}$ is the [[dependent pushout]] of the [[dependent pullback]]
$$\lambda j.\lambda p.\pi_1(\pi_2(p)(j)):\prod_{j:I} \left(\bigcap_{i:I} R(i)\right) \to R(j)$$
of the embeddings into $T$,
$$\bigcup_{i:I} R(i) \coloneqq \bigsqcup_{i:I}^{\bigcap_{i:I} R(i), \lambda j.\lambda p.\pi_1(\pi_2(p)(j))} R(i)$$
\end{definition}

\begin{definition}
Given a type $T$, a **singleton subtype** of $T$ is a [[contractible type]] $S$ with an embedding $i:S \hookrightarrow T$.
\end{definition}

\begin{definition}
Given a type $T$ and structural subtypes $R$ and $S$ with embeddings $i_R:R \hookrightarrow T$ and $i_S:S \hookrightarrow T$, the **[[cartesian product]]** of $R$ and $S$ is simply the [[product]] $R \times S$. The associated [[embedding]] $i_{R \times S}:(R \times S) \hookrightarrow (T \times T)$ is defined by
$$i_{R \times S}(a) \coloneqq (i_R(\pi_1(a)), i_S(\pi_2(a)))$$
for all $a:R \times S$, where $\pi_1$ and $\pi_2$ are the first and second product projection functions defined in the [[inference rules]] of the negative [[product type]] $R \times S$.
\end{definition}

## Decidable subtypes

Given a Russell type of propositions, a [[subtype]] $P:A \to \mathrm{Prop}$ is a **decidable subtype** if for all $x:A$, $P(x)$ satisfies the [[law of excluded middle]]

$$\mathrm{isDecidable}(P) \equiv \prod_{x:A} P(x) \vee \neg P(x)$$

Similarly, given a Tarski type of propositions, a [[subtype]] $P:A \to \mathrm{Prop}$ is a **decidable subtype** if for all $x:A$, $P(x)$ satisfies the [[law of excluded middle]]

$$\mathrm{isDecidable}(P) \equiv \prod_{x:A} \mathrm{El}(P(x) \vee \neg P(x))$$

For all types $A$, there exists an equivalence between the type of decidable subtypes of $A$ and the type of [[boolean-valued functions]] of $A$

$$\mathrm{toDecidSub}:\left(A \to \mathrm{Bool}\right) \simeq \left(\sum_{P:A \to \mathrm{Prop}} \mathrm{isDecidable}(P)\right)$$

The membership relation of decidable subsets is defined by whether the boolean-valued function evaluations is true

$$x \in P \coloneqq \mathrm{toDecidSub}^{-1}(P)(x) =_\mathrm{true}$$

Decidable subtypes are closed under the empty subtype, the improper subtype, and unions with a disjoint singleton subtype:

The empty subtype and improper subtype are given by

$$\emptyset_A \coloneqq \mathrm{toDecidSub}(\lambda x:A.\mathrm{false}) \qquad \im_A \coloneqq \mathrm{toDecidSub}(\lambda x:A.\mathrm{true})$$

Singleton subtypes correspond to boolean-valued functions for which [[uniqueness quantifier|there exists a unique]] $x:A$ such that $f(x)$ is true, and is given by

$$\sum_{f:A \to \mathrm{bool}} \exists!x:A.f(x) =_\mathrm{bool} \mathrm{true}$$

Unions of decidable subtypes correspond to lambda abstraction of the disjunction of boolean-valued function evaluations

$$P \cup Q \coloneqq \mathrm{toDecidSub}(\lambda x:A.\mathrm{toDecidSub}^{-1}(P)(x) \vee \mathrm{toDecidSub}^{-1}(Q)(x))$$

and intersections of decidable subtypes correspond to lambda abstraction of the conjunction of boolean-valued function evaluations

$$P \cap Q \coloneqq \mathrm{toDecidSub}(\lambda x:A.\mathrm{toDecidSub}^{-1}(P)(x) \wedge \mathrm{toDecidSub}^{-1}(Q)(x))$$

Two decidable subtypes are disjoint if their intersection is the empty subtype.

## See also ##

* [[subobject]]

* [[embedding]], [[embedding type]]

* [[n-monomorphism]]

* [[coercion]]

* [[decidable subtype]]

## References
 {#References}

* {#HofmannPierce96} [[Martin Hofmann]], [[Benjamin Pierce]], *Positive Subtyping*, Information and Computation **126** 1 (1996) 11-33 [[doi:10.1006/inco.1996.0031](https://doi.org/10.1006/inco.1996.0031)]

  > (including early discussion of [[lenses in computer science]])

* [[Univalent Foundations Project]], [p. 111](https://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf#page=123) of: _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_ (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

* {#Rijke19} [[Egbert Rijke]], §12.2 in: _[[Introduction to Homotopy Type Theory]]_ (2019)  [[web](http://www.andrew.cmu.edu/user/erijke/hott/), [pdf](http://www.andrew.cmu.edu/user/erijke/hott/hott_intro.pdf), [GitHub](https://github.com/EgbertRijke/HoTT-Intro)]

* [[Martín Escardó]], *[The subtype classifier and other classifiers](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html#subtypeclassifier)*, 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)]

* [[Felix Cherubini]], [[Thierry Coquand]], [[Matthias Hutzler]], *A Foundation for Synthetic Algebraic Geometry* (2023) [[arXiv:2307.00073](https://arxiv.org/abs/2307.00073)]

  > (in the context of [[synthetic algebraic geometry]])

In [[Agda]]:

* [[UniMath project]], *[foundation-core.subtypes.html](https://unimath.github.io/agda-unimath/foundation-core.subtypes.html)*

* [[Ettore Aldrovandi]], *[Subtypes](https://www.math.fsu.edu/~ealdrov/teaching/2020-21/fall/MAS5932/agda/more-logic.html#subtypes)*, §4 in: *Propositions, Sets and Logic--Ⅱ* [[web](https://www.math.fsu.edu/~ealdrov/teaching/2020-21/fall/MAS5932/agda/more-logic.html)]

In [[Lean]]:

* [[Kyle Miller]], [pp. 22](https://math.berkeley.edu/~kmill/talks/2020-06-26-lean-seminar.pdf#page=22) in: *Doing math the Lean way*, Berkeley Lean Seminar (2020) [[pdf](https://math.berkeley.edu/~kmill/talks/2020-06-26-lean-seminar.pdf#page=22), [[Miller-MathTheLeanWay.pdf:file]]]

[[!redirects subtype]]
[[!redirects subtypes]]
[[!redirects subtyping]]

[[!redirects material subtype]]
[[!redirects material subtypes]]

[[!redirects structural subtype]]
[[!redirects structural subtypes]]

[[!redirects supertype]]
[[!redirects supertypes]]

[[!redirects type of subtypes]]

[[!redirects decidable subtype]]
[[!redirects decidable subtypes]]


succedent

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

#Contents#
* table of contents
{:toc}

## Idea

A [[judgement]] on the right of the "$\vdash$"-symbol of a [[sequent]]/[[hypothetical judgement]]: a [[judgement]] that is a consequence of the [[antecedent]] judgements on the left of the symbol.

If there is only a single succedent it is also called a _[[consequent]]_.

## Related concepts

## Related concepts

[[antecedents]] $\vdash$ **succedents**

[[antecedents]] $\vdash$ [[consequent]]

[[!redirects succedents]]