separator

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### Context
#### Category theory
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[[!include category theory - contents]]
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# Separators
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## Idea

An [[object]] $S$ (or family $\mathcal{S}$ of objects) in a [[category]] $\mathcal{C}$ is called a _separator_ or _generator_ if [[generalized elements]] with [[domain]] $S$ (or domain from $\mathcal{S}$) are sufficient to distinguish [[morphisms]] in $\mathcal{C}$.

The [[duality|dual]] notion is that of a _[[coseparator]]_.

## Caution on terminology {#Terminology}

The term 'generator' is slightly more ambiguous because of the use of 'generators' in [[generators and relations]]. That said, there is a link between these two senses provided by theorem \ref{motive} (q.v.).

## Definitions

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

An [[object]] $S \in \mathcal{C}$ of a [[category]] $\mathcal{C}$ is called a __separator__ or a __generator__ or a __separating object__ or a __generating object__, or is said to __separate morphisms__ if:

* for every pair of [[parallel morphisms]] $f,g \colon X \to Y$ in $\mathcal{C}$, if $f\circ e = g\circ e$ for every morphism $e\colon S \to X$, then $f = g$

=--

Equivalently, we have that $S$ is a separator if, for every object $X$ in $\mathcal{C}$, morphisms $f:S\rightarrow X$ are [[jointly epimorphic family|jointly epic]]. Assuming that $\mathcal{C}$ is [[locally small|locally small category]], we have equivalently that $S$ is a separator if the [[hom functor]] $Hom(S,-) \colon \mathcal{C} \to $ [[Set]] is [[faithful functor|faithful]].

More generally:

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

A [[family]] $\mathcal{S} = (S_a)_{(a \in A)}$ of [[objects]] of a [[category]] $\mathcal{C}$ is a __separating family__ or a __generating family__ if:

* for every pair of [[parallel morphisms]] $f,g \colon X \to Y$ in $\mathcal{C}$, if $f \circ e = g \circ e$ for every $e \colon S_a \to X$ sourced in the family, then $f = g$.
=--

Assuming again that $\mathcal{C}$ is [[locally small category|locally small]], we have equivalently that $\mathcal{S}$ is a separating family if the family of [[hom functors]] $Hom(S_a,-) \colon \mathcal{C} \to $ [[Set]] (for $a \in A$) is [[jointly faithful family of functors|jointly faithful]].

Since repetition is irrelevant in a separating family, we may also speak of a _separating [[class]]_ instead of a separating family.

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

A __separating set__ is a [[size issues|small]] separating class.
=--

### In fibered categories
  {#fibered}

The notion of separating family can be generalized from categories to [[fibered categories]] in such a way that the [[family fibration]] of a category $\mathbf{C}$
has a separating family if and only if $\mathbf{C}$ has a small separating family.
+-- {: .num_defn}
###### Definition
A separating family in a [[fibered category]] $P:\mathbf{E}\to \mathbf{B}$ is an object $S\in \mathbf{E}$ such that for every parallel pair $f,g:A\to B$ in $E$ with $f\neq g$ and $P(f) = P(g)$ there exist arrows $c: X\to S$ and $h:X\to A$ (constituting a span) such that $c$ is $P$-cartesian, and $f h \neq g h$ .
=--

See Definition B2.4.1 in the [[Elephant]].

## Examples and applications

* In [[Set]], any [[inhabited set]] is a separator; in particular, the [[point]] is a separator.

*  More generally, in any [[well-pointed category]], any [[terminal object]] is a separator.  More generally still, in any represented [[concrete category]], the representing object is a separator.

* The standard example of a separator in the category of $R$-[[modules]] over a [[ring]] $R$ is any [[free module]] $R^I$ (for $I$ an [[inhabited set]]) and $R$ (which is $R^I$ for $I$ a [[point]]) in particular.  If a separator is a finitely generated [[projective object]] in the category of $R$-modules, then one sometimes says (especially in the older literature, e.g. Freyd's *Abelian Categories*) that the separator is a _progenerator_.  Progenerators are important in classical Morita theory, see [[Morita equivalence]].

* The existence of a small separating family is one of the conditions in [[Giraud's theorem]] characterizing [[Grothendieck topos]]es.

* The existence of a small (co)separating family is one of the conditions in one version of the [[adjoint functor theorem]].

* In $Set^{op}$, the two-element set $2$ is a separator, thus every [[continuous functor]] $F:Set\to C$ into a locally small category $C$ has a [[left adjoint]] by the special [[adjoint functor theorem]].

* On the other hand, the opposite of the category of groups does not admit a separator, since there exists a continuous functor $ML: Group\to Set$ which does not have a left adjoint (see Example 3.1 on the page [[adjoint functor theorem]]).

## Strengthened separators

+-- {: .num_theorem #motive}
###### Motivating theorem

If $C$ is [[locally small category|locally small]] and has all small [[coproduct]]s, then a set-indexed family $(S_a)_{(a\colon A)}$  is separating  if and only if, for every $X\in C$, the canonical morphism
$$ \varepsilon_X\colon \coprod_{a\colon A, f\colon S_a \to X} S_a \longrightarrow X $$
is an [[epimorphism]].
=--

This theorem explains a likely origin of the term "generator" or "generating family". For example, in linear algebra, one says that a set of morphisms $f_a: S_a \to X$ spans or generates $X$ if the induced map $\oplus S_a \to X$ maps epimorphically onto $X$.

More generally:

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

If $\mathcal{E}$ is a subclass of epimorphisms, we say that $(S_a)_{(a\colon A)}$ is an _$\mathcal{E}$-separator_ or __$\mathcal{E}$-generator__ if each morphism $\varepsilon_X$ (as above) is in $\mathcal{E}$.
=--

The weakest commonly-seen strengthened notion is that of __extremal separator__, i.e. separator where all maps $\varepsilon_X$ are [[extremal epimorphisms]]. The notion of extremal separator admits an
equivalent reformulation not referencing coproducts:

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

If $C$ is [[locally small category|locally small]] and has all small [[coproduct]]s, then a set-indexed family $(S_i)_{(i\colon I)}$ is an extremal separator if and only if
the functors $C(S_i,-):C\to\mathrm{Set}$ are jointly faithful and jointly [[conservative functor|conservative]].
=--

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

Assume first that the family $(S_i)_{(i\colon I)}$ is an extremal separator.
The functors $C(S_i,-):C\to\mathrm{Set}$ are jointly faithful for every separator.
To see that they are also jointly conservative, let $f:A\to B$ such that all
$C(S_i,f)$ are bijective. Then $\varepsilon_B$ factors through $f$ since all its
components do, which implies that $f$ is an extremal epi since $\varepsilon_B$ is one by assumption. It remains to show that $f$ is a monomorphism. For this, let $u,v:X\to A$ such that $f u = f v$. Then we have $f u h = f v h$ for all $i\in I$ and $h:S_i\to X$,
which implies $u h = v h$ since the $C(S_i,f)$ are bijective, and we conclude that $u=v$
since $(S_i)_{(i\colon I)}$ is separating.

Conversely, assume that the functors $C(S_i,-)$ are jointly faithful and jointly
conservative. Given $A\in C$, joint faithfulness implies that $\varepsilon_A$ is epic.
To see that it is extremally so, assume a factorization $\varepsilon_A = m g$ with $m$ monic. We have to show that
$m$ is an isomorphism, and for this it is sufficient to show that all $C(S_i,m)$
are bijections. Injectivity is clear since $m$ is monic, and surjectivity follows since every $h:S_i\to A$ factors through $\varepsilon_A$.
=--

The concepts "strong separator" and "regular separator" corresponding to the
notions of [[strong epimorphism]] and [[regular epimorphism]] do not admit such a reformulation, but the following result shows that they are equivalent to extremal separators in reasonable categories.

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

Assume that $C$ is [[locally small category|locally small]] and has all small [[coproduct]]s.

1. If $C$ is balanced, then every separator is extremal.

2. If $C$ has pullbacks, then every extremal separator is strong.

3. If $C$ is [[regular category|regular]], then every strong separator is regular.

The converse implications do always hold.
=--

+-- {: .proof}
###### Proof
This is a direct consequence of the facts that

1. in a balanced category every epi is extremal,

2. in a category with pullbacks, every extremal epi is strong, and

3. in a regular category every strong epi is regular.

=--

+-- {: .un_remark}
###### Remarks

1. Proposition \ref{Extremal} gives rise to a notion of extremal separator that makes sense independently of the existence of coproducts. In fact claim 1 of the preceding result holds in this more general setting, since every faithful functor out of a balanced category is conservative.

2. Most of the literature uses the term "strong separator" (or strong generator) for what we call an extremal separator. [Adamek and Rosicky (Section 0.6)](#AdamekRosicky) also comment on this mismatch, writing
"It would be more reasonable, but unfortunately less standard, to call [a strong generator] an extremal generator".
   However, item 2 of the preceding result shows that this discrepancy disappears and the terms coincide in presence of pullbacks (and coproducts).

3. In the Elephant, Johnstone uses "separator" in the same sense as we do, and writes "generator" for extremal separators, in the more general sense not assuming coproducts. Since he always assumes finite limits, he can use a simplified criterion only requiring joint conservativity of the hom-functors (since a conservative functor $F:C\to D$ is automatically faithful whenever $C$ has equalizers and $F$ preserves them).
=--

### Dense separators

Finally, the strongest kind of separator commonly seen is that of __dense separator__.

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

A __dense separator__ in a category $C$ is a family $(S_i)_{(i\colon I)}$ of objects
such that the generated full subcategory is [[dense subcategory|dense]].
=--

Every dense separator is an extremal separator, and it is also strong and regular whenever those words make sense, i.e. $C$ is locally small and has small coproducts. If $C$ furthermore has pullbacks and the coproducts are pullback-stable, then every regular separator is dense (see [Borceux I, Proposition 4.5.6](#Borceux)).
To see that the pullback-stability condition is necessary, consider the category of abelian groups. Here, the free group on one generator is a regular, but not a dense separator.

### Giraud's axioms

[[Grothendieck topos#Giraud|Giraud's axioms]] characterize Grothendieck toposes as locally small regular categories with effective equivalence relations and disjoint and pullback-stable coproducts admitting a small separator. The previously stated and cited results show that in fact every such separator is dense (the effectivity and disjointness assumptions don't play a role for this conclusion).

## Related concepts

* [[coseparator]]

## References

* {#Borceux} [[Francis Borceux]], _Handbook of Categorical Algebra I_, Cambridge University Press, 1994

* {#AdamekRosicky} [[Jiří Adámek]] and [[Jiří Rosicky]], Locally presentable and accessible categories, Cambridge University Press

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sequence

# Sequences
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## Definitions

A __sequence__ is a [[function]] whose [[source|domain]] is a [[subset]] of the set $\mathbb{N}$ of [[natural numbers]] (or more generally a subset of the set $\mathbb{Z}$ of [[integers]]; cf. *[[bi-infinite sequence]]* and the further generalizations [below](#Generalizations)).  Often one means an __infinite sequence__, which is a sequence whose domain is infinite.  Sequences (especially finite ones) are often called __[[lists]]__ in computer science.  (In [[constructive mathematics]], the domain of a sequence must be a [[decidable subset]] of $\mathbb{Z}$.)

Sequences may also be indicated by functions $\omega \to X$ where $\omega$ is the first countably infinite [[ordinal]]: the key piece of structure on $\mathbb{N}$ relevant for the study of sequences, particularly in analysis, is the order structure.

Up to [[bijection]], the only possible [[domains]] are those of the form
$$ \{i\colon \mathbb{Z} \;|\; 0 \leq i \lt n\} $$
for $n = 0, 1, 2, \ldots, \infty$; other domains are used for notational convenience.

An alternative generalisation takes the domain to be a set of [[ordinal numbers]], without loss of generality the set
$$ \{i\colon \Ord \;|\; i \lt \alpha\} $$
for $\alpha$ some specific ordinal number (or the [[proper class]] $\Ord$ of all ordinal numbers, if one wishes to allow for a proper class).

A _subsequence_ of a sequence $a = a_n: \mathbb{N} \to X$ is a composition

$$\mathbb{N} \stackrel{i}{\to} \mathbb{N} \stackrel{a}{\to} X$$

where $i$ is an order-preserving monomorphism. The salient point is that $i$ be [[cofinal diagram|cofinal]] as an embedding.

## Notation

One normally writes the value of the sequence $a$ at the argument $i$ as $a_i$ rather than $a(i)$.  Similarly, given a term $a[i]$ with the free variable $i$, one often defines a sequence whose values equal those terms as $(a[i])_{i \lt n}$, $\{a[i]\}_i$, or the like.  In fact, one even often says literally 'Let $(a_i)$ be a sequence.' even though 'Let $a$ be a sequence.' would be less of an abuse of notation.  This is all because notation for sequences arose before [[functions]] were considered in their full generality, and one distinguished a 'sequence' (whose domain was a set of integers) from a 'function' (whose domain was an interval in the real line or a region in the complex plane).  Early mathematicians also often conflated the sequence (the function itself) with its range (a subset of the function\'s [[target]]); hence the use of curly braces.  All of this applies in greater generality to [[families]] with index sets other than $\mathbb{N}$.

## Generalisations
 {#Generalization}

### Nets

Infinite sequences are often used in [[topology]], but for topology in general, one needs to generalise to [[nets]], also called _Moore--Smith sequences_.  Here one replaces the domain $\mathbb{N}$ by any arbitrary [[direction|directed set]].

### Sequential nets

Recall that [[weak countable choice]] is a rather weak version of the [[axiom of choice]] that is accepted even in most schools of [[constructive mathematics]]; it follows separately from both [[excluded middle]] and [[countable choice]].  However, when it fails (as it does in the [[internal language]] of some widely studied [[toposes]], such as the [[topos of sheaves]] over the [[real line]]), then some important results about sequences fail, including many standard results in [[topology]].  In this case, we may want a slight generalisation that we call _sequential nets_.

A __[[sequential net]]__ is a [[multi-valued function]] from $\mathbb{N}$ (or a [[decidable subset|decidable]] [[subset]] thereof) to $X$, that is a [[span]] $\mathbb{N} \leftarrow A \rightarrow X$ where the map $A \to \mathbb{N}$ is a [[surjection]] (or has a decidable range).  Note that $A$ inherits the structure of a directed set via $A \to \mathbb{N}$, so that $A \to X$ is a net.  As a net, every sequential net is equivalent (in the sense of corresponding to the same [[filter]]) to some sequence, if you assume WCC.  Without WCC, however, this equivalence fails.

(Using a multi-valued function here is a special case of an alternative definition of [[net]] that uses only [[partially ordered]] directed sets; see [[net]].  In some [[foundations of mathematics]], we can get the same result by defining a sequential net to be a __presequence__: a [[prefunction]], which is like a function but need not preserve [[equality]], from $\mathbb{N}$ or a decidable subset thereof.)

Without WCC, many of the usual properties of [[metric spaces]] and other [[sequential spaces]] fail, but they continue to hold using sequential nets in the place of sequences.  For example, every (located Dedekind) [[real number]] may be represented as a sequential Cauchy net, even when they might not all be represented as Cauchy sequences; see [[Cauchy real number]].

## Sequence types

In [[dependent type theory]], a sequence type is simply the [[function type]] $\mathbb{N} \to A$, and thus comes with the following rules:

Formation rules for sequence types:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathbb{N} \to A \; \mathrm{type}}$$

Introduction rules for sequence types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, n:\mathbb{N} \vdash a(n):A}{\Gamma \vdash \lambda(n:\mathbb{N}).a(n):\mathbb{N} \to A}$$

Elimination rules for sequence types:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, a:\mathbb{N} \to A, n:\mathbb{N} \vdash \mathrm{ev}(a, n):A}$$

Computation rules for sequence types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, n:\mathbb{N} \vdash a(n):A}{\Gamma, m:\mathbb{N} \vdash \beta_\Pi(m):\mathrm{ev}(\lambda(n:\mathbb{N}).a(n), m) =_{A} a(m)}$$

Uniqueness rules for sequence types:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, a:\mathbb{N} \to A \vdash \eta_\Pi(a):a =_{\mathbb{N} \to A} \lambda(n:\mathbb{N}).a(n)}$$

Sequence types also have their own extensionality principle, called [[sequence extensionality]]. This states that given two sequences $a:\mathbb{N} \to A$ and $b:\mathbb{N} \to A$ there is an [[equivalence of types]] between the [[identity type]] $a =_{\mathbb{N} \to A} b$ and the [[dependent sequence type]] $(n:\mathbb{N}) \to (a(n) =_{A} b(n))$:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, a:\mathbb{N} \to A, b:\mathbb{N} \to A \vdash \mathrm{seqext}(a, b):(a =_{\mathbb{N} \to A} b) \simeq (n:\mathbb{N}) \to (a(n) =_{A} b(n))}$$

Sequence types are used in [[strongly predicative mathematics]], where one does not have [[function types]], to construct the [[real numbers]].

## Sequence spaces

In [[functional analysis]], one considers [[topological vector spaces]] of infinite sequences; these are the [[sequence spaces]].  (Actually, these generalise quite nicely to [[net]] spaces.)

## Related concepts

* [[limit of a sequence]]

* [[sequentially compact space]]

* [[sequence algebra]]

* [[sequential net]]

* [[tuple]], **sequence**, [[function]]

* [[dependent tuple]], [[dependent sequence]], [[dependent function]]

Not all that related, but similar sounding:

* [[sequential limit]]

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sequence algebra

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### Context
#### Algebra
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[[!include higher algebra - contents]]
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#### Analysis
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[[!include analysis - contents]]
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# Contents
* table of contents
{: toc}

## Definition

For a [[commutative ring]] $R$, the **sequence algebra** is the [[commutative algebra|commutative $R$-algebra]] $\mathbb{N} \to R$, with canonical [[injection]] $c \colon R \to (\mathbb{N} \to R)$ which sends each [[element]] $r$ in $R$ to a [[constant function]] $c(r)$. The [[commutative algebra]] operations are pointwise defined as:

$$c(r)(n) \coloneqq r$$
$$0(n) \coloneqq 0$$
$$(x + y)(n) \coloneqq x(n) + y(n)$$
$$(-x)(n) \coloneqq -x(n)$$
$$(r x)(n) \coloneqq r x(n)$$
$$(1)(n) \coloneqq 1$$
$$(x \cdot y)(n) \coloneqq x(n) \cdot y(n)$$

and thus the sequence set $\mathbb{N} \to R$ is a commutative $R$-algebra.

## Polynomials

Given a commutative ring $R$, a **sequential polynomial** is a [[sequence]] $x \colon \mathbb{N} \to R$ where there exists a [[natural number]] $N$ such that for all $i \gt N$ we have $x(i) = 0$.

## Operators

### Left shift operator

Given a commutative ring $R$ and a sequence $x \colon \mathbb{N} \to R$ of terms in $R$, the **left shift operator**
$$T \colon (\mathbb{N} \to R) \to (\mathbb{N} \to R)$$
is defined as
$$T x(i) \coloneqq x(i + 1)$$
for $i \in \mathbb{N}$.

### Series operator

Given a commutative ring $R$ and a sequence $x:\mathbb{N} \to R$ of terms in $R$, the **series operator**
$$\Sigma:(\mathbb{N} \to R) \to (\mathbb{N} \to R)$$
is inductively defined as
$$\Sigma x(0) \coloneqq x(0)$$
$$\Sigma x(i + 1) \coloneqq \Sigma x(i) + x(i + 1)$$
for $i \in \mathbb{N}$.

For any such sequence $x:\mathbb{N} \to R$, $\Sigma x$ is called a **[[series]]**.

### Inverse series operator

Given a commutative ring $R$ and a sequence $x:\mathbb{N} \to R$ of terms in $R$, the **inverse series operator**
$$\Sigma^{-1}:(\mathbb{N} \to R) \to (\mathbb{N} \to R)$$
is inductively defined as
$$\Sigma^{-1} x(0) \coloneqq x(0)$$
$$\Sigma^{-1} x(i + 1) \coloneqq \Sigma^{-1} x(i) - x(i + 1)$$
for $i \in \mathbb{N}$.

For any such sequence $x:\mathbb{N} \to R$, $\Sigma^{-1}\Sigma x = x$ and  $\Sigma\Sigma^{-1}x = x$.

### Infinite product operator

Given a commutative ring $R$ and a sequence $x:\mathbb{N} \to R$ of terms in $R$, the **infinite product operator**
$$\Pi:(\mathbb{N} \to R) \to (\mathbb{N} \to R)$$
is inductively defined as
$$\Pi x(0) \coloneqq x(0)$$
$$\Pi x(i + 1) \coloneqq \Pi x(i) \cdot x(i + 1)$$
for $i \in \mathbb{N}$.

For any such sequence $x:\mathbb{N} \to R$, $\Pi x$ is called an **[[infinite product]]**.

### Discrete convolution

Given a commutative ring $R$ and sequence $x:\mathbb{N} \to R$ and $y:\mathbb{N} \to R$, the **discrete convolution**
$$(-)*(-):(\mathbb{N} \to R) \times (\mathbb{N} \to R) \to (\mathbb{N} \to R)$$
is defined as
$$(x*y)(i) \coloneqq \sum_{n:[0,i]} x(i) y(i - n)$$
for $i \in \mathbb{N}$.

Given a commutative ring $R$ and sequence $x:\mathbb{N} \to R$ and $y:\mathbb{N} \to R$, the **Cauchy product** of the two series $\Sigma x \cdot \Sigma x$ is the discrete convolution of $x$ and $y$.

$$\Sigma x \cdot \Sigma y \coloneqq x * y$$

### Sequential derivative

Given a commutative ring $R$ and a sequence $x:\mathbb{N} \to R$ of terms in $R$, the **sequential derivative**
$$\mathcal{D}:(\mathbb{N} \to R) \to (\mathbb{N} \to R)$$
is defined as
$$\mathcal{D} x(i) \coloneqq (i + 1) x(i + 1)$$
for $i \in \mathbb{N}$.

The sequential derivative is a [[derivation]], where linearity is satisfied by addition and scalar multiplication of sequences and the [[Leibniz rule]] is satisfied for the Cauchy product of sequences. As a result, every sequence algebra is a [[differential algebra]].

### Right shift operators

Given a commutative ring $R$ and a sequence $x:\mathbb{N} \to R$, the set of right shifted sequences of $x$ is the [[fiber]] of the left shift operator at $x$:

$$\{y:\mathbb{N} \to R \vert T y = x\}$$

A right shift operator is an element of the above set.

Every sequence in $R$ has a right shift operator $T^{-1}_s:(\mathbb{N} \to R) \to (\mathbb{N} \to R)$ for every term $r \in R$,
defined as
$$T^{-1}_r x(0) = r$$
$$T^{-1}_r x(i + i) \coloneqq x(i)$$
for $i \in \mathbb{N}$.

### Sequential antiderivatives

Given a commutative ring $R$ and a sequence $x:\mathbb{N} \to R$, the set of sequential antiderivatives of $x$ is the [[fiber]] of the sequential derivative at $x$:

$$\{y:\mathbb{N} \to R \vert \mathcal{D} y = x\}$$

A __sequential antiderivative__ is a element of the above set.

If $R$ is a commutative $\mathbb{Q}$-algebra, then every sequence in $R$ has an antiderivative operator $\mathcal{D}^{-1}_r:(\mathbb{N} \to R) \to (\mathbb{N} \to R)$ for every term $r \in R$,
defined as
$$\mathcal{D}^{-1}_r x(0) = r$$
$$\mathcal{D}^{-1}_r x(i + i) \coloneqq \frac{1}{i} x(i)$$
for $i \in \mathbb{N}$.

## Sequence functions

Given a commutative ring $R$, a **sequence function** is a function $f:R \to (\mathbb{N} \to R)$.

### Functional shift operator

Given a commutative ring $R$ and a sequence function $f:R \to (\mathbb{N} \to R)$, the **functional shift operator**
$$\mathcal{T}:R \to (R \to (\mathbb{N} \to R)) \to (R \to (\mathbb{N} \to R))$$
is defined as
$$\mathcal{T}_c f(r)(i) \coloneqq f(r - c)(i)$$
for $c \in R$, and $r \in R$, and $i \in \mathbb{N}$.

### Power sequence functions

Given a commutative ring $R$ and a sequence $x:\mathbb{N} \to R$ of terms in $R$, the **power sequence function**
$$\mathcal{P}:(\mathbb{N} \to R) \to (R \to (\mathbb{N} \to R))$$
is defined as
$$\mathcal{P}_x (r)(i) \coloneqq x(i) r^i$$
for $i \in \mathbb{N}$ and $r \in R$.

### Functional derivative

Given a commutative ring $R$, the **functional derivative** of the power sequence function $f:R \to (\mathbb{N} \to R)$
$$\tilde{D}:(R \to (\mathbb{N} \to R)) \to (R \to (\mathbb{N} \to R))$$
is defined as
$$(\tilde{D} \mathcal{P}_x)(r)(i) \coloneqq (\mathcal{D} x(i)) r^i $$

### Power series

For any such sequence $x:\mathbb{N} \to R$, $\Sigma\mathcal{P}_x(r)$ is called a **formal [[power series]]** or **formal [[Maclaurin series]]**. For any such sequence $x:\mathbb{N} \to R$ and element $c \in R$, $\Sigma \mathcal{T}_c \mathcal{P}_x (r)$ is called a **formal [[Taylor series]]**.

### Analytic functions

Suppose $R$ is a commutative ring with a [[partial function]]
$$\lim_{n \to \infty} (-)(n) \colon (\mathbb{N} \to R) \to R$$
making $R$ into a [[sequentially Hausdorff space]].

Then, a [[partial function]] $f \colon R \to R$ is an **analytic function** if there exists an element $c \in R$ and sequence $x \colon \mathbb{N} \to R$ such that

$$f(r) = \lim_{n \to \infty} \Sigma \mathcal{T}_c \mathcal{P}_x (r)(n)$$

for all elements $r \in \dom f$.

## See also

* [[sequence]]

* [[sequence space]]

* [[function algebra]]

* [[differential algebra]]

* [[sequentially Hausdorff space]]

* [[analytic function]]

[[!redirects sequence algebras]]


sequence calculus > history

< [[sequent calculus]]


sequence extensionality

+-- {: .rightHandSide}
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### Context
#### Set theory
+-- {: .hide}
[[!include set theory - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

\tableofcontents

## Definition

### In set theory

Let $\mathbb{N}$ be the set of natural numbers, and let $A$ be an arbitrary set. Then **sequence extensionality** states that for every [[sequence]] $a:\mathbb{N} \to A$ and $b:\mathbb{N} \to A$, $a =_{\mathbb{N} \to A} b$ if and only if $a(n) =_A b(n)$ for all natural numbers $n \in \mathbb{N}$.

### In category theory

Sequence extensionality in an [[arithmetic pretopos]] states that for every morphism $a:\mathbb{N} \to A$ and $b:\mathbb{N} \to A$, $a =_{\mathbb{N} \to A} b$ if and only if $a \circ n =_{\mathbb{1} \to A} b \circ n$ for all morphisms $n:\mathbb{1} \to \mathbb{N}$.

This definition makes sense in any finitely complete category with a parameterized [[natural numbers object]].

### In dependent type theory

In [[dependent type theory]], sequence extensionality states that given two sequences $a:\mathbb{N} \to A$ and $b:\mathbb{N} \to A$ there is an [[equivalence of types]] between the [[identity type]] $a =_{\mathbb{N} \to A} b$ and the [[dependent sequence type]] $(n:\mathbb{N}) \to (a(n) =_{A} b(n))$:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, a:\mathbb{N} \to A, b:\mathbb{N} \to A \vdash \mathrm{seqext}(a, b):(a =_{\mathbb{N} \to A} b) \simeq (n:\mathbb{N}) \to (a(n) =_{A} b(n))}$$

There is also a version of sequence extensionality for [[dependent sequence types]] called dependent sequence extensionality, which states that given two dependent sequences $a:(n:\mathbb{N}) \to A(n)$ and $b:(n:\mathbb{N}) \to A(n)$ there is an [[equivalence of types]] between the [[identity type]] $a =_{(n:\mathbb{N}) \to A(n)} b$ and the [[dependent sequence type]] $(n:\mathbb{N}) \to (a(n) =_{A(n)} b(n))$:
$$\frac{\Gamma, n:\mathbb{N} \vdash A(n) \; \mathrm{type}}{\Gamma,a:(n:\mathbb{N}) \to A(n), b:(n:\mathbb{N}) \to A(n) \vdash \mathrm{dseqext}(a, b):(a =_{(n:\mathbb{N}) \to A(n)} b) \simeq (n:\mathbb{N}) \to (a(n) =_{A(n)} b(n))}$$

## Properties

While [[function extensionality]] implies sequence extensionality, sequence extensionality is weaker than function extensionality. However, sequence extensionality is usually more relevant in [[dependent type theories]] which are [[strongly predicative]], which usually have [[dependent sequence types]] but do not have general [[dependent function types]].

## See also

* [[extensionality]]
* [[function extensionality]]

[[!redirects sequence extensionality]]
[[!redirects dependent sequence extensionality]]


sequence operad

\tableofcontents

## Idea

The __sequence operad__ is an [[E-infinity operad]],
valued in [[cochain complexes]],
that acts on [[simplicial cochains]]
of any [[simplicial set]].

Furthermore, this operad has an explicit presentation
in terms of a countable collection of generators and relations.

The operations in this operad are [[generalized cup products|generalized Steenrod cup-$i$ products]]
(denoted by $x \cup_i y$).

## Definition

The __sequence operad__ $Seq$ is the [[suboperad]]
of the [[endomorphism operad]]
of the [[simplicial cochains]] functor
$$S^* \colon sSet \to coCh$$
such that $Seq(n)$ is the subobject of $Hom((S^*)^{\otimes n},S^*)$
consisting of Steenrod's [[generalized cup products]].

## Main property

Theorem 2.15 in [McClure and Smith](#Cochain)
shows that $Seq$ is an [[E-infinity operad]] in [[cochain complexes]].

By definition, the operad $Seq$ acts on [[simplicial cochains]]
of any [[simplicial set]],
in particular, it acts on [[singular cochains]] of a [[topological space]].

This provides a simple and concrete model of the [[E-infinity algebra]]
structure on [[simplicial cochains]].

## Generalizations to $E_n$-algebras

One can easily identify a suboperad $Seq_n$ of the operad $Seq$
by imposing a simple combinatorial condition on the multiindices
of [[generalized cup products]] (Definition 3.2 [here](#Cochain)).

The resulting operad is an [[E_n-operad]],
i.e., it is weakly equivalent to the singular cochains on the little
$n$-cubes operad (Theorem 3.5 [here](#Cochain)).

## References

* {#Cochain} [[James E. McClure]], [[Jeffrey H. Smith]],
_Multivariable cochain operations and little $n$-cubes_,
[arXiv:math/0106024v3](https://arxiv.org/abs/math/0106024v3).


sequence space

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### Context
#### Functional analysis
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#### Topology
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=--

# Sequence spaces
* table of contents
{: toc}

## Sequence spaces

A classical _sequence space_ is a [[vector space]] of [[sequences]] of [[real numbers]], equipped with a [[p-norm]] that makes it a [[normed vector space]]. More generally one may consider spaces of functions on any set.

Specific sequence spaces are usually known through their symbolic names, such as '$c_0$' and '$l^p$', that appear below.  The term 'sequence space' is useful as a general name without symbols in it.

The sequences spaces are basic examples of [[topological vector spaces]].  They all have a discrete flavour that (maybe) makes them easy to understand, but they are not actually [[discrete spaces]].

The generalization of sequences space to spaces of functions on more general [[measure spaces]] are the _[[Lebesgue spaces]]_.

## Definitions

Fix a [[set]] $N$; typically, $N$ is the set $\mathbb{N}$ of [[natural numbers]], but this is not necessary for the basic concepts.  Sometimes one uses the set $\mathbb{Z}$ of [[integers]] (which is the [[underlying set]] of an [[abelian group]], useful for some purposes), which of course is [[bijective]] with $\mathbb{N}$.  For the simplest examples, let $N$ be a [[finite set]].

Also fix a [[topological vector space]] $K$; typically, $K$ is either the space $\mathbb{C}$ of [[complex numbers]] or the space $\mathbb{R}$ of [[real numbers]].  We will assume below that $K$ is at least a [[Banach space]]; but since much of the point of the sequence spaces is to be simple examples of Banach spaces, you probably want something familiar as $K$.

We will think of a [[function]] from $N$ to $K$ as a __$K$-valued $N$-sequence__, or simply a __sequence__.  The various sequence spaces will be [[subsets]] of the [[function set]] $K^N$ of all sequences.  In general, if '$X$' is the symbol for a sequence space, then we may specify $N$ and $K$ by writing '$X(N,K)$' (or a variation thereon), but often this is suppressed.

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

$l^1$ is the space of __absolutely summable sequences__:
$$ \sum_k {|a_k|} \lt \infty .$$
We equip $l^1$ with the __$l^1$-norm__
$$ {\|a\|_1} \coloneqq \sum_k {|a_k|} .$$
=--

This is a [[Banach space]].

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

$l^2$ is the space of __absolutely square-summable sequences__ (or, over a [[real field]], simply _square-summable sequences_):
$$ \sum_k {|a_k|^2} \lt \infty .$$
We equip $l^2$ with the __$l^2$-norm__
$$ {\|a\|_2} \coloneqq \sqrt{\sum_k {|a_k|^2}} .$$
=--

This is also a Banach space; in fact, it\'s a [[Hilbert space]] (assuming that $K$ is).  Furthermore, *every* Hilbert space (over $K$ a [[field]]) arises in this way, up to [[isometric isomorphism]], using an [[orthonormal basis]] for $N$.

More generally, for $0 \lt p \lt \infty$:

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

$l^p$ is the space of __absolutely $p$th-power--summable sequences__:
$$ \sum_k {|a_k|^p} \lt \infty .$$
We equip $l^p$ with the __$l^p$-norm__
$$ {\|a\|_p} \coloneqq \sqrt[p]{\sum_k {|a_k|^p}} .$$
=--

This is at least an $F$-[[F-space|space]], which is a Banach space iff $p \geq 1$.  (For $p \lt 1$, the 'norm' is not really a norm in the sense of a [[normed vector space]].)

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

$l^\infty$ is the space of __absolutely bounded sequences__:
$$ \sup_k {|a_k|} \lt \infty .$$
We equip $l^\infty$ with the __supremum norm__:
$$ {\|a\|_\infty} \coloneqq \sup_k {|a_k|} .$$
=--

This is also a Banach space.

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

$c_c$ (or $c_{00}$) is the space of __almost-zero sequences__:
$$ \ess\forall k,\; a_k = 0 ,$$
where '$\ess\forall$' means 'for all but finitely many' ($\tilde{K}$-[[K-finite|finite]] in [[constructive mathematics]]).
We equip $c_c$ with the topology of [[compact convergence]] (here, convergence on finite subsets).
=--

This is a [[locally convex space]].

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

$c_0$ is the space of __zero-limit sequences__:
$$ \forall \epsilon,\; \ess\forall k,\; {|a_k|} \lt \epsilon ,$$
where as usual $\epsilon$ is a [[positive number]] and again '$\ess\forall$' means 'for all but finitely many'.
We equip $c_0$ with the [[supremum norm]].
=--

This is also a locally convex space, in fact a [[Banach space]].

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

$c_\infty$ is the space of __convergent sequences__:
$$ \exists L,\; \forall \epsilon,\; \ess\forall k,\; {|a_k - L|} \lt \epsilon ,$$
where $L$ is an element of $K$ and the other notation is as in $c_0$ above.
We also equip $c_\infty$ with the supremum norm.
=--

This is also a [[Banach space]].  $c_\infty$ is also written simply '$c$', but this can be confusing; see the [Generalisations](#Generalisations) below.

There is some argument to be made that an element of $c_\infty$ should be a sequence with the [[extra structure]] of a specific limit $L$, rather than a sequence with the [[extra property]] that some limit exists.  This makes no difference if $N$ is [[infinite set|infinite]]; but if $N$ is finite then the version of $c_\infty$ with extra structure is the $l^\infty$-[[l-infinity-direct sum|direct sum]] of the [[ground field]] and the version of $c_\infty$ with extra property.

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

$c_b$ is the space of __absolutely bounded sequences__:
$$ \sup_k {|a_k|} \lt \infty .$$
We equip $c_b$ with the supremum norm too.
=--

This is yet another [[Banach space]].  Indeed, $c_b = l^\infty$, two different ways of thinking about the same thing.  (But they generalise differently.)

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

Finally, $N^K$ is the space of all sequences.  We equip $N^K$ with the [[product topology]], also called the topology of [[pointwise convergence]].
=--

This should probably be denoted '$c$', in line with the [generalisation](#Generalisations) below; but that symbol is often used for $c_\infty$, so it would be confusing.

## Properties

These properties all use the version of $c_\infty$ with extra property.

For $0 \lt p \lt q \lt \infty$, we have $c_c \subseteq l^p \subseteq l^q \subseteq c_0$, with each space [[dense subspace|dense]] in the next (using the topology of the next).  This continues: $c_0 \subseteq c_\infty \subseteq c_b = l^\infty$, but now each space, far from being dense, is a [[closed subspace]] of the next (with the [[induced topology]]).  Finally, $l^\infty = c_b \subseteq K^N$

When $N$ is [[finite set|finite]], these spaces are all the same, being just the [[cartesian spaces]] $K^N$; when $N$ is [[infinite set|infinite]], the inclusions above are all proper (at least if $K$ is nontrivial).

The various [[direct sums of Banach spaces]] follow the sequence spaces $l^p$ for $1 \leq p \leq \infty$.

The [[Riesz representation theorems]] give many nice results for the [[dual vector space|dual spaces]] of the sequence spaces:

*  the dual of $c_0$ is $l^1$,
*  the dual of $l^p$ is $l^q$ for $p + q = p q$ and $1 \lt p , q \lt \infty$,
*  the dual of $l^1$ is $l^\infty$,
*  the dual of $l^\infty$ is $l^1$ in [[dream mathematics]], but something much larger in [[classical mathematics]].

## Generalisations
   {#Generalisations}

The sequence spaces $l^p$ generalise to the [[Lebesgue spaces]] $L^p$ on arbitrary [[measure spaces]].  In fact, $l^p(N)$ is simply $L^p(N,\mu)$, where $\mu$ is [[counting measure]].

The sequence spaces $c_c$, $c_0$, $c_\infty$, $c_b$, and $K^N$ generalise to the spaces $C_c$, $C_0$, $C_\infty$, $C_b$, and $C$ of [[continuous maps]] on a [[local compactum]].  In fact, $c_*(N)$ is simply $C_*(N,\tau)$, where $\tau$ is the [[discrete topology]].  (Note that one never uses the symbol '$C$' for '$C_\infty$' with capital letters.)

A common setting for both of these generalisations is a (locally compact Hausdorff) [[topological group]].  While $l^\infty$ and $c_b$ are the same, $L^\infty$ and $C_b$ are the same *only* if the group is discrete.  (Otherwise $C_b \subset L^\infty$ properly.)

## In constructive mathematics

The spaces $c_c$, $c_0$, and $c_\infty$ work just fine in [[constructive mathematics]] (as does $K^N$, since it has no interesting structure anyway).  For $l^p$, we need $N$ to have [[decidable equality]] to define $\|a\|_p$; even so, $\|a\|_p$ (even when bounded) is only a [[lower real number]], so we usually require it be [[located real number|located]] to have an element of $l^p$.  With these caveats, $l^p$ works just fine for $0 \lt p \lt \infty$.  For $c_b = l^\infty$, we cannot get a [[Banach space]] with [[located real number|located]] [[norms]], as is usually required for constructive [[functional analysis]] ... well, unless we require $N$ to be [[finite set|finite]] (in the strictest sense), which leaves out the motivating example.  Nevertheless, we can still treat $l^\infty$ as a *semicontinuous* Banach space, that is one where the norms may be any bounded [[lower reals]]; for that matter, we can also consider semicontinuous versions of $l^p$.  (Another way to treat $l^\infty$ may be formally, as the [[dual vector space|dual]] of $l^1$; I don\'t know how well this works.)

## See also

* [[sequence]]

* [[sequence algebra]]

## References

At least for the term 'sequence space', try [Wikipedia](https://en.wikipedia.org/wiki/Sequence_space) and _[[HAF]]_.

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[[!redirects sequence spaces]]

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sequent

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#### Deduction and Induction
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[[!include deduction and induction - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

#Sequent#
* table of contents
{:toc}

## Idea

In formal [[logic]] a _sequent_ ([Gentzen 35](#Gentzen), [Martin-Löf 83](#Martin-Löf)) or _hypothetical judgement_ ([Pfenning, Davies 00](Pfenning-Davies)) is an expression in the [[metalanguage]]
which is a string of symbols of the form

$$
  Antecedent \vdash Succedent
$$

where

1. $Antecedent$ are symbols indicating [[judgements]] called the _[[antecedents]]_ or _[[context]]_,

1. $Succedent$ are symbols indicating [[judgements]] called the _[[succedents]]_ or (if it is just a single judgement) the _[[consequent]]_

1. the **consequence sign** or **turnstile**-symbol "$\vdash$" expresses that $Succedent$ is a _consequence_ of $Antecedent$:

   " $Antecedent$ yields $Succedent$."

   or

   "With $Antecedent$ the $Succedent$ can be [[proof|proven]]."

   or

   "$Antecedent$, con-sequent-ly $Succedent$."

   Or similar.

Historically the "consequence" here was early on transmuted to "sequenz" ([Gentzen](#Gentzen)) and then later to "sequent". See the section _[History](#History)_ below.

In systems of [[formal logic]] such as [[natural deduction]]/[[type theory]] such sequents express rules for explicit _symbol manipulation_ admitted in the system rather than formal [[implications]] _within_ the system. The latter instead are expressed by [[terms]] of [[function type]] $t : \phi \to \psi$. But the [[term introduction rule]] for terms of function types say that given one, one is allowed to get the other.

Typically one allows a list of expressions on both sides of the turnstile-symbols as in

$$
 Antec_1, \cdots, Antec_k \vdash Succ_1, \cdot, Succ_l
$$

often abbreviated as

$$
  \vec Antec \vdash \vec Succ
$$

in which case on the left a [[conjunction]] of the [[antecedents]] and on the right a [[disjunction]] of [[succedents]] is understood, as in

"If $Antec_1$ and $Antec_2$... are given then one of $Succ_1, \cdots, Succ_l$ is a consequence".

An **[[intuitionistic logic|intuitionistic]] sequent** has at most one succedent, which is then called the **[[consequent]]**.  Often "intuitionistic sequent" is used to mean one with *exactly* one succedent, although technically it would make more sense to call those [[minimal logic|minimal]] sequents.

Another variant is that in some frameworks the [[antecedent]] and [[succedent]] displayed are required to be [[propositions]] and the [[free variables]] of the [[context]] are instead displayed beneath the turnstile as in

$$
  \vec \phi(x)\, true \vdash_{\vec x} \vec \psi(x) \, true
  \,.
$$

If the framework regards [[propositions as types]] then this is the same as writing

$$
  \vec x. \vec \phi(x) \vdash \vec \psi(x)
  \,.
$$

Finally one can of course consider sequences of sequents

$$
  (\Gamma_1 \vdash \Delta_1), \cdots, (\Gamma_n \vdash \Delta_n)
$$

and if these are read [[disjunction|disjunctively]] it is like a higher-order sequent without [[antecedent]] and called a *[[hypersequent]]*.

Rules for formal manipulation of sequents are called _[[sequent calculi]]_ or _[[deduction calculi]]_. See there for more details.

## Definition

The precise nature of sequents has been formalized differently in different systems of [[formal logic]]. We discuss a few

### Intuitionistic sequents

(...) ([Martin-Löf](#Martin-Löf)) (...)

### Gentzen's sequents

(...) ([Gentzen](#Gentzen)) (...)

### Sequents in focalized calculi

(...) [Simmons](#Simmons) (...)

## Semantics

We discuss aspects of the [[categorical semantics]] of sequents, hence their interpretation when the ambient formal logic is regarded as the [[internal language]] of a [[category]].

### In homotopy type theory

Under the [[categorical semantics]] of [[homotopy type theory]]
sequents in the type theory pretty accurately correspond to
[[morphisms]] in the [[(∞,1)-topos]]. We indicate how this works, first for type declarations, then for terms of [[dependent types]].

Let $\mathbf{H}$ be an [[(∞,1)-topos]]. Write $Type \in \mathbf{H}$ for the [[universe in a topos|internal universe]] of  [[small objects]] of $\mathbf{H}$, called the _[[object classifier]]_.

This is defined as the [[representable functor|representing object]]
for the [[core]] of the [[small object|small]] [[codomain fibration]], exhibited by an  [[equivalence of ∞-groupoids]]

$$
  name
  :
  Core(\mathbf{H}_{/X})^{small} \stackrel{\simeq}{\to} \mathbf{H}(X,Type)
  \,.
$$

This equivalence sends an $X$-family $(E \to X) \in \mathbf{H}_{/X}$ to its "name", denoted

$$
  X \stackrel{\vdash E}{\to} Type
  \,,
$$

which is the morphism characterized by fitting into an
[[(∞,1)-pullback|∞-pullback]] square of the form

$$
  \array{
     E &\to& \widehat{Type}
     \\
     \downarrow &\swArrow& \downarrow
     \\
     X &\underset{\vdash E}{\to}& Type
  }
  \,,
$$

If here we simply replace, notationally, the arrow "$\to$" by the turnstile "$\vdash$", display a generic [[generalized element]] $x$ of $X$ and then write $E(x)$ to highlight the dependence of the [[fibers]] of $E$ on $x$ in $X$, then the symbols "$X \stackrel{\vdash E}{\to} Type$" become the
sequent "$x : X \vdash E(x) : Type$".
This sequent is the [[syntax]] of which the morphism is the [[categorical semantics]].

Similarly, if $X \stackrel{t}{\to}_X E$
is a [[section]] of $E$ over $X$, hence
a [[generalized element]] in the [[slice (∞,1)-topos]] $\mathbf{H}_{/X}$,
then by replacing the arrow-symbol by a
 turnstile-symbol we get "$x : X \vdash t(x) : E(x)$".
This is the sequent for the [[term]] $t$ of the [[dependent type]] $E$.

In summary we have under the [[relation between category theory and type theory]] the notational correspondence:

**morphisms to sequents.**

| $\,$ | [[types]] | [[terms]] |
|--|--|--|
| [[(∞,1)-topos theory|∞-topos theory]] | $\;\;\;\;X \stackrel{\vdash \;\;\;\;E}{\to} \;\;\Type$ | $\;\;\;\;X \stackrel{\vdash \;\;\;t}{\to} {}_X \;\;E$ |
|[[homotopy type theory]] | $x : X \vdash E(x) : Type$ | $x : X \vdash t(x) : E(x) $ |

## History
 {#History}

The notion of sequent was introduced in section 2.3 of ([Gentzen 1935](#Gentzen)) (called _Sequenz_ there). In ([Martin-Löf 1984, pages 29-30](#Martin-Löf)) it says

> The forms of hypothetical judgement $[$ have $]$ the form

>$A_1 true, \cdots, A_n true \vdash A prop$

>which says that $A$ is a proposition under the assumptions that $A_1, \cdots, A_n$ are all true, and, on the other hand, the form

> $A_1 true, \cdots, A_n true \vdash A true$

> which says that the proposition A is true under the assumptions that $A_1, \cdots, A_n$ are all true. Here I am using the vertical bar for the relation of logical consequence, that is, for what Gentzen expressed by means of the arrow $\to$ in his sequence calculus, and for which the double arrow $\Rightarrow$ is also a common notation. It is the relation of logical consequence, which must be carefully distinguished from implication. What stands to the left of the consequence sign, we call the hypotheses, in which case what follows the consequence sign is called the thesis, or we call the **judgements that precede the consequence sign** the antecedents and the **judgement that follows after the consequence sign** the consequent. This is the terminology which Gentzen took over from the scholastics, except that, for some reason, he changed consequent into succedent and consequence into sequence, Ger. Sequenz, usually improperly rendered by sequent in English.

## Related concepts

* [[sequent calculus]]

* [[cut rule]]

[[!include logic symbols -- table]]

## References

In section 2.3

* [[Gerhard Gentzen]]  _Untersuchungen über das logische Schließen I_ _Mathematische Zeitschrift_ 39(1), Springer-Verlag 1935. <http://dx.doi.org/10.1007/BF01201353>.
 {#Gentzen}

what today is called a _sequent_ is introduced under _Sequenz_ (Ger: _sequence_), apparently derived from _Konsequenz_ (Ger: _consequence_) and denoted by a single arrow "$\to$".

In the lectures

* [[Per Martin-Löf]], _On the meaning of logical constants and the justifications of the logical laws_, leture series in Siena (1983) ([web](http://docenti.lett.unisi.it/files/4/1/1/6/martinlof4.pdf))
 {#Martin-Löf83}

where the author provides a modern foundation for logic based on a clear separation of the notions of [[judgment]] and [[proposition]] (see at [[Martin-Löf dependent type theory]]) the author says (pages 29-30) that "the forms of hypothetical judgements that I shall need" are [[Gentzen]]'s sequents without the symmetry between [[antecedent]] and [[succedent]] that Gentzen used.

Referring explicitly to these lectures, these are are then just called _hypothetical judgements_ in section 3 of

* [[Frank Pfenning]], Rowan Davies, _A judgemental reconstruction of modal logic_ (2000) ([pdf](http://www.cs.cmu.edu/~fp/papers/mscs00.pdf))
 {#Pfenning-Davies}

In section D1.1 of

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_

sequents are introduced in the context of a basic introduction to [[formal logic]]. There the the notation $\phi \vdash_{\vec c} \psi$ is used where $\phi$ is required to be a [[proposition]] and the [[context]] [[variables]] $\vec x$ are typeset below the turnstile. From the [[categorical semantics]] in section D1.2 it is clear that in the sense of [[dependent type theory]] these variables are to stand to the left of the turnstile.

See also

* Robert J. Simmons, _Structural focalization_ ([arXiv:1109.6273](http://arxiv.org/abs/1109.6273))
 {#Simmons}

[[!redirects sequents]]

[[!redirects turnstile]]


sequent calculus

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

# Sequent calculus
* table of contents
{: toc}

## Idea

A sequent calculus is a [[deductive system]] for [[predicate logic]] (or something similar) in which the basic [[judgment]] is a [[sequent]]
$$ \phi_1, \phi_2, \ldots \vdash \psi_1, \psi_2, \ldots ,$$
expressing the idea that some statement $\psi_i$ must be true if every statement $\phi_i$ is true.  The original sequent calculi were developed by [[Gerhard Gentzen]] in 1933 building on anticipatory ideas by [[Paul Hertz]].

A central property of sequent calculi, which distinguishes them from systems of [[natural deduction]], is that it allows an easier analysis of proofs: the [subformula property](#subformulaprop) that it enjoys allows easy [[induction]] over proof-steps. The reason is roughly that, using the language of [[natural deduction]], in sequent calculus "every rule is an [[introduction rule]]" which introduces a term on _either_ side of a [[sequent]] with no [[elimination rules]]. This means that working backward every "un-application" of such a rule makes the sequent necessarily simpler.

## Definitions

We will start with an arbitrary [[signature (in logic)|signature]] $\Sigma$; the simplest case (for [[propositional logic]]) uses the empty signature (with no [[types]]).  We will assume unlimited [[variables]] for [[propositions]] and unlimited variables for each type.

\begin{remark}
We will be defining several terms below; but keep in mind that these are definitions *for sequent calculus*, since these terms may also be used more generally.
\end{remark}

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

A __[[context]]__ over $\Sigma$ is a (finite) [[list]] $\vec{x}\colon \vec{T}$ or
$$ x_1\colon T_1, x_2\colon T_2, \ldots ,$$
where the $x_i$ are [[variables]] and the $T_i$ are (formulas for) [[types]] over $\Sigma$.
=--

For the empty signature, the only context is empty; if $\Sigma$ consists of a single type, then a context is effectively a list of variables.

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

Given a context $\Gamma$ over $\Sigma$, a __cedent__ in $\Gamma$ over $\Sigma$ is simply a [[list]] $\vec{\phi}$ or
$$ \phi_1, \phi_2, \ldots $$
of (formulas for) [[propositions]] in $\Gamma$ over $\Sigma$.
=--

(Recall that a proposition in the context $\Gamma$ allows the variables introduced by $\Gamma$, but no others, to be free variables.  Such a proposition in context is also called a [[predicate]].)

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

A __sequent__ over $\Sigma$ consists of a context $\Gamma$ over $\Sigma$ and two cedents in $\Gamma$, called the __[[antecedent]]__ and the __[[succedent]]__.  This is written
$$ \vec{\phi} \vdash_{\vec{x}\colon \vec{T}} \vec{\psi} ,$$
where $\vec{x}\colon \vec{T}$ is the context $\Gamma$, with the antecedent on the left and the succedent on the right.
=--

\begin{remark}

If $\Sigma$ has a single type (a so-called 'untyped' signature) and one tacitly assumes that the single type is [[inhabited type|inhabited]], then one may (up to equivalence in a certain sense) reconstruct the context of any sequent from the cedents, by examining which variables appear; this is even possible if $\Sigma$ has more than one type (all assumed inhabited), although more difficult if certain abuses of notation are used.  Then the context can be ignored.  This is not really the right way to do things, but (especially for untyped signatures) it is traditional.

Confusingly, the cedents are sometimes also called 'contexts'; then one speaks of the __left context__ (the antecedent), the __right context__ (the succedent).  This is most easily done when ingoring what we have been calling the context; I\'m not even sure what term would then be used for that context (maybe the __type context__?).  See also also the [interpretation of sequents as hypothetical judgements](sequent#Gentzen2MartinLof) (which I hope to write soon), which shows how the type context, antecedent, and succedent are all used to form the context of a related [[hypothetical judgement]].

One also sometimes uses 'antecedent' or 'succedent' for an *individual* proposition on the relevant side of the sequent; see also the use of 'consequent' in [minimal sequents](#minimaletc) below.

\end{remark}

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

A __[[theory]]__ over $\Sigma$ is an arbitrary [[subset|set]] of sequents over $\Sigma$, called the __[[axioms]]__ of the theory.
=--

We will explain below how the sequent calculus may be used to prove [[theorems]] from the axioms; each theorem will also be a sequent.

We may also consider particular kinds of sequents or theories, by form or content.

+-- {: .num_defn #minimaletc}
###### Definitions (by form)

A __closed sequent__ is one in which the context is empty; a __closed theory__ is one in which every sequent is closed.

An __intuitionistic sequent__ is one in which the succedent consists of at most one proposition; an __intuitionistic theory__ is one in which every sequent is intuitionistic.

A __minimal sequent__ is one in which the succedent consists of exactly one proposition (then called the __[[consequent]]__); a __minimal theory__ is one in which every sequent is minimal.

A __dual-intuitionistic sequent__ is one in which the antecedent consists of at most one proposition; a __dual-intuitionistic theory__ is one in which every sequent is dual-intuitionistic.

A __dual-minimal sequent__ is one in which the antecedent consists of exactly one proposition; a __dual-minimal theory__ is one in which every sequent is dual-minimal.

A __classical sequent__ is simply a sequent, emphasising that we are not restricting to (dual)-intuitionistic or (dual)-minimal sequents; similarly, a __classical theory__ is any theory.
=--

These (except the first) are so-called because of their relationship to [[intuitionistic logic]], [[minimal logic]], [[classical logic]], etc.  Any of these [[logics]] may be presented using any sort of sequent, but Gentzen\'s original sequent calculi presented each logic using only corresponding sequents.

+-- {: .num_defn}
###### Definitions (by content)

A __regular sequent__ is one in which every proposition is given by a formula in [[regular logic]]; a __regular theory__ is one in which every sequent is regular.

A __coherent sequent__ is one in which every proposition is given by a formula in [[coherent logic]]; a __coherent theory__ is one in which every sequent is coherent.
=--

Obviously, this list could be continued.

## Rules of proof

The basic building blocks of a proof in sequent calculus are **[[rule of inference|inference rules]]** or **rewrite rules**, which are rules for inferring the validity of certain sequents from other sequents.  One writes

$$
  \frac{\vec{\sigma}}{\tau}
$$

where $\vec{\sigma}$ is a (possibly empty) [[list]] of sequents and $\tau$ is a single sequent, to indicate that from the validity of the sequents in $\vec{\sigma}$ that of $\tau$ may be inferred, or that $\vec{\sigma}$ may be _rewritten_ to $\tau$.

A **[[deduction|derivation]]** or **proof tree** is a compound of rewrite rules, such as

$$
  \frac{\frac{\sigma_1 \;\; \sigma_2}{\sigma_3} \;\; \sigma_4 }{\sigma_5}
  \,.
$$

We will give the rules in several classes.

+-- {: .num_defn}
###### Rules (structural rules)

The **[[identity rule]]** is
  $$
    \frac{}{\alpha \vdash_{\vec{x}\colon \vec{T}} \alpha}
     \,,
  $$
stating that, in any context $\Gamma$, any proposition in $\Gamma$ follows from itself, always.

The **[[substitution rule]]** is
  $$
    \frac{
      \vec{\phi} \vdash_{\vec{x}\colon \vec{T}} \vec{\psi}
    }{
      \vec{\phi}[\vec{s}/\vec{x}] \vdash_{\vec{y}\colon \vec{U}} \psi[\vec{s}/\vec{x}]
    }
    \,,
  $$
or
  $$
    \frac{
      \vec{\phi} \vdash_{\Gamma} \vec{\psi}
    }{
      a^*\vec{\phi} \vdash_{\Delta} a^*\vec{\psi}
    }
    \,,
  $$
  where $a$ is any [[interpretation]] of $\Gamma$ in $\Delta$.  Explicitly, such an interpretation is a list $\vec{s}$ of [[terms]] (of the same length as the list which is the context $\Gamma$), where each term $s_i$ is a term over $\Sigma$ of type $T_i$ in the context $\Delta$.  Then $a^*\phi$, or $\phi[\vec{s}/\vec{x}]$, is obtained from $\phi$ by substituting each $s_i$ for the corresponding $x_i$, and $a^*\vec{\phi}$ (and $a^*\vec{\psi}$) are obtained by applying this substitution to every proposition in the list.  Of course, this rule is vacuous if $\Sigma$ has no terms.

The **[[cut rule]]** is
  $$
    \frac{
      (\vec{\phi} \vdash_{\vec{x}\colon \vec{T}} \vec{\psi}, \alpha) \;\;\; (\alpha, \vec{\chi} \vdash_{\vec{x}\colon \vec{T}} \vec{\omega})
    }{
      \vec{\phi}, \vec{\chi} \vdash_{\vec{x}\colon \vec{T}} \vec{\psi}, \vec{\omega}
   }\,;
  $$
the proposition $\alpha$ has been 'cut'.

The __[[exchange rules]]__ are
  $$
    \frac{
      \vec{\phi}, \alpha, \beta, \vec{\psi} \vdash_{\vec{x}\colon \vec{T}} \vec{\chi}, \vec{\omega}
    }{
      \vec{\phi}, \beta, \alpha, \vec{\psi} \vdash_{\vec{x}\colon \vec{T}} \vec{\chi}, \vec{\omega}
    }
    \,
  $$
and
  $$
    \frac{
      \vec{\phi}, \vec{\psi} \vdash_{\vec{x}\colon \vec{T}} \vec{\chi}, \alpha, \beta, \vec{\omega}
    }{
      \vec{\phi}, \vec{\psi} \vdash_{\vec{x}\colon \vec{T}} \vec{\chi}, \beta, \alpha, \vec{\omega}
    }
    \,;
  $$
more generally, we may apply any [[permutation]] of the antecedent and succedent (because every permutation is a [[composite]] of [[transposition]]s).

The __[[weakening rules]]__ are
  $$
    \frac{
      \vec{\phi}, \vec{\psi} \vdash_{\vec{x}\colon \vec{T}} \vec{\chi}
    }{
      \vec{\phi}, \alpha, \vec{\psi} \vdash_{\vec{x}\colon \vec{T}} \vec{\chi}
    }
    \,
  $$
and
  $$
    \frac{
      \vec{\phi} \vdash_{\vec{x}\colon \vec{T}} \vec{\psi}, \vec{\chi}
    }{
      \vec{\phi} \vdash_{\vec{x}\colon \vec{T}} \vec{\psi}, \alpha, \vec{\chi}
    }
    \,;
  $$
more generally, we may insert any sequence of propositions anywhere in the antecedent and succedent.

The __[[contraction rules]]__ are
  $$
    \frac{
      \vec{\phi}, \alpha, \alpha, \vec{\psi} \vdash_{\vec{x}\colon \vec{T}} \vec{\chi}
    }{
      \vec{\phi}, \alpha, \vec{\psi} \vdash_{\vec{x}\colon \vec{T}} \vec{\chi}
    }
  $$
and
  $$
    \frac{
      \vec{\phi} \vdash_{\vec{x}\colon \vec{T}} \vec{\psi}, \alpha, \alpha, \vec{\chi}
    }{
      \vec{\phi} \vdash_{\vec{x}\colon \vec{T}} \vec{\psi}, \alpha, \vec{\chi}
    }
    \,;
  $$
in the absence of the exchange rule, it is important that we can remove duplications only one proposition at a time (as shown here) and only if they are adjacent (as shown here).
=--

Some or all of these rules may be dropped in a [[substructural logic]].  However, even so, the first three rules can generally be proved (except for the identity rule for atomic propositions); see [[cut elimination]].

+-- {: .num_defn }
###### Rules (logical rules)

The **[[truth]] rule** is
  $$
    \frac{
    }{
      \vec{\phi} \vdash_{\vec{x}\colon \vec{T}} \vec{\psi}, \top, \vec{\chi}
    }
    \,;
  $$
that is, any sequent is valid if the necessarily true statement is in the succedent.

[[de Morgan duality|Dually]], the **[[falsehood]] rule** is
  $$
    \frac{
    }{
      \vec{\phi}, \bot, \vec{\psi} \vdash_{\vec{x}\colon \vec{T}} \vec{\chi}
    }
    \,;
  $$
that is, any sequent is valid if the necessarily false statement is in the antecedent.

The **binary [[conjunction]] rules** are
  $$
    \frac{
      \vec{\phi}, \alpha, \vec{\psi} \vdash_{\vec{x}\colon \vec{T}} \vec{\chi}
    }{
      \vec{\phi}, \alpha \wedge \beta, \vec{\psi} \vdash_{\vec{x}\colon \vec{T}} \vec{\chi}
    }
    \,
  $$
and
  $$
    \frac{
      \vec{\phi}, \beta, \vec{\psi} \vdash_{\vec{x}\colon \vec{T}} \vec{\chi}
    }{
      \vec{\phi}, \alpha \wedge \beta, \vec{\psi} \vdash_{\vec{x}\colon \vec{T}} \vec{\chi}
    }
    \,
  $$
on the left and
  $$
    \frac{
      (\vec{\phi} \vdash_{\vec{x}\colon \vec{T}} \vec{\psi}, \alpha, \vec{\chi})
        \;\;\;
      (\vec{\phi} \vdash_{\vec{x}\colon \vec{T}} \vec{\psi}, \beta, \vec{\chi})
    }{
      \vec{\phi} \vdash_{\vec{x}\colon \vec{T}} \vec{\psi}, \alpha \wedge \beta, \vec{\chi}
    }
    \,
  $$
on the right.

Dually, the **binary [[disjunction]] rules** are
  $$
    \frac{
      (\vec{\phi}, \alpha, \vec{\psi} \vdash_{\vec{x}\colon \vec{T}} \vec{\chi})
        \;\;\;
      (\vec{\phi}, \beta, \vec{\psi} \vdash_{\vec{x}\colon \vec{T}} \vec{\chi})
    }{
      \vec{\phi}, \alpha \vee \beta, \vec{\psi} \vdash_{\vec{x}\colon \vec{T}} \vec{\chi}
    }
    \,
  $$
on the left and
  $$
    \frac{
      \vec{\phi} \vdash_{\vec{x}\colon \vec{T}} \vec{\psi}, \alpha, \vec{\chi}
    }{
      \vec{\phi} \vdash_{\vec{x}\colon \vec{T}} \vec{\psi}, \alpha \vee \beta, \vec{\chi}
    }
    \,
  $$
and
  $$
    \frac{
      \vec{\phi} \vdash_{\vec{x}\colon \vec{T}} \vec{\psi}, \beta, \vec{\chi}
    }{
      \vec{\phi} \vdash_{\vec{x}\colon \vec{T}} \vec{\psi}, \alpha \vee \beta, \vec{\chi}
    }
    \,
  $$
on the right.

The **[[negation]] rules** are
  $$
    \frac{
      \vec{\phi} \vdash_{\vec{x}\colon \vec{T}} \alpha, \vec{\psi}
    }{
      \vec{\phi}, \neg{\alpha} \vdash_{\vec{x}\colon \vec{T}} \vec{\psi}
    }
    \,
  $$
and
  $$
    \frac{
      \vec{\phi}, \alpha \vdash_{\vec{x}\colon \vec{T}} \vec{\psi}
    }{
      \vec{\phi} \vdash_{\vec{x}\colon \vec{T}} \neg{\alpha}, \vec{\psi}
    }
    \,;
  $$
in other words, a proposition may be moved to the other side if it is replaced by its negation.

The **[[conditional]] rules** are
  $$
    \frac{
      (\vec{\phi} \vdash_{\vec{x}\colon \vec{T}} \alpha)
        \;\;\;
      (\beta \vdash_{\vec{x}\colon \vec{T}} \vec{\psi})
    }{
      \vec{\phi}, \alpha \to \beta \vdash_{\vec{x}\colon \vec{T}} \vec{\psi}
    }
    \,
  $$
and
  $$
    \frac{
      \vec{\phi}, \alpha \vdash_{\vec{x}\colon \vec{T}} \beta, \vec{\psi}
    }{
      \vec{\phi} \vdash_{\vec{x}\colon \vec{T}} \alpha \to \beta, \vec{\psi}
    }
    \,.
  $$

Dually, the **[[relative complement]] rules** are
  $$
    \frac{
      \vec{\phi}, \alpha \vdash_{\vec{x}\colon \vec{T}} \beta, \vec{\psi}
    }{
      \vec{\phi}, \alpha \setminus \beta \vdash_{\vec{x}\colon \vec{T}} \vec{\psi}
    }
    \,
  $$
and
  $$
    \frac{
      (\vec{\phi} \vdash_{\vec{x}\colon \vec{T}} \alpha)
        \;\;\;
      (\beta \vdash_{\vec{x}\colon \vec{T}} \vec{\psi})
    }{
      \vec{\phi} \vdash_{\vec{x}\colon \vec{T}} \alpha \setminus \beta, \vec{\psi}
    }
    \,.
  $$

The **[[equality]] rules** are
  $$
    \frac{
      (\vec{\phi} \vdash_{\vec{x}\colon \vec{T}} \alpha[s/y])
        \;\;\;
      (\alpha[t/y] \vdash_{\vec{x}\colon \vec{T}} \vec{\psi})
    }{
      \vec{\phi}, (s = t) \vdash_{\vec{x}\colon \vec{T}} \vec{\psi}
    }
    \,,
  $$
where $s$ and $t$ are any [[terms]] of the same [[type]] $U$ in the context $\vec{x}\colon \vec{T}$ and $\alpha$ is a proposition in the context $(\vec{x}\colon \vec{T}, y\colon U)$, and
  $$
    \frac{
    }{
      \vec{\phi} \vdash_{\vec{x}\colon \vec{T}} \vec{\psi}, (s = s)
    }
    \,,
  $$
where $s$ is any [[term]] of any type in the context $\vec{x}\colon \vec{T}$.

One could (but rarely does) introduce dual [[apartness]] rules.

The **[[universal quantification]] rules** are
  $$
    \frac{
      \vec{\phi}, \alpha[s/y], \vec{\psi} \vdash_{\vec{x}\colon \vec{T}} \vec{\chi}
    }{
      \vec{\phi}, \forall(y\colon U) \alpha, \vec{\psi} \vdash_{\vec{x}\colon \vec{T}} \vec{\chi}
    }
    \,,
  $$
where $s$ is a term of type $U$ in the context $\vec{x}\colon \vec{T}$ and $\alpha$ is a proposition in the [[extended context]] $(\vec{x}\colon \vec{T}, y\colon U)$, and
  $$
    \frac{
      \vec{\phi}[\hat{y}] \vdash_{\vec{x}\colon \vec{T}, y\colon U} \vec{\psi}[\hat{y}], \alpha, \vec{\chi}[\hat{y}]
    }{
      \vec{\phi} \vdash_{\vec{x}\colon \vec{T}} \vec{\psi}, \forall(y\colon U) \alpha, \vec{\chi}
    }
    \,,
  $$
where $\phi[\hat{y}]$ is the proposition in the context $(\vec{x}\colon \vec{T}, y\colon U)$ produced by [[weakening rule|weakening]] (in the type context) the proposition $\phi$ in the context $\vec{x}\colon \vec{T}$ (and similarly for a list of propositions).

Dually, the **[[existential quantification]] rules** are
  $$
    \frac{
      \vec{\phi}[\hat{y}], \alpha, \vec{\psi}[\hat{y}] \vdash_{\vec{x}\colon \vec{T}, y\colon U} \vec{\chi}[\hat{y}]
    }{
      \vec{\phi}, \exists(y\colon U) \alpha, \vec{\psi} \vdash_{\vec{x}\colon \vec{T}} \vec{\chi}
    }
    \,
  $$
and
  $$
    \frac{
      \vec{\phi} \vdash_{\vec{x}\colon \vec{T}} \vec{\psi}, \alpha[s/y], \vec{\chi}
    }{
      \vec{\phi} \vdash_{\vec{x}\colon \vec{T}} \vec{\psi}, \exists(y\colon U) \alpha, \vec{\chi}
    }
    \,.
  $$
=--

We have written the rules _additively_, which works best when using (dual)-intuitionistic or (dual)-minimal sequents, but it is also possible to write them in a slightly different _multiplicative_ manner.  This makes a difference if some of the [[weakening rule]] and [[contraction rule]] are abandoned; [[linear logic]] uses both rules but for different connectives.  Similarly, we have written the rules to apply as much as possible without the [[exchange rule]], but the remaining asymmetries (in the rules for $\neg$, $\to$, $\setminus$, and $=$) mean that these diverge into left and right operations in [[noncommutative logic]].  (Compare the difference between [[left duals]] and [[right duals]] in a non-[[symmetric monoidal category|symmetric]] [[monoidal category]].)

Gentzen originally did not include $\top$ or $\bot$, but if $\top$ is defined as $\alpha \to \alpha$ for any atomic $\alpha$ and $\bot$ is defined as $\neg{\top}$, then their rules can be proved.  Similarly, he did not use $\setminus$; we may define $\alpha \setminus \beta$ to mean $\alpha \wedge \neg{\beta}$.  It would also be possible to leave out $\to$, defining $\alpha \to \beta$ as $\neg{\alpha} \vee \beta$.  With or without these optional operations and rules, the resulting logic is [[classical logic|classical]].

If we use only those rules that can be stated using intuitionistic sequents, then the result is [[intuitionistic logic]]; this is again true with or without $\top$, $\bot$, or $\setminus$.  However, if we leave out $\to$, then we cannot reconstruct it; the definition of $\to$ using $\neg$ and $\vee$ is not valid intuitionistically.  On the other hand, if we include $\bot$ (and optionally $\top$) but leave out $\neg$ and $\setminus$, then we get intuitionistic logic using all (classical) sequents except the right conditional rule, which must remain intuitionistic.  (Conversely, we could get classical logic using only intuitionistic sequents and adding the law of [[excluded middle]] as an axiom.)

If we use only those rules that can be stated using minimal sequents (so necessarily leaving out $\setminus$), then the result is still intuitionistic logic; but if we also leave out $\bot$, then the result is [[minimal logic]].  Dual results hold for dual-intuitionistic and dual-minimal sequents.

## Cut-free proofs
{#subformulaprop}

The [[cut rule]] expresses the composition of proofs.  Gentzen's main result ([Gentzen, Haupsatz](#Gentzen)) is that any derivation that uses the cut rule can be transformed into one that doesn't -- the _[[cut-elimination theorem]]_.  This yields a [[normal form|normalization]] algorithm for proofs, which provided much of the inspiration behind [[Lambek]]'s approach to categorical logic.  Similarly, any derivation that uses the identity rule can be transformed into one that uses it only for atomic propositions (those provided by the signature $\Sigma$ and equality).

The most important property of cut-free proofs is that every formula occurring anywhere in a proof is a subformula of a formula contained in the conclusion of the proof (the [[subformula property]]).  This makes induction over proof-trees much more straightforward than with [[natural deduction]] or other systems.

See [[cut elimination]].

## Related entries

* [[natural deduction]]

* [[cut rule]]

* [[hypersequent calculus]]

* [[Paul Hertz]]

## References

Sequent calculus was introduced in

* {#Gentzen} [[Gerhard Gentzen]],  _Untersuchungen über das logische Schließen I_ _Mathematische Zeitschrift_ **39** 1 (1935) [[doi:10.1007/BF01201353](http://dx.doi.org/10.1007/BF01201353)]

which appeared in English translation as:

* [[Gerhard Gentzen]], *Investigations into Logical Deduction*, in M. E. Szabo (ed.), *The Collected Papers of Gerhard Gentzen*, Studies in Logic and the Foundations of Mathematics **55**, Springer (1969) 68-131 [[ISBN:978-0-444-53419-4](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/55), [pdf](https://logic-teaching.github.io/prop/texts/Gentzen%201969%20-%20Investigations%20into%20Logical%20Deduction.pdf)]

Introduction:

* [[Paul Downen]], [[Zena M. Ariola]], *A tutorial on computational classical logic and the sequent calculus*, Journal of Functional Programming **28** (2018) e3 [[doi:10.1017/S0956796818000023](https://doi.org/10.1017/S0956796818000023), [pdf](https://www.pauldownen.com/publications/sequent-intro.pdf)]

Textbook accounts:

* Morten Heine Sørensen, Pawel Urzyczyn, Chapter 7 of: *Lectures on the Curry-Howard isomorphism*, Studies in Logic **149** (2006) [[ISBN:9780444520777](https://www.elsevier.com/books/lectures-on-the-curry-howard-isomorphism/sorensen/978-0-444-52077-7), [pdf](https://disi.unitn.it/~bernardi/RSISE11/Papers/curry-howard.pdf)]

* Sara Negri, Jan von Plato, §1, §2 in: *Structural Proof Theory*, Cambridge University Press (2010) [[doi:10.1017/CBO9780511527340](https://doi.org/10.1017/CBO9780511527340), [§1 pdf](https://www.logicmatters.net/resources/pdfs/NegriPlato1.pdf), [§2 pdf](https://www.logicmatters.net/resources/pdfs/NegriPlato1.pdf)]

For the basic definitions in a context of [[topos theory]]:

* [[Peter Johnstone]], Defs. D1.1.5 and D1.3 in: _[[Sketches of an Elephant]]_

See also:

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

* [Sequent Calculus Primer](http://sakharov.net/sequent.html)

Discussion for [[type theory]]:

* Stéphane Lengrand,  Roy Dyckhoff, James McKinna, _A Sequent Calculus for Type Theory_ ([inria](http://hal.inria.fr/hal-00150286/), [pdf](http://138.251.206.45/~rd/publications/42070441.pdf))

[[!redirects sequent calculus]]
[[!redirects sequent calculi]]
[[!redirects sequent calculuses]]


sequential convergence space > history

< [[sequential convergence space]]
[[!redirects sequential convergence space -- history]]


sequential limit

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#### Limits and colimits
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## Definition

A **sequential (co)limit** is a [[limit]]/[[colimit]] whose [[diagram]] category is a nonzero [[ordinal]] or its [[opposite poset|opposite]] (regarded as a [[poset]], regarded as a [[category]]). For instance over a [[tower diagram]].

 Sometimes the term is used even more specifically for a (co)limit over the ordinal $\omega$.

Thus, a sequential limit is a special case of a [[directed limit]]. See there for more details.

## Related concepts

* [[tower of fibrations]]

* [[lim^1 and Milnor sequences]]

## References

Discussion of [[sequential colimits]] (in the generality of [[homotopy colimits]]) in [[homotopy type theory]]:

* [[Egbert Rijke]], *Sequential colimits* [[pdf](https://www.andrew.cmu.edu/user/erijke/hott/sequences.pdf)], Lecture 15 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/)]

It could also be found in section 26 of the draft of the textbook:

* {#RijkeDraft22} [[Egbert Rijke]] (2022), *[[Introduction to Homotopy Type Theory]]*, draft. ([pdf](https://raw.githubusercontent.com/martinescardo/HoTTEST-Summer-School/main/HoTT/hott-intro.pdf))

[[!redirects sequential limit]]
[[!redirects sequential limits]]
[[!redirects cosequential limit]]
[[!redirects cosequential limits]]
[[!redirects sequential colimit]]
[[!redirects sequential colimits]]

[[!redirects sequential diagram]]
[[!redirects sequential diagrams]]


sequential multicategory

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##  Idea

The 2-category [[Cat|$Cat$]] of [[categories]] embeds [[fully faithfully]] into the 2-category $Multicat$ of [[multicategories]] in two ways:

1. As the unary multicategories (i.e. in which every multimorphism has unary domain).
2. Via the "discrete cocones" construction, in which we define the multimorphisms of the multicategory $C_\blacktriangleright$ induced by a category $C$ by
\begin{equation}
C_\blacktriangleright(X_1, \ldots, X_n; Y) := C(X_1, Y) \times \cdots \times C(X_n, Y)
\end{equation}

A multicategory isomorphic to one induced by the discrete cocones construction is called a **sequential multicategory** (since its multimorphisms are sequences of unary morphisms).

## Properties

- Every sequential multicategory is [[symmetric multicategory|symmetric]].

- A sequential multicategory $C_\blacktriangleright$ is [[representable multicategory|representable]] if and only if $C$ has coproducts (Example 8.8(2) of [Hermida 2000](#Hermida)).

- A category $C$ is [[preadditive category|preadditive]] if and only if its corresponding sequential multicategory $C_\blacktriangleright$ is [[cartesian multicategory|cartesian]] (see [Pisani 2013](#Pisani13) and [Pisani 2014](#Pisani14)); and is hence [[additive category|additive]] if and only if its corresponding multicategory is cartesian and representable.

- The discrete cocones construction $(-)_\blacktriangleright : Cat \to Multicat$ is left adjoint to the [[category of monoids]] construction $Mon : Multicat \to Cat$, and this restricts to [[cartesian multicategories]]. In fact, this is a consequence of the more general fact that $Multicat$ is [[copowered]] over $Cat$: see [Pisani 2013](#Pisani13) and [Pisani 2014](#Pisani14).

## References

The discrete cocones construction was introduced in Example 2.2(2) of:

* {#Hermida} [[Claudio Hermida]], _Representable multicategories_, Adv. Math. 151 (2000), no. 2, 164-225 ([pdf](http://wslc.math.ist.utl.pt/s84.www/cs/claudio/articles/rep-mult.pdf))

and studied extensively in:

* {#Pisani13} [[Claudio Pisani]], _Some remarks on multicategories and additive categories,_ [arXiv:1304.3033](https://arxiv.org/abs/1304.3033) (2013).

* {#Pisani14} [[Claudio Pisani]], _Sequential multicategories_, Theory and Applications of Categories 29.19 (2014), [arXiv:1402.0253](https://arxiv.org/abs/1402.0253)

The discrete cocones construction is also discussed in Example 2.1.6 of [[Higher Operads, Higher Categories]].

[[!redirects sequential multicategories]]


sequential net

\tableofcontents

## Definition

### In set theory

A __sequential net__ is a [[multi-valued function]] from $\mathbb{N}$ (or a [[decidable subset|decidable]] [[subset]] thereof) to $X$, that is a [[span]] $\mathbb{N} \leftarrow A \rightarrow X$ where the map $A \to \mathbb{N}$ is a [[surjection]] (or has a decidable range).

### In type theory

A **sequential net** in a type $B$ can be defined as

* a [[span]] with domain of the [[natural numbers]], whose function into the natural numbers is a surjection, which consists of
  * a type $C$
  * a family of elements $z:C \vdash g(z):\mathbb{N}$
  * a family of elements $z:C \vdash h(z):B$
  * a family of witnesses $x:\mathbb{N} \vdash \mathrm{isInhab}(x):\left[\sum_{z:B} g(z) =_\mathbb{N} x\right]$ that each dependent type $\sum_{z:B} g(z) =_\mathbb{N} x$ is [[inhabited]] for all natural numbers $x:\mathbb{N}$
* a [[multivalued function]] with domain of the [[natural numbers]], which consists of
  * a type family $x:\mathbb{N} \vdash P(x)$
  * a family of elements $x:\mathbb{N}, p:P(x) \vdash f(x, p):B$
  * a family of witnesses $x:\mathbb{N} \vdash \mathrm{isInhab}(x):\left[P(x)\right]$ that each dependent type $P(x)$ is [[inhabited]] for all natural numbers $x:\mathbb{N}$
* a total [[correspondence]] with domain of the [[natural numbers]], consisting of
  * a type family $x:\mathbb{N}, y:B \vdash R(x, y)$
  * a family of witnesses $x:\mathbb{N} \vdash \mathrm{isInhab}(x):\left[\sum_{y:B} R(x, y)\right]$ that each dependent type $\sum_{y:B} R(x, y)$ is [[inhabited]] for all natural numbers $x:\mathbb{N}$

## Properties

Given a sequential net $A$, $A$ inherits the structure of a [[directed set]] via $A \to \mathbb{N}$, so that $A \to X$ is a [[net]]. In the context of [[weak countable choice]], as a net, every sequential net is equivalent (in the sense of corresponding to the same [[filter]]) to some sequence.

Without [[weak countable choice]], many of the usual properties of [[metric spaces]] and other [[sequential spaces]] fail, but they continue to hold using sequential nets in the place of sequences. For example, every (located Dedekind) [[real number]] may be represented as a sequential Cauchy net, even when they might not all be represented as Cauchy sequences; see [[Cauchy real number]].

## Dependent sequential nets

A dependent sequential net is like a sequential net, but where we generalize from a single type $B$ to a family of types $x:\mathbb{N} \vdash B(x)$ indexed by the natural numbers.

A **dependent sequential net** in a type family $x:\mathbb{N} \vdash B(x)$ can be defined as

* a dependent multivalued function with domain of the [[natural numbers]], which consists of
  * a type family $x:\mathbb{N} \vdash P(x)$
  * a family of elements $x:\mathbb{N}, p:P(x) \vdash f(x, p):B(x)$
  * a family of witnesses $x:\mathbb{N} \vdash \mathrm{isInhab}(x):\left[P(x)\right]$ that each dependent type $P(x)$ is [[inhabited]] for all natural numbers $x:\mathbb{N}$
* a total [[dependent correspondence]] with domain of the [[natural numbers]], consisting of
  * a type family $x:\mathbb{N}, y:B(x) \vdash R(x, y)$
  * a family of witnesses $x:\mathbb{N} \vdash \mathrm{isInhab}(x):\left[\sum_{y:B(x)} R(x, y)\right]$ that each dependent type $\sum_{y:B(x)} R(x, y)$ is [[inhabited]] for all natural numbers $x:\mathbb{N}$

## See also

* [[sequence]], [[dependent sequence]]

* [[net]]

* [[weak countable choice]], [[countable choice]], [[dependent choice]]

[[!redirects sequential net]]
[[!redirects sequential nets]]

[[!redirects dependent sequential net]]
[[!redirects dependent sequential nets]]


sequential preconvergence space

[[!redirects sequential preconvergence spaces]]

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## Idea ##

By a *sequential preconvergence space* we shall mean the most general kind of [[topological space]] where the notion of [[convergence]] and [[limit of a sequence|limit]] for [[sequences]] makes sense. The possibly more evident term "[[sequential convergence space]]" for this notion is already taken to mean a synonym of *[[subsequential space]]*, but that definition is not the most general definition for which the notions of convergence and limit makes sense for sequences.

## Definition ##

A set $S$ is a **sequential preconvergence space** if it comes with a [[binary relation]] $x \to c$ between the set of all [[sequences]] in $S$ and $S$ itself, for $x \in S^\mathbb{N}$ and $c \in S$. A sequential preconvergence space is **[[sequentially Hausdorff]]** if the binary relation is a [[functional relation]], and every sequentially Hausdorff sequential preconvergence space has a [[partial function]]
$$\lim_{n \to \infty} (-)_n:(\mathbb{N} \to S) \to S$$

## See also ##

* [[limit of a sequence]]
* [[sequentially Hausdorff space]]
* [[subsequential space]]
* [[preconvergence space]]
* [[metric space]]
* [[uniform space]]
* [[Cauchy space]]
* [[Booij premetric space]]