predicate logic

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# Contents
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## Idea

__Predicate logic__, also called __first-order logic__ and sometimes abbreviated __FOL__, is the usual sort of [[logic]] used in the [[foundations of mathematics]].

In contrast to [[0th-order logic]], first-order logic admits [[variables]] in [[predicates]] [[bound variable|bound]] by _[[quantification|quantifiers]]_ ("[[universal quantifier|for all]]" $\forall$ and "[[existential quantifier|there exists]] $\exists$).

(This means that the [[categorical semantics]] of 1st order logic is given by _[[hyperdoctrines]]_.)

However, in contrast to [[higher-order logic]], first-order logic does not allow variables that stand for predicates themselves.  (This distinction can become somewhat confusing when the first-order theory in question is a [[material set theory]], such as [[ZFC]], in which the variables stand for "sets" which behave very much like predicates.)

A __predicate calculus__ is simply a system for describing and working with predicate logic.  The precise form of such a calculus (and hence of the logic itself) depends on whether one is using [[classical logic]], [[intuitionistic logic]], [[linear logic]], etc; see those articles for details.

[[type theory|Typed]] predicate logic may be called [[type theory#Logic over type theory|first-order logic over type theory]]. Typed predicate [[intuitionistic logic]] is often identified with the [[internal logic]] of a [[Heyting category]], while typed predicate [[classical logic]] is often identified with the [[internal logic]] of a [[Boolean category]]. Any untyped predicate logic could be thought of as a [[simple type theory|simply typed]] predicate logic with only one type $V$, known as the __[[domain of discourse]]__ or the __universe of discourse__.

For many (perhaps most?) authors, predicate logic is really __predicate logic with [[equality]]__. That means that one predicate symbol with two arguments $=(x,y)$ (also  written $x=y$) is distinguished, and has to satisfy reflexivity, symmetry, transitivity and the rule of substitution: $(x = y \wedge \phi(x))\implies \phi(y)$ for any predicate $\phi$ with one argument. However, some forms of predicate logic do not include an equality primitive, such as [[FOLDS]] (in whose name 'FOL' stands for 'first-order logic').  In some first-order theories, such as [[ZFC]], equality can be defined and so is not needed in the logic itself.

In constructive mathematics, one could also have a __predicate logic with [[apartness]]__, which has one predicate symbol with two arguments $\#(x,y)$ (also  written $x\#y$) is distinguished, and has to satisfy irreflexivity, symmetry, and comparison. By definition of an [[apartness relation]], its negation is an [[equivalence relation]], which we could take to be [[equality]] $x = y \coloneqq \neg(x \# y)$ satisfying the rule of substitution: $(\neg(x \# y) \wedge \phi(x))\implies \phi(y)$ for any predicate $\phi$ with one argument. This makes the domain of discourse into an [[apartness space]].

## Properties

[[Lindström's theorems]] give important abstract characterizations of classical untyped first-order logic.

## Related concepts

* [[logic]]

  * [[propositional logic]] (0th order)

  * **predicate logic** (1st order)

  * [[second-order logic]]

  * [[higher-order logic]]

  * [[predicate logic as a dependent type theory]]

  * [[rule C]]

* [[first-order set theory]]

* [[type theory]]

* [[Lindström's theorem]]

## References

### General

Textbook accounts:

In view of the [[Curry-Howard correspondence]]:

* Morten Heine Sørensen, Pawel Urzyczyn, §9 of: *Lectures on the Curry-Howard isomorphism*, Studies in Logic **149**, Elsevier (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)]

The [[syntax - semantics duality]] in first order logic is discussed in

* [[Steve Awodey]], [[Henrik Forssell]], _First-Order Logical Duality_ .([arXiv:1008.3145](http://arxiv.org/abs/1008.3145))

On the historical development:

* José Ferreirós, _The Road to Modern Logic - An Interpretation_ , The Bulletin of Symbolic Logic **7** no. 4 (2001) pp.441-484. ([draft](http://personal.us.es/josef/BSL0704-001.pdf))

### From type theory
 {#ReferencesFromTypeTheory}

First order logic within [[type theory]]/[[homotopy type theory]] via _[[propositions as types]]_ is discussed in

* [[Univalent Foundations Project]], section 3 of  _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_

[[!redirects Predicate logic]]
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[[!redirects first-order language]]
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[[!redirects FOL]]

[[!redirects Predicate logic with equality]]
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predicative mathematics

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#### Mathematics
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# Predicative mathematics
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## Idea

__Predicative mathematics__ is a way of doing [[mathematics]] without allowing impredicative definitions.

Informally, a definition is _impredicative_ if it refers to a totality which includes the thing being defined.  For example, the definition of a particular real number $x$ as the least upper bound of a given set $A$ is impredicative, because it characterizes $x$ as a particular element of some set (the upper bounds of $A$) which includes $x$.  Possibly $x$ can be defined in some other way and then proved to be the least upper bound of $A$, but to define it thus by fiat is impredicative.

There are (at least) two broad schools of the [[foundations]] of predicative mathematics that don\'t talk much to each other: one school that uses lower-order forms of [[higher-order logic]] (or the [[set theory]] that this justifies) and [[classical logic]], and a [[constructive mathematics|constructive]] school that uses first-order [[type theory]] (or the set theory that this justifies) and [[intuitionistic logic]]. The common ground is that both schools reject [[power sets]]; other axioms may vary. Here we tend to think of what predicative mathematics allows [[Set|the category of sets]] to be.

Constructive predicativists sometimes accept principles (such as [[function sets]] and the [[axiom of replacement]]) that classical predicativists must reject because they imply impredicative results using [[excluded middle]].  Such mathematics may be called _weakly predicative_.

### Power classes vs no power sets at all

Another division within predicative mathematics is whether [[power sets]] can be formed but are [[proper classes]] instead of sets or [[large sets]] instead of [[small sets]], or whether [[power sets]] simply cannot be formed at all.

In the former case, almost everything which could be done in impredicative mathematics could be done in predicative mathematics, only with the requirement that size issues have to be taken into consideration at all times. This is the case with predicative constructive [[dependent type theories]] which have [[type universes]], where given the type universe $U$ one could construct a large [[type of propositions]] by $\Omega \coloneqq \sum_{A:U} \mathrm{isProp}(A)$ with power sets as function sets with codomain $\Omega$, as well as various notions of [[class theory]] such as [[Morse-Kelley class theory]], where power sets exist as [[power classes]]. Sometimes, one would have an entire hierarchy of power sets $\mathcal{P}_n(S)$ for hierarchy level $n$, as is common in [[dependent type theory]] with a cumulative hierarchy of universes, with $\mathcal{P}_{n + 1}(S)$ larger than $\mathcal{P}_n(S)$ for all $n$. This is also the case with [[Bertrand Russell]]'s original predicative ramified hierarchy of types in [[Principia Mathematica]].

When power sets don't exist at all, whether as a set or a proper class, this results in strictly weaker foundations, since in this case one simply cannot form various mathematical structures which require the use of [[power sets]], such as the [[Dedekind real numbers]], [[topological spaces]], [[frames]], and [[locales]]. This is usually the case for predicative mathematics done internally in a [[Heyting pretopos|Heyting]] or [[Boolean pretopos]], as well as for predicative [[material set theories]] like [[Kripke–Platek set theory]] which do not have an internal notion of [[class]]. In [[dependent type theory]], this notion of predicativity requires not having any [[type universes]] in the type theory itself, since otherwise $\sum_{A:U} \mathrm{isProp}(A)$ is a large [[type of propositions]]. In addition, unlike for [[set theory]], not having power sets in [[dependent type theory]] results in additional structure like [[formal topologies]] or [[inductive covers]] not being definable in the type theory, since without universes or types of propositions one cannot define relations between elements and subtypes.

## Impredicative axioms

Not all of these axioms are rejected by all predicativists, but they at least come under some suspicion.

### Infinity

The [[axiom of infinity]] is not usually considered impredicative, but we list it anyway, as it is needed for the others to have force.  Mathematics that does not require this axiom, [[finite mathematics]], can be interpreted in a predicative framework even if it uses many of the axioms below.

### Power set

The axiom that any set has a [[power set]] is perhaps the fundamental feature missing from predicative mathematics.  In particular, the sequence
$$ \mathbb{N}, \mathcal{P}\mathbb{N}, \mathcal{P}\mathcal{P}\mathbb{N}, \ldots $$
may be accepted in part, but not forever.

The failure of the power set axiom means that the category of sets is not an [[elementary topos]].

### Unbounded separation

The constructive school generally accepts the [[axiom of replacement]] but not unbounded forms of the [[axiom of separation]].  (This choice is not available to the classical school, since replacement and [[excluded middle]] together imply full separation.)

Naïvely, the axiom of separation says that, if $A$ is a set and $P$ is a [[function]] from $A$ to the set of [[truth value]]s, then there is a set
$$ \{ A | P \} = \{ x \in A \;|\; P(x) \} .$$
To be precise, however, this $P$ should be written as a [[predicate]] in the language of set theory.  The form of separation justified by type theory and such structural set theories as [[ETCS]] requires the [[quantifiers]] in this predicate to be guarded by sets; unbounded separation is the generalisation of this to arbitrary quantifiers.

Arguably, the impredicative core of both separation and power sets (in the presence of bounded separation) is [[limited separation]]: separation in which the quantifiers in $P$ may be guarded by power classes.

*We need more on this, particularly with regards to the classical school and replacement.*

### Function sets

One sometimes speaks of forbidding [[function set]]s instead of power sets.  That is, it is the sequence
$$ \mathbb{N}, \mathbb{N}^{\mathbb{N}}, \mathbb{N}^{\mathbb{N}^{\mathbb{N}}}, \ldots $$
that is avoided.

Of course, function sets can be constructed out of power sets (using *bounded* separation), so forbidding function sets certainly forbids power sets.  The converse holds if there is a set $\Omega$ of [[truth value]]s.

With [[excluded middle]], the set of truth values is easy to achieve, as $\{0,1\}$; in particular, if you have $\mathbb{N}$, then you certainly have $\Omega$.  So the classical school of predicativism rejects function sets.

The constructive school, however, often accepts function sets (thus being *weakly* predicative).  In this school, the sequence above is fine.  Actually, the slightly stronger axiom of [[subset collection]] is adopted by [[Peter Aczel]]\'s $CZF$ and justified by [[Per Martin-Löf]]\'s $ITT$.

[[Brouwer]], on the other hand, did not accept the sequence above, although his followers differ on when (if ever) it stops.

### Bijection sets

In the presence of the [[axiom of choice]], [[bijection sets]] are in bijection with [[power sets]], and so are impredicative.

### Univalence axiom

One consequence of the rejection of [[bijection sets]] in predicative mathematics with the axiom of choice is that the univalence axiom of a universe of sets is no longer available, since it postulates a bijection between equality and bijection sets.

### Excluded middle

Classical predicativists of course accept [[excluded middle]]; otherwise they would be [[constructivism|constructivists]].  But from the perspective of weakly predicative constructive mathematics, excluded middle is impredicative, since it implies power sets (given function sets) and unbounded separation (given replacement).

### The axiom of choice

Some classical predicativists accept the [[axiom of choice]]. But from the perspective of weakly predicative constructive mathematics, the axiom of choice is impredicative, since it implies excluded middle, and thus power sets (given function sets) and unbounded separation (given replacement).

### Propositional resizing

In the constructive school, one would sometimes have multiple sets of propositions, but those only represent the set of $U$-small [[propositions]] or [[subsingletons]] $\Omega_U$, given a [[universe]] $U$, rather than the set of all propositions $\Omega$. In general, given any two universes $U$ and $V$, one cannot prove that $\Omega_U$ is equivalent to $\Omega_V$. This is common in type theoretic models of constructive mathematics.

The axiom of [[propositional resizing]] is then statement that given any two universes $U$ and $V$, the sets $\Omega_U$ and $\Omega_V$ of propositions in $U$ and $V$ respectively are in [[bijection]] with each other $\Omega_U \cong \Omega_V$. This axiom implies that there is only one set of propositions up to bijection, which in combination with the existence of [[function sets]] imply [[power sets]], so is impredicative.

### Ill-founded structures

Most foundations of mathematics are predicative in one sense: no set may belong to itself.  This (or rather, a certain strengthening of this) is the [[axiom of foundation]].  An alternative is the axiom of antifoundation, which explicitly allows for and tames such sets as $\bullet$, where $\bullet = \{\bullet\}$.  Indeed, this equation is a perfectly good way to *define* $\bullet$ using antifoundation, yet this is about as impredicative as a definition can get.

Once one accepts the [[axiom of infinity]], there\'s not much objection to accepting more general [[inductive type]]s such as [[inductive-inductive types]] and [[quotient inductive types]]; these are sets that are defined recursively much like a [[natural numbers object]].  Categorially, we may see these as [[initial algebra]]s of certain functors on $Set$.  [[coinductive type|Coinductive types]], which are the [[terminal coalgebra|final coalgebra]]s of these functors, also exist in impredicative theories, but not predicatively.

### Polymorphic definitions

A different sort of impredicativity is to be found in some [[type theory|type theories]], in which, roughly speaking, one is allowed to define "functions" whose "domain" is the class of all types.  For instance, one might be able to form a type called $\forall \alpha:Type, \alpha\to\alpha$, which has the property that an inhabitant of that type is a "function" which assigns to *every* type $\alpha$, an endomorphism of $\alpha$.  This is clearly impredicative, since the type $\forall \alpha:Type, \alpha\to\alpha$ is also a possible value of $\alpha$.

The philosophy behind this sort of impredicative definition is that any inhabitant of such a type must be defined "uniformly" enough that it uses no details about the type $\alpha$, and thus can equally well be applied to any $\alpha$.  For instance, consider the operation which assigns to every $\alpha$ the identity $id\colon \alpha\to\alpha$; this is defined in exactly the same way for every $\alpha$, and hence inhabits the type $\forall \alpha:Type, \alpha\to\alpha$.

This sort of impredicativism can be shown to be incompatible with impredicative set-theoretic axioms such as power sets; see [this paper](https://www.cl.cam.ac.uk/~amp12/papers/nontpt/nontpt.pdf) of Andy Pitts.  Since such type theories generally do have function types, it follows that they cannot be classical.

## The category of sets

So what is the category of sets in predicative mathematics?

At bottom, let us suppose that $Set$ is a [[Heyting category|Heyting]] [[pretopos]]; this is a category whose [[internal logic]] is first-order and contains only constructions that don\'t require any of the above axioms.

Since we\'re not doing finite mathematics, we may also include a [[natural numbers object]].  In fact, we could include more general [[inductive type]]s, since these are no harder to justify philosophically than $\mathbb{N}$, although the proofs that these exist if $\mathbb{N}$ does rely on possibly impredicative axioms.  Then $Set$ is a Heyting $W$-pretopos.

If you accept function sets, then $Set$ is [[locally cartesian closed category|locally cartesian closed]] and thus a $\Pi$-pretopos.  In this case, $\mathb{N}$ is enough to get all $W$-[[W-type|types]], so we have a $\Pi$-$W$-pretopos.  If you accept excluded middle, then $Set$ is a [[Boolean category|Boolean]] pretopos or even a Boolean $W$-pretopos.  But a Boolean $\Pi$-pretopos is necessarily a [[topos]], which would make the theory impredicative.

However, $Set$ is still a [[Grothendieck topos]], defined as a category of sheaves or in terms of Giraud\'s characterisation.  We require the existence of power sets to prove the theorem that such a category is an elementary topos, so predicatively a Grothendieck topos may not be an elementary topos at all.

## The real numbers

An important question in predicative mathematics is the status of the set $\mathbb{R}$ of [[real number]]s.  This set is often constructed as a subset $R_D$ of $\mathcal{P}(\mathbb{N}) \times \mathcal{P}(\mathbb{N})$ or as a [[subquotient]] $R_C$ of $\mathbb{N}^{\mathbb{N}}$, neither of which can be formed in an arbitrary Heyting $W$-pretopos. The latter can be formed in a $\Pi$-$W$-pretopos, but it is not necessarily correct.

The constructive school of predicativism can construct $\mathbb{R}$ in various ways. One method is to use $R_C$ directly, but this will only go so far unless something is done to prove that it is Dedekind-complete. This will follow from [[weak countable choice]] ($WCC$), which is accepted by some constructive schools. Using [[subset collection]], a variation on $R_C$ is possible which can be proved Dedekind-complete *without* $WCC$.

However, not all constructive predicative mathematics accept [[subset collection]] or [[weak countable choice]]. Another method is to use a predicative version of $R_D$, where given a [[sigma-frame|$\sigma$-frame]] $\Sigma$, $\Sigma$-[[Dedekind cuts]] are defined as pairs of functions $L, U$ from the [[rational numbers]] $\mathbb{Q}$ to $\Sigma$, which represent the open subsets of $\mathbb{Q}$, rather than pairs of functions into the class of propositions. $R_D$ is defined as the set of all $\Sigma$-Dedekind cuts, or as the [[sigma-Dedekind complete|$\Sigma$-Dedekind complete]] [[archimedean field]]. This will ensure that the set of real numbers is a set rather than a [[proper class]], at the cost of Dedekind completeness, which always results in a proper class in constructive predicative mathematics. Nevertheless, a significant portion of [[real analysis]] could be developed using this approach, such as [[differential calculus]], [[integral calculus]], and [[differential geometry]].

It is also possible to assert the existence of $\mathbb{R}$ by fiat, much like $\mathbb{N}$ exists by the axiom of infinity.  This is the approach taken by the classical school; they use $\mathcal{P}\mathbb{N}$ instead of $\mathbb{R}$ directly, but these are isomorphic by excluded middle.  This is natural from the perspective of predicative set theory as a weak form of higher-order logic; you assert the existence of $\mathbb{N}$, $\mathcal{P}\mathbb{N}$, and maybe $\mathcal{P}\mathcal{P}\mathbb{N}$, then stop.

One could also assert $\mathbb{R}$ by fiat via the [[universal property]] of the real numbers as the [[terminal object|terminal]] [[Archimedean ordered field]]. From this characterization, the real numbers are a [[terminal coalgebra]] of the [[identity functor|identity endofunctor]] $X \mapsto X$ on the [[category]] of [[Archimedean ordered fields]], and in a sense is impredicative, since coinductively defined sets are impredicative.

If [[quotient sets]] exist and [[sequence algebras]] of Archimedean ordered fields exist, then it is provable that $\mathbb{R}$ is Cauchy complete. From the definition of terminal object, $\mathbb{R}$ is an algebra of the endofunctor $X \mapsto \mathcal{C}(X)$ in the category of Archimedean ordered fields which takes Archimedean ordered fields $X$ to the Archimedean ordered field $\mathcal{C}(X)$ of equivalence classes of Cauchy sequences in $X$. Every algebra of the endofunctor $X \mapsto \mathcal{C}(X)$ in the category of Archimedean ordered fields is a Cauchy complete Archimedean ordered field.

Similarly, if [[power sets]] of Archimedean ordered fields exist, then it is provable that $\mathbb{R}$ is Dedekind complete. From the definition of terminal object, $\mathbb{R}$ is an algebra of the endofunctor $X \mapsto \mathcal{D}(X)$ in the category of Archimedean ordered fields which takes Archimedean ordered fields $X$ to the Archimedean ordered field $\mathcal{C}(X)$ of two-sided Dedekind cuts in $X$. Every algebra of the endofunctor $X \mapsto \mathcal{D}(X)$ in the category of Archimedean ordered fields is a Dedekind complete Archimedean ordered field.

There is also the question of what exactly it means to say that $\mathbb{R}$ exists; is it a set or a [[proper class]]? Without function sets, the distinction between these is not clear-cut; higher-order logic suggests a hierarchy of more and more proper (less and less [[small category|small]]) classes rather than a single unified notion of set and class, which is similar to the concept of a [[universe in a topos]] for impredicative set theory. If you allow $\mathbb{R}$ only as a proper class, then you are doing predicative mathematics so long as you don't have [[power classes]], if you allow $\mathbb{R}$ as a set, then as long as you don't allow all [[power sets]] or [[power classes]], you are still doing predicative mathematics in the same sense that foundations in which only the [[limited principle of omniscience]] holds but not full [[excluded middle]] is still constructive mathematics.

## Formalising mathematics

How much of mathematics can be done predicatively?

A surprisingly large amount of mathematics can be formalised, using various coding tricks, in a theory in which $\mathbb{N}$ is a set but $\mathcal{P}\mathbb{N}$ is a proper class.  This is somewhat easier in [[Nik Weaver]]\'s 'conceptualist' approach, which accepts $\mathcal{P}\mathcal{P}\mathbb{N}$ as a proper class (so that $\mathcal{P}N$ and $\mathbb{R}$ are small); the encoding is not really more complicated than what is usually done in [[material set theory]] for ordered pairs and the like.  Note that these are conservative over [[Peano arithmetic]] ($PA$); that is, anything expressible in $PA$ and provable in these systems is provable in $PA$ (which certainly cannot be said of [[ZFC]] or [[ETCS]], which prove the consistency of $PA$).

Constructive mathematics generally requires great care with anything after the middle of the 19th century other than basic [[discrete mathematics]], but requiring it to be predicative does not usually add much difficulty, as long as function sets are allowed.  This even extends to [[category theory]], which is not usually contemplated in the classical approach.  (However, the internal logic of a $\Pi$-$W$-pretopos is certainly not conservative over $PA$; it already proves consistency of the latter.)

## Related concepts

* [[predicative topos]]

## References

*  [[Solomon Feferman]]; [Relationships between Constructive, Predicative and Classical Systems of Analysis](http://hlombardi.free.fr/FefermanRelationships.pdf) (PDF).

*  [[Nik Weaver]], [papers on conceptualism](https://web.archive.org/web/20100616093000/https://www.math.wustl.edu/~nweaver/conceptualism.html).

*  from the Standford Encyclopedia of Philosophy:
   *  [Predicativity in constructive set theory](http://plato.stanford.edu/entries/set-theory-constructive/#PreConSetThe).
   *  [Predicativism](http://plato.stanford.edu/entries/philosophy-mathematics/index.html#Pre)

Discussion of [[predicative toposes]] is in

* [[Ieke Moerdijk]], [[Erik Palmgren]], _Type theories, toposes and constructive set theory: predicative aspects of [[algebraic set theory]]_ Ann. Pure Appl. Logic, 114(1-3):155–201, 2002

* [[Benno van den Berg]], _Predicative toposes_ ([arXiv:1207.0959](http://arxiv.org/abs/1207.0959))
 {#vdBerg}

Constructive predicative definitions of the real numbers are discussed in

*  [[Paul Taylor]]\'s [page on Dedekind cuts](http://www.paultaylor.eu/ASD/dedras/classical)

* Univalent Foundations Project, [[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]] (2013)

## Discussion

A discussion was had on this page, now [archived at the nForum](https://nforum.ncatlab.org/discussion/15265/predicative-mathematics/?Focus=103587#Comment_103587)

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predicative topos

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#### Topos Theory
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#### Foundations
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# Contents
* table of contents
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## Idea

In traditional mathematics and [[set theory]], the [[category]] [[Set]] of all [[sets]] is the archetypical [[topos]]. However, in [[predicative mathematics]] with its notion of sets, these do not form a topos since they do not satisfy the [[powerset axiom]]. In [[constructive mathematics]], we may be *weakly* predicative and *almost* form a topos; the resulting weaker structure which [[Bishop sets]] (for example) form is a  _predicative topos_ ([van den Berg](#vdBerg)).

## Definition

There are various notions of predicative topos in the literature (see references below); here we discuss the notion advocated by [[Benno van den Berg]].

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###### Definition

A **predicative topos** is a [[ΠW-pretopos]] satisfying [[WISC]].

=--

This is ([van den Berg, def. 6.1](#vdBerg)), who however calls what we call WISC the "axiom of multiple choice", and the usual [[axiom of multiple choice]] the "strong axiom of multiple choice".

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###### Remark

In other words, the [[axiom of multiple choice]] appears in different strengths in the literature; see at _[[WISC]]_. Accordingly there is the notion of strong/weak predicative topos if it satisfies the strong/weak version of this axiom; van den Berg's notion of predicative topos is the weak version.

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Note that an impredicative [[elementary topos]] is *not* necessarily a predicative topos in this sense.  A topos is always a $\Pi$-pretopos, and if it has an [[NNO]] then it is a $\Pi W$-pretopos, but it need not satisfy WISC ([Roberts 2015]({#Roberts15})).

A different approach to a notion of predicative topos is to assert, rather than a weak choice principle, the existence of some categorical version of [[higher inductive types]].  But this would also exclude some elementary toposes.

## Properties

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###### Theorem

A [[category]] of [[internal sheaves]] over an [[internal site]] in a predicative topos is again a predicative topos.

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This is ([van den Berg, theorem 7.5](#vdBerg)), using [vdBerg-Moerdijk, theorem 4.21](#vdBergMoerdijk)

## Examples

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###### Theorem

The [[category]] of [[Bishop sets]] in [[Martin-Löf dependent type theory]] is a [strong predicative topos.

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This is ([van den Berg, theorem 6.2](#vdBerg)), based on ([Moerdijk-Palmgren 99, section 7](#MoerdijkPalmgren99)) and ([Moerdijk-Palmgren 00, section 10](#MoerdijkPalmgren00)).

Not every [[topos]] is a predicative topos, or even every $Set$-topos (i.e. a topos with a [[geometric morphism]] to the category $Set$, see ([Roberts 2015](#Roberts15))), but every _[[Grothendieck topos]]_ -- a [[bounded geometric morphism|bounded]] $Set$-topos -- is.

## References

The goal of formulating a notion of _predicative topos_ was stated and pursued in a series of articles containing

* [[Ieke Moerdijk]], [[Erik Palmgren]], _Wellfounded trees in categories_ (1999) ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.44.6700))
 {#MoerdijkPalmgren99}

* [[Ieke Moerdijk]], [[Erik Palmgren]], _Type Theories, Toposes and Constructive Set Theory: Predicative Aspects of AST_ (2000) ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.34.8934))
 {#MoerdijkPalmgren00}

* [[Benno van den Berg]], [[Ieke Moerdijk]], _Aspects of predicative algebraic set theory III: Sheaves. Proceedings of the London Mathematical Society., (2012) ([arXiv:0912.1242](https://arxiv.org/abs/0912.1242))
 {#vdBergMoerdijk}

The definition then appears in

* [[Benno van den Berg]], _Predicative toposes_ ([arXiv:1207.0959](http://arxiv.org/abs/1207.0959))
 {#vdBerg}

Discussion of the [[category of sets]] as a $\Pi$-W-topos when formalized via [[h-sets]] in [[homotopy type theory]] is in

* [[Egbert Rijke]], [[Bas Spitters]], _Sets in homotopy type theory_ ([arXiv:1305.3835](http://arxiv.org/abs/1305.3835))

A non-example constructed via the internal logic of an unbounded $Set$-topos is given in

* {#Roberts15} [[David Roberts]], _The weak choice principle WISC may fail in the category of sets_, [Studia Logica](http://link.springer.com/journal/11225) Volume 103 (2015) Issue 5, pp 1005-1017, doi:[10.1007/s11225-015-9603-6](http://dx.doi.org/10.1007/s11225-015-9603-6) [arXiv:1311.3074](http://arxiv.org/abs/1311.3074)

[[!redirects predicative topos]]
[[!redirects predicative topoi]]
[[!redirects predicative toposes]]

[[!redirects strong predicative topos]]
[[!redirects strong predicative topoi]]
[[!redirects strong predicative toposes]]

[[!redirects predicativity]]


predicativism > history

< [[predicativism]]
[[!redirects predicativism -- history]]


Predrag Cvitanović

* [Wikipedia entry](https://en.wikipedia.org/wiki/Predrag_Cvitanovi%C4%87)

## Selected writings

On [[Lie algebra weight systems]] in [[Yang-Mills theory]] [[Feynman amplitudes]]:

* [[Predrag Cvitanović]], _Group theory for Feynman diagrams in non-Abelian gauge theories_, Phys. Rev. D14 (1976) 1536-1553 ([doi:10.1103/PhysRevD.14.1536](https://doi.org/10.1103/PhysRevD.14.1536), [spire:108133](http://inspirehep.net/record/108133), [[CvitanovicWeights76.pdf:file]])

On [[string diagram]]/[[Penrose notation]] for [[Lie theory]] in [[physics]] ([[gauge theory]], [[Yang-Mills theory]], ...):

* [[Predrag Cvitanović]], _Group Theory: Birdtracks, Lie's, and Exceptional Groups_, Princeton University Press July 2008 ([PUP](https://press.princeton.edu/books/paperback/9780691202983/group-theory), [birdtracks.eu](http://birdtracks.eu/), [pdf](http://www.birdtracks.eu/version9.0/GroupTheory.pdf))

Further discussed also by [Peigné (2022)](string+diagram#Peigné22).

category: people

[[!redirects Predrag Cvitanovic]]


predual

# Preduals
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## Idea

In any context in which one speaks of [[duals]], $x$ is a __predual__ of $a$ if $a$ is the dual of $x$.  If there is more than one sort of dual (left and right, algebraic and topological, etc), then the same goes for preduals.

The term is most commonly encountered when "dual" is meant in the sense of [[dual object in a closed category]]. In particular, it is common parlance in the context of [[Banach spaces]]; see also the discussion below on von Neumann algebras.

## Definitions

The Idea section above gives a fine definition, except for the word 'is' in the phrase '$x$ is the dual of $a$'. One ought to say: given an [[object]] $a$, a __predual__ of $a$ is an object $x$ and an [[isomorphism]] $i\colon a \to x^*$, where $x^*$ is the dual (whatever that means in the appropriate context) of $x$.  (Or if the objects in question form some kind of [[higher category]], then $i$ is an [[equivalence]].)

Similarly, given a [[morphism]] $f\colon a \to b$ and isomorphisms $i\colon x^* \to a$ and $j\colon y^* \to b$, a __predual__ of $f$ (relative to these preduals of $a$ and $b$) is a morphism $u\colon y \to x$ making this [[commutative diagram]] (or in a higher category, with a $2$-equivalence filling it):
$$ \array {
   x^*                      & \overset{i}\to  & a \\
   \mathllap{u^*}\downarrow &                 & \downarrow\mathrlap{f} \\
   y^*                      & \underset{j}\to & b
   } $$
(We may continue to preduals of $2$-morphisms etc, if appropriate.)

One can, of course, apply the term "predual" to any of the contexts of "dual" listed in [[category with duals]] (before the examples of [[star-autonomous category]] and [[closed category]], which were said not to really belong). But this is less commonly encountered in practice, since it reduces to a concept with some other standard name, e.g., a left predual of an object is the same as its right dual.

## Of von Neumann algebras

Preduals are particularly studied for [[von Neumann algebras]].  Indeed, a von Neumann algebra may be defined as a $C^*$-[[C-star-algebra|algebra]] with a predual of its underlying [[Banach space]].  We have the nice theorem that this predual is essentially unique, so we speak of [[the]] predual of a von Neumann algebra.  Similarly, a [[morphism]] of von Neumann algebras is a morphism of $C^*$-algebras with a predual (relative to the algebras\' unique preduals) of its underlying morphism of Banach spaces; this is again unique.

Since the preduals are unique, we may give them symbols; if the dual of $A$ is denoted $A^*$, then we use $A_*$ for the predual of $A$.  (If one uses a different symbol for duals, then preduals follow that.)

[[!redirects predual]]
[[!redirects preduals]]


prefibered category

## Definition

A category $\pi:E\to B$ over $B$ is a __prefibered category__ if for every morphism
$f:b\to b'$ there is a weakly [[cartesian morphism]] $u:e\to e'$ with $\pi(e) = f$.

This differs from a [[fibered category]] by not requiring that a composition of
weakly cartesian morphisms is weakly cartesian, or equivalently that *strongly* cartesian liftings exist.

Just as a fibred category corresponds to a [[pseudofunctor]] $B^{op}\to Cat$, a prefibred category corresponds to a normal [[lax functor]] $B^{op}\to Cat$. For constrast, every functor into $B$ is a normal lax functor $B^{op} \to Prof$, see at [[displayed category]].

Prefibered (prefibred) category (Rus. предрасслоённая категория, Fr. catégorie préfibrée)

## See also

* [[foliated category]]
* [[fibered category]]
* [[displayed category]]

## References

* [[A. Grothendieck]], M. Raynaud et al. _Revêtements étales et groupe fondamental_ ([[SGA1]]), Lecture Notes in Mathematics __224__, Springer 1971 (retyped as [math.AG/0206203](http://arxiv.org/abs/math/0206203); published  version Documents Mathématiques __3__, Société Mathématique de France, Paris 2003)

[[!redirects prefibred category]]
[[!redirects prefibered categories]]
[[!redirects prefibred categories]]
[[!redirects prefibration]]


prefield ring

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## Idea ##

In the same vein that [[commutative rings]] are to [[integral domains]] and [[GCD rings]] are to [[GCD domains]], [[prefield rings]] are to [[fields]].

## Definition ##

A commutative ring $R$ is a **prefield ring** if the [[ring of fractions]] of $R$ is isomorphic to $R$. Equivalently, a commutative ring $R$ is a prefield ring if for all elements $a \in R$, $a$ is a [[cancellative element]] if and only if $a$ is a [[unit]]; the [[monoid of cancellative elements]] $\mathrm{Can}(F)$ is equivalent to the [[group of units]] $R^\times$.

## Examples ##

* The [[rational numbers]] $\mathbb{Q}$ are a prefield ring.

* A classical field $F$ is a prefield ring whose monoid of cancellative elements is the set of elements [[not equal to]] zero.

* A [[Heyting field]] $F$ is a prefield ring whose monoid of cancellative elements is the multiplicative monoid of elements [[tight apartness relation|apart from]] zero.

* The [[trivial ring]] $0$ is the unique prefield ring up to unique isomorphism such that zero is in the monoid of cancellative elements. The trivial ring is also the terminal prefield.

* Given any positive integer $n$, the [[integers modulo n]] $\mathbb{Z}/n\mathbb{Z}$ is a prefield ring whose monoid of cancellative elements consists of all integers $m$ modulo $n$ which are [[coprime]] with $n$.

* Let $F$ be a [[discrete field]] and let $\overline{F}$ be the [[algebraic closure]] of $F$. Given any non-zero [[polynomial]] $p \in \overline{F}[x]$, the [[quotient ring]] $\overline{F}[x]/p \overline{F}[x]$ is a prefield ring whose monoid of cancellative elements consists of polynomials $q \in \overline{F}[x]$ modulo $p$ such that the [[greatest common divisor]] of $p$ and $q$ is an element of the [[group of units]] $\gcd(p, q) \in \overline{F}[x]^\times$.

* Let $R$ be a [[unique factorization domain]] such that for every [[irreducible element]] $x \in R$, the [[ideal]] $x R$ is a [[maximal ideal]]. Given any non-zero element $x \in R$, the [[quotient ring]] $R/x R$ is a prefield ring whose monoid of cancellative elements consists of elements $y \in R$ modulo $x$ such that the [[greatest common divisor]] of $x$ and $y$ is an element of the [[group of units]] $\gcd(x, y) \in R^\times$.

* A prefield ring $F$ is a [[local ring]] if the set of non-cancellative elements, the [[zero divisors]], is the [[Jacobson radical]] $J(F)$.

* Non-example: the [[integers]] $\mathbb{Z}$ are not a prefield ring.

## See also ##

* [[commutative ring]]

* [[GCD ring]]

* [[Bézout ring]]

* [[unique factorization ring]]

* [[Euclidean ring]]

* [[ring of fractions]]

* [[prevector space]]

* [[field]]

[[!redirects prefield]]
[[!redirects prefields]]

[[!redirects prefield ring]]
[[!redirects prefield rings]]


prefix order

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# Prefix Orders
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## Idea ##

By a  _prefix order_ we mean a [[partial order]] which is also downwards [[total order|totally ordered]].

The concept was introduced in ([Cuijpers 13a](#Cuijpers13a), [Cuijpers 13b](#Cuipers13b)) in the context of [[semantics]] for [[programming languages]], and in particular hybrid and cyber-physical systems, as need arose to generalize the notion of (execution) tree to something that would include a mix of continuous and discrete progress of time. After a lot of experimentation, [Cuijpers 13a](#Cuijpers13a) discovered that the generalization needed was something that expresses two things: time branches when it goes forward (due to different possible courses of action a system may take), but seems linear when looking backward.

A different way of looking at it, is that executions of systems have a natural prefix order on them. Trying to come up with a generic definition of what a 'prefix' is, Cuijpers could not find anything in literature and came up with the definition below. As it turns out, in roughly the same period, [Ferlez-Cleaveland-Marcus 14](#FerlezCleavelandMarcus14) discovered the definition of "pointed prefix orders" and named them "generalized execution trees".

## Definition ##

### As a set with extra structure

A prefix order may be understood as a [[set]] with [[extra structure]].

Given a [[set]] $S$, a __prefix order__ on $S$ is a (binary) [[relation]] $\leq$ with the following properties:

* [[reflexive relation|reflexivity]]: $x \leq x$ always;

* [[transitive relation|transitivity]]: if $x \leq y \leq z$, then $x \leq z$;

* [[antisymmetric relation|antisymmetry]]: if $x \leq y \leq x$, then $x = y$.

* [[downward totality]]: if $x \leq z$ and $y \leq z$, then $x \leq y$ or $y \leq x$.

A __prefix ordered set__ is a set equipped with a prefix order.

### As a partial order with downward totality

A prefix ordered set is precisely a [[poset]] satisfying the extra condition that
$x \leq z$ and $y \leq z$ implies that $x \leq y$ or $y \leq x$.

### History preserving functions

The morphisms of prefix ordered sets are [[history preserving function]]s.
Given $x \in S$, the __history__ of $x$ is the set $x^- = \{ y \mid y \leq x \}$.
A __history preserving function__ $f$ from a prefix ordered set $S$ to a prefix ordered set $T$ is a [[function]] from $S$ to $T$ (seen as [[structured sets]]) such that $f(x^-) = f(x)^-$.

Equivalently, it is a [[monotone function]] between $S$ and $T$ (interpreted as [[posets]]) satisfying the additional property that for every $x \in S$ and $y \in T$ such that $y \leq f(x)$ there exists a $x' \in S$ with $x' \leq x$ and $f(x') = y$.

In this way, prefix ordered sets form a [[category]] [[Pfx]]. Furthermore, they are sometimes also viewed as a [[concrete category]] over [[Pos]] (rather than [[Set]]), with history preserving maps preserving the way in which executions of a system unfold, while order preserving maps preserve only the way in which the executions are observed by external observers. (For an example of why this distinction is useful, see also the article on [[embedding open maps]] as a generalization of [[bisimulation]])

## Properties ##

* Every prefix order is isomorphic to its set of histories under subsets, leading to the intuition that the elements of a prefix order 'are' their histories. Formally: given a prefix ordered set $S$, the set $H = \{ x^- \mid x \in S \}$ is prefix ordered and isomorphic to $S$ (with 'taking the history of an element' and 'taking the maximum of a history' being the witnessing isomorphisms).

* Products of prefix orders are not the usual Cartesian products, but rather are formed by all possible 'mergings' of the histories of elements of the combined prefix orders.

* In the work of Cleaveland, a different type of products was defined, but at this point I do not know of a notion of morphism on prefix ordered sets that would result in that product.

## Examples ##

* Be careful. Products of prefix orders are not the usual Cartesian products, rather the histories of a prefix order are 'merged'. When taking the product of a countable and an uncountable history, unexpected phenomena may happen that leave one with an empty set. A 'cure' for this seems to be to restrict to prefix orders in which each history has a minimum.

## Related concepts

* [[partial order]]
* In some books on order theory, a partially ordered set that is downward well-ordered is used as a characterization of a [[tree]]. Prefix orders relax this definition to downward total orders.
* Any partial order can be viewed as an [[Alexandroff topology]]. When viewing prefix orders as an Alexandroff topology, the history preserving maps translate to continuous open maps.

## References
 {#References}

* {#Cuijpers13a} [[Pieter Cuijpers]], _Prefix Orders as a General Model of Dynamics_, Proceedings of the [9th International Workshop on Developments in Computational Models (DCM)](http://www.dcm-workshop.org.uk/2013/) p. 25–29 ([pdf](http://www.dcm-workshop.org.uk/2013/dcm-2013-pre-proc.pdf#page=30))

* {#Cuijpers13b} [[Pieter Cuijpers]], _The Categorical Limit of a Sequence of Dynamical Systems_, [EPTCS 120 : Proceedings EXPRESS/SOS 2013](https://www.win.tue.nl/expresssos2013/), p. 78-92, 2013 ([arXiv:1307.7445](https://arxiv.org/abs/1307.7445), [doi:10.4204/EPTCS.120.7]())

* {#FerlezCleavelandMarcus14} James Ferlez, Rance Cleaveland, Steve Marcus, _Generalized Synchronization Trees_, LLNCS 8412: Proceedings of FOSSACS'14, 2014, p. 304–319 (10.1007/978-3-642-54830-7_20)

[[!redirects prefix ordered]]
[[!redirects downward totality]]
[[!redirects downward total relation]]
[[!redirects history preserving function]]
[[!redirects Pfx]]


preframe

# Preframes
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## Definition

A **preframe** is a [[poset]] with finite [[meets]] and [[directed set|directed]] [[joins]], and in which the finite meets distribute over the directed joins.

## Remarks

A preframe is a [[frame]] if and only if it is also a [[distributive lattice]], i.e. it has finite joins which are distributed over by finite meets.

Preframes are a useful technical tool in the study of [[proper map]]s of [[locales]].

Since preframes involve operations of arbitrarily large arity, it is not automatic that they are [[monadic|monadic functor]] over Set or that they can be presented by [[generators and relations]].  It is, however, true; see

* [[Peter Johnstone]] and [[Steven Vickers]], "Preframe presentations present", [MR](http://www.ams.org/mathscinet-getitem?mr=1173013).

[[!redirects preframe]]
[[!redirects preframes]]


prefunction

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## Idea ##

In certain [[foundations of mathematics]], the "[[functions]]" between [[sets]] or [[types]] or [[presets]] do not preserve the [[equality]] in the [[set]]/[[type]]/[[preset]]. Many times, the functions that do not preserve equality are called "prefunctions", and a function is then defined to be a prefunction that satisfies [[function extensionality]].

## Definition

### In set theory

Suppose that $X$ and $Y$ are [[sets]], a *prefunction* from $X$ to $Y$ is a [[mapping]] that does not preserve [[equality]]: it is [[not]] [[true]] that $a = b$ always [[implication|implies]] $f(a) = f(b)$. A function is defined as a prefunction that preserves equality; such a prefunction is said to be __extensional__.

For example, consider the identity prefunction on the set of pairs of positive numbers, $Z^+ \times Z^+$ and the set of positive fractions $Q^+$. From $Z^+ \times Z^+$ to $Q^+$, this is a function, since $a/b = c/d$ if $(a,b) = (c,d)$. But from $Q^+$ to $Z^+ \times Z^+$, it is not a function, since (for example) $2/4 = 3/6$ but $(2,4) \neq (3,6)$. A related example is the operation of taking the numerator of a (positive) fraction; from $Q^+$ to $Z^+$, we may view this as a prefunction but not as a function, although it is a function on $Z^+ \times Z^+$.

Given sets $X$ and $Y$, the [[function set]] from $X$ to $Y$ is a [[subset]] of this set of prefunctions between $X$ and $Y$. Composition of prefunctions is also possible, but likewise does not preserve equality.

### In type theory

In many [[foundations of mathematics|foundations]] based on [[type theory]], such as in [[Martin-Löf type theory]], all [[types]] come equipped with an [[identity type]] which behaves similarly as [[equality]] does in [[Sets|sets]]. These types, therefore, are not [[presets]] in the strict sense, in that the latter do not carry any equality at all. However, [[function type|functions]] between such types usually do not satisfy [[function extensionality]], so that in the set-theoretic sense they are still like prefunctions.

## Categorical semantics

The difference between functions and prefunctions in sets is modeled in [[category theory]] ([[categorical semantics]]) as the difference between a [[concrete category]] and a [[category]] with a functor $U:C \to Set$ which is not faithful.

## See also ##

* [[preset]]

* [[universe]]

* [[function extensionality]]

* [[type theory]], [[homotopy type theory]]

[[!redirects prefunction]]
[[!redirects prefunctions]]
[[!redirects pre-function]]
[[!redirects pre-functions]]


pregroup grammar

[[!redirects pregroup]]
[[!redirects pregroups]]
[[!redirects pregroup grammars]]
[[!redirects Pregroup grammar]]
[[!redirects Pregroup grammars]]

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## Overview

**Pregroup grammar** is a mathematical model of natural language grammar introduced by [Lambek (1999)](#Lambek99); it is part of the [[categorial grammar]] tradition.

## Pregroups

### Definition ###

A **preordered monoid** is a tuple $(P, \leq, \cdot, 1)$ where $(P, \leq)$ is a [[preorder]] and $(P, \cdot, 1)$ is a [[monoid]] such that the multiplication is monotone, i.e.

* $a \leq c \wedge b \leq d \implies a \cdot b \leq c \cdot d \quad$ (monotony)

or equivalently:

* $b \leq d \implies a \cdot b \cdot c \leq a \cdot d \cdot c \quad$ (substitution)

for all $a, b, c, d \in P$. A **pomonoid** is a preordered monoid that is furthermore a [[poset]], i.e.

* $a \leq b \leq a \implies a = b \quad$ (antisymmetry)

A **pregroup** is a pomonoid such that every object $t \in P$ has [[category with duals|left and right adjoints]] $t^l, t^r \in P$.
Explicitly we have the following four axioms:

* $t^l \cdot t \leq 1, \quad t \cdot t^r \leq 1 \quad$ (contraction)
* $1 \leq t \cdot t^l, \quad 1 \leq t^r \cdot t \quad$ (expansion)

In other words, a pregroup is a [[rigid monoidal category]] which is [[thin category|thin]] and [[skeletal category|skeletal]].
Note that a commutative pregroup is simply an [[ordered group]].
Here are a few variants of pregroups that appear in the literature, see [Coecke 2013](#Coecke13):

* If we drop the anti-symmetry axiom for [[posets]], we get a **quasi-pregroup**. These are useful as they avoid the equality $a \leq a \cdot a^l \cdot a \leq a \implies a = a \cdot a^l \cdot a$ for all $a \in P$.
* If we drop the two expansion axioms, we get a **protogroup**.
* If we drop the axioms for the right adjoints, we get a **left pregroup**. Symmetrically, if we drop the left adjoints we get a **right pregroup**.

### Monotone unbounded functions ###

Take the natural numbers $\mathbb{N}$ and define the set $P$ of [[monotone function|monotone functions]] $f : \mathbb{N} \to \mathbb{N}$ that are furthermore unbounded, i.e. for all $n \in \mathbb{N}$ there is an $m \in \mathbb{N}$ with $n \leq f(m)$.
$P$ is a pomonoid with composition as product, identity as unit and $f \leq g$ iff $f(n) \leq g(n)$ for all $n \in \mathbb{N}$. To show that $P$ is a left pregroup, define:

$$f^l(m) = \text{min} \{n \in \mathbb{N} \vert m \leq f(n)\}$$

we get:

$$(f^l \cdot f)(m) = f^l(f(m)) = \text{min}\{n \in \mathbb{N} \vert f(m) \leq f(n)\} \leq m = 1(m)$$

and

$$1(m) = m \leq f(\text{min} \{ n \in \mathbb{N} \vert m \leq f(n)\}) = f(f^l(m)) = (f \cdot f^l) (m)$$

for all $m \in \mathbb{N}$, i.e. $f^l \cdot f \leq 1 \leq f \cdot f^l$.

If we replace the natural numbers $\mathbb{N}$ by the integers $\mathbb{Z}$ then the set of unbounded monotone functions $f : \mathbb{Z} \to \mathbb{Z}$ is both a left and a right pregroup where:

$$f^r(m) = \text{max} \{n \in \mathbb{N} \vert f(n) \leq m\}$$

For example, if $f(n) = 2 n$ then $f^l(n) = \lceil \frac{n + 1}{2} \rceil \neq \lceil \frac{n}{2} \rceil = f^r(n)$.

**Proposition ([Buszkowski 2001](#Buszkowski01)):** Every pregroup is a pregroup of functions, i.e. a sub-pomonoid of $X \to X$ for some poset $X$.

### Free pregroups and the switching lemma ###

In [[linguistics|linguistic]] applications, one starts with a [[partial order|poset]] $B = \{ s, n, \dots \}$ of **basic types** (e.g. sentence and noun) and then takes the free pregroup $P_B$ generated by $B$.
For a basic type $b \in B$, iterated adjoints $\dots, b^{ll}, b^l, b, b^r, b^{rr}, \dots \in P_B$ are called **simple types**; an arbitrary type $t \in P_B$ may then be given as a sequence of simple types.
The following lemma makes the parsing problem for free pregroups decidable --- i.e. given a sentence type $s \in P_B$ and the types for a sequence of words $t_1, \dots, t_n \in P_B$, is $t_1 \dots  t_n \leq s$?

**Switching lemma ([Lambek 1999](#Lambek99)):** For any pair of types $t, t' \in P_B$, if $t \leq t'$ then there is a type $t'' \in P_B$ such that $t \leq t''$ without expansions and $t'' \leq t'$ without contractions, i.e. free protogroups recognise the same language as free pregroups.

### From pregroups to compact 2-categories ###

In [Preller, Lambek 2007](#PrellerLambek07), Preller and Lambek generalise the free pregroup construction above to free compact 2-categories in order to capture [[proof relevance]].
This allows to distinguish between the distinct parsings of ambiguous phrases such as
"(men and women) who like maths" vs. "men and (women who like maths)".

Note that the compact 2-categories of Lambek and Preller differ from [[compact closed 2-category]] in that they are not required to be symmetric.
A non-symmetric compact closed 2-category with one object is simply a [[rigid monoidal category]].

Given a poset of basic types $B$, the objects of the free rigid monoidal category $C_B$ are the same as that of the free (quasi-)pregroup, the arrows may be described as planar [[string diagram|string diagrams]]. Given two types $t, t' \in P_B$, we have that $t \leq t'$ if and only if there is an arrow $r : t \to t'$ in $C_B$.

## Pregroup grammars as free rigid monoidal categories {#PregroupGrammar}

A **pregroup grammar** is a tuple $G = (B, \Sigma, \Delta, s)$ where $B$ and $\Sigma$ are finite sets called the _basic types_ and the _vocabulary_ respectively, $\Delta \subseteq \Sigma \times P_B$ is a relation called the _dictionary_ and $s \in P_B$ is a designated _sentence type_.
We require that $\Delta$ is finite, i.e. the set of possible types $\Delta(w) \subseteq P_B$ is finite for each word $w \in \Sigma$.
The _language_ of $G$ is then given by:

$$L(G) = \big\{ w_1 \dots w_n \in \Sigma^n \vert n \in \mathbb{N}, \quad \exists t \in \prod_{i \leq n} \Delta(w_i) \quad t \leq s \big\}$$

i.e. a sequence of words $w_1 \dots w_n \in \Sigma^n$ is said to be grammatical whenever there is a dictionary entry $(w_i, t_i) \in \Delta$ for each word $w_i$ such that $t_1 \dots t_n \leq s$.
One may simplify this by redefining $L(G) = \big\{ w_1 \dots w_n \in \Sigma^n \vert C_G(w_1 \dots w_n, s) \neq \emptyset \big\}$ where $C_G$ is the free [[rigid monoidal category]] with:

* generating objects the [[disjoint union]] $B + \Sigma$,
* generating arrows the dictionary entries $(w, t) \in \Delta$ with $dom(w, t) = w$ and $cod(w, t) = t$.

That is, a sequence of words is grammatical whenever there exists a string diagram going from the sequence of words to the sentence type.
Note that traditionally the identity wires for words $w \in \Sigma$ are omitted, hence dictionary entries are depicted as triangles with no input.
Contractions are depicted as cups, e.g. from $(Alice, n), (loves, n^r s n^l), (Bob, n) \in \Delta$ and $n n^r s n^l n \leq s$ we get the following diagram:

\begin{centre}
    \begin{tikzpicture}
		\node (0) at (-6, 0) {};
		\node (1) at (-5, 1) {};
		\node (2) at (-4, 0) {};
		\node (3) at (-5, 0) {};
		\node (4) at (-3, 0) {};
		\node (5) at (-2, 1) {};
		\node (6) at (-1, 0) {};
		\node (7) at (-2.5, 0) {};
		\node (8) at (-1.5, 0) {};
		\node (9) at (0, 0) {};
		\node (10) at (1, 1) {};
		\node (11) at (2, 0) {};
		\node (12) at (1, 0) {};
		\node (13) at (-5, 0.25) {Alice};
		\node (14) at (-2, 0.25) {loves};
		\node (15) at (1, 0.25) {Bob};
		\node (16) at (-5.25, -0.5) {$n$};
		\node (17) at (-3, -0.5) {$n^r$};
		\node (18) at (-1, -0.5) {$n^l$};
		\node (19) at (1.25, -0.5) {$n$};
		\node (20) at (-2, 0) {};
		\node (21) at (-2, -1.5) {};
		\node (22) at (-1.75, -1.25) {$s$};

		\draw [bend right=90, looseness=1.50] (3.center) to (7.center);
		\draw [bend right=90, looseness=1.50] (8.center) to (12.center);
		\draw (0.center) to (1.center);
		\draw (1.center) to (2.center);
		\draw (2.center) to (0.center);
		\draw (4.center) to (5.center);
		\draw (5.center) to (6.center);
		\draw (6.center) to (4.center);
		\draw (9.center) to (10.center);
		\draw (10.center) to (11.center);
		\draw (11.center) to (9.center);
		\draw (20.center) to (21.center);
\end{tikzpicture}
\end{centre}

## Pregroup semantics as strong monoidal functors

One may give a [[semantics]] to a pregroup grammar $G = (B, \Sigma, \Delta, s)$ by defining a strong [[monoidal functor]] $F : C_G \to S$, where $C_G$ is the free [[rigid monoidal category]] described in [section 2](#PregroupGrammar). $S$ is a suitable [[rigid monoidal category]], e.g. $\text{FdVect}$ or $\text{Rel}$, depending on the application.
Note the similarity with a [[Lawvere theory]] as a monoidal functor from a syntactic category to $\text{Set}$.

We require the image for all words $w \in \Sigma$ to be the monoidal unit $F(w) = I$, hence the image for each dictionary entry $(w, t) \in \Delta$ is given by a state $F(w, t) : I \to F(t)$.
The meaning $F(r) : I \to F(s)$ for a sentence $w_1 \dots w_n \in \Sigma^n$ with grammatical reduction $r : w_1 \dots w_n \to s$ may then be computed from the individual meanings $F(w_i, t_i) : I \to F(t_i)$ of the words, following [[Gottlob Frege|Frege's]] principle of [[compositionality]].
This has been developed in [Preller 2005](#Preller05) as well as in a series of papers by [[Bob Coecke]] and others, see [[categorical compositional distributional semantics]].

## Expressive power and extensions

**Theorem ([Buszkowski, Moroz 2008](#BuszkowskiMoroz08)):** For each pregroup grammar $G$, there is a [[context-free grammar]] $G'$ such that $L(G) = L(G')$. Furthermore, $G'$ may be computed in time polynomial in the size of $G$.

The opposite direction also holds, hence pregroup grammar and [[context-free grammar]] are said to be **weakly equivalent**: the translation preserves only the generated languages, it does not preserve the structure of syntax trees.

Since it is generally accepted that natural languages go beyond context-free languages this result illustrates the lack of expressive power of pregroup grammars whence it becomes necessary to enhance the original grammar model:

One way ([Kobele 2005](#Kobele05)) is to consider **product pregroup grammars** $G_1\times G_2$: these assign to a string $w$ a tuple $(t_1,t_2)$ of syntactic types with grammaticality of $(w,(t_1,t_2))$ checked by parallel computation of the grammaticality of $(w,t_1)$ in $G_1$ and of $(w,t_2)$ in $G_2$. $(w,(t_1,t_2))$ is then grammatical iff $(w,t_1)$ and $(w,t_2)$ are i.e. $L(G_1\times G_2)= L(G_1)\cap L(G_2)$ (cf. [Kobele&Kracht (2005)](#KobeleKracht05), prop. 4). This  permits to describe e.g. the context-sensitive rules of Swiss German ([Lambek (2008)](#Lambek08)). The checking of syntactic types is still given by computation in a pregroup, namely the product pregroup $P_1\times P_2$, but $P_1\times P_2$ is not free!

Kobele and Kracht ([2005](KobeleKracht05)) use this together with the above result by Buszkowski and the classical result in formal language theory that any type 0 language $L$ is the homomorphic image $h(L_1\cap L_2)$ of the intersection of two (deterministic) context-free languages $L_1,L_2$ in order to show that _pregroup grammars generate every type 0 language_ when type checking is done in _arbitrary_ pregroups.

Since it is usually assumed that natural languages do not go beyond [[mildly context-sensitive grammar|mildly context-sensitive languages]] one sees that (homomorphic images) of products of arbitrary pregroup grammars have excessive power and it becomes necessary to cut back this power. One way to do this is by restricting $G_1$ to be of a type simpler than context-free e.g. a [[Dyck language]] thought of now as a control language supervising the "context-free" computations in $G_2$. This idea was explored in Kobele ([2005](Kobele05)).

Another approach to adapt the pregroup grammar formalism to mildly context-sensitive languages was proposed by [Genkin et al. (2010)](#GenkinEtAl) which augment pregroup grammars with buffers.

## References

* {#Lambek99} [[Joachim Lambek]], _Type Grammar Revisited_, Logical Aspects of Computational Linguistics (1999)

* {#Preller05} [[Anne Preller]], _Category Theoretical Semantics for Pregroup Grammars_, Logical Aspects of Computational Linguistics, 5th International Conference, LACL 2005 ([pdf](https://www.researchgate.net/publication/220718685_Category_Theoretical_Semantics_for_Pregroup_Grammars))

* {#PrellerLambek07} [[Anne Preller]], [[Joachim Lambek]], _Free Compact 2-Categories_, Mathematical Structures in Computer Science, Cambridge University Press (CUP), 2007 ([pdf](https://hal-lirmm.ccsd.cnrs.fr/lirmm-00137681v2/document))

* {#Lambek08} [[Joachim Lambek]], _From Word to Sentence: A Computational Algebraic Approach to Grammar_, Polimetrica 2008 ([pdf](http://www.math.mcgill.ca/barr/lambek/pdffiles/2008lambek.pdf))

* {#BuszkowskiMoroz08} [[Wojciech Buszkowski]], [[Katarzyna Moroz]], _Pregroup Grammars and Context-free Grammars_, Computational Algebraic Approaches to Natural Language, Polimetrica (2008) ([pdf](https://pdfs.semanticscholar.org/1924/30f2252b6e0a7f982a3ae69a3ccf9c2981c0.pdf))

* {#Buszkowski01} [[Wojciech Buszkowski]]. _Lambek grammars based on pregroups._ Logical Aspects of Computational Linguistics, LNAI 2099: 95–109, 2001.

* {#Coecke13} [[Bob Coecke]], _An alternative Gospel of structure: order, composition, processes_, Introductory chapter to C. Heunen, M. Sadrzadeh, and E. Grefenstette. Quantum Physics and Linguistics: A Compositional, Diagrammatic Discourse. Oxford University Press, 2013 ([arxiv:1307.4038](https://arxiv.org/abs/1307.4038))

* {#Kobele05} [[Gregory M. Kobele]], _Pregroups, Products, and Generative Power_ , handout Chieti workshop on pregroups May 2005.

* {#KobeleKracht05} [[Gregory M. Kobele]], [[Marcus Kracht]], _On Pregroups, Freedom, and (Virtual) Conceptual Necessity_ , U. Penn Working Papers in Linguistics **10** no.1 (2005).

* {#GenkinEtAl10} [[Daniel Genkin]], [[Nissim Francez]], and [[Michael Kaminski]]. _Mildly Context-Sensitive Languages via Buffer Augmented Pregroup Grammars_, In Essays in Memory of Amir Pnueli, 2010.


pregroupoid

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### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
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# Contents
* table of contents
{:toc}

## Definition

In [[intensional type theory]] such as [[homotopy type theory]] a **pregroupoid** is a [[precategory]] $C$ with a family of functions $(-)^{{-1}_{a, b}}:Mor_C(a, b) \to Mor_C(b, a)$ for [[objects]] $a:Ob(C)$ and $b:Ob(C)$ and [[identifications]] $\lambda(a, b,
 f):f^{{-1}_{a, b}} \circ_{a, b, a} f = id_a$ and $\rho(a, b, f):f \circ_{b, a, b} f^{{-1}_{a, b}} = id_b$ for [[objects]] $a:Ob(C)$ and $b:Ob(C)$ and [[morphisms]]  $f:Mor_C(a, b)$.

Equivalently, it is a [[dagger precategory]] with identifications $\lambda(a, b,
 f):f^{\dagger_{a, b}} \circ_{a, b, a} f = id_a$ and $\rho(a, b, f):f \circ_{b, a, b} f^{\dagger_{a, b}} = id_b$ for [[objects]] $a:Ob(C)$ and $b:Ob(C)$ and [[morphisms]] $f:Mor_C(a, b)$.

## Examples

* A pregroupoid whose sets of morphisms are all subsingletons is a [[symmetric proset]].

* A pregroupoid whose object type is a set is a [[strict groupoid]].

* A pregroupoid whose canonical functions
$$idtotruncmor(a, b):a =_C b \to \vert Mor_C(a, b) \vert$$
are equivalences of types for all objects $a:Ob(C)$ and $b:Ob(C)$ is a [[skeleton|skeletal groupoid]].

* A pregroupoid whose canonical functions
$$idtomor(a, b):a =_C b \to Mor_C(a, b)$$
are equivalences of types for all objects $a:Ob(C)$ and $b:Ob(C)$ is a [[univalent groupoid]]. Every 1-truncated type is equivalent to a univalent groupoid.

* Every [[gaunt category|gaunt pregroupoid]] is a [[set]].

## See also

* [[precategory]]

* [[dagger precategory]]

* [[univalent groupoid]]

## References

* {#UFP13} Univalent Foundations Project, [[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]] (2013)

[[!redirects pregroupoids]]


preimage

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### Context
#### Type theory
+-- {: .hide}
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=--
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#Contents#
* table of contents
{:toc}

## Definition

Given a [[function]] $f: X \to Y$ and a [[subset]] $S$ of $Y$, the __preimage__ (sometimes also called the __[[inverse image]]__, though that may mean something different) of $S$ under $f$ is a [[subset]] of $X$, consisting of those arguments whose values belong to $S$.

That is,
$$ f^*(S) = \{ a: X \;|\; f(a) \in S \} .$$
A traditional notation for $f^*$ is $f^{-1}$, but this can conflict with the notation for an [[inverse function]] of $f$ (which indeed might not even exist). In fact $f^\ast$ is borrowed from a notation for [[pullbacks]]; indeed, a preimage is an example of a pullback:

$$\array{
f^\ast(S) & \hookrightarrow & X \\
\downarrow & (pb) & \downarrow \mathrlap{f} \\
S & \hookrightarrow & Y
}$$

Notice also that $f^\ast$ may be regarded as an operator $P(Y) \to P(X)$ between [[power sets]]. Power sets $P(X)$ are [[exponential objects]] $2^X$ in the [[topos]] $Set$; under this identification the pre-image operator $f^\ast$ is thereby identified with the map $2^f: 2^Y \to 2^X$ (variously called "pulling back along $f$ or substituting along $f$) obtained by [[currying]] the composite map

$$2^Y \times X \stackrel{1 \times f}{\to} 2^Y \times Y \stackrel{eval}{\to} 2.$$

The appearance of the asterisk as a *superscript* in $f^\ast$ serves as a reminder of the [[contravariant functor|contravariance]] of the map $f \mapsto f^\ast = 2^f$. Similarly, one uses a subscript notation such as $f_*$ (or sometimes $f_!$) for the direct [[image]], considered as an operator $f_\ast: 2^X \to 2^Y$ in the *covariant* direction.

Naturally all of this generalizes to the context of [[toposes]], where the set $2$ is replaced by the [[subobject classifier]] $\Omega$ and $f^\ast = \Omega^f$, with a pullback description similar to the above.

## Properties

* [[interactions of images and pre-images with unions and intersections]]:

  * [[pre-images preserve unions and intersections]] (a general reason for this being that unions are colimits, intersections are limits, and $f^\ast$ is simultaneously a left- and a right-adjoint: $f^\ast$ is right-adjoint to the [[existential quantifier]] $\exists_f$ and left-adjoint to the [[universal quantifier]] $\forall_f$)

As emphasized by [Lawvere](#Lawvere73), the quantifiers $\exists_f, \forall_f$ are vastly generalized by the concept of [[enriched category theory|enriched]] [[Kan extensions]] which provide left and right adjoints to pulling-back operators $V^f: V^D \to V^C$ for $V$-enriched functors $f: C \to D$.

## Iterated preimages

For [[partial function|partial]] [[endofunctions]], one could also consider iterating the construction of the preimage of the partial endofunction.

For every [[set]] $T$ and [[subset]] $S \subseteq T$, let $f:S \to T$ be a function from the subset $S$ to $T$. Given any subset $R \subseteq T$, the preimage $f^{-1}(R)$ by definition is a subset of $S$ and thus a subset of $T$. One could restrict the domain of $f$ to $f^{-1}(R)$ and the codomain of $f$ to $R$, and find the preimage of $f^{-1}(R)$ under $f$, or the 2-fold iterated preimage of $R$ under $f$:

$$f^{-2}(R) \coloneqq \{x \in S \vert \exists b \in f^{-1}(R).f(x) = b\}$$

One could repeat this definition indefinitely, which could be formalised by the indexed sets $f^{-n}(R)$ representing the __$n$-fold iterated preimage of $R$ under $f$__.

$$f^{-0}(R) \coloneqq R$$

and for $n \in \mathbb{N}$,

$$f^{-(n+1)}(S) \coloneqq \{x \in S \vert \exists b \in f^{-(n)}(R).f(x) = b\}$$

One example of an iterated preimage is the set of [[iterated differentiable functions]] and the iterated [[continuously differentiable functions]] $C^n(\mathbb{R})$, which are the $n$-th iterated preimage of all functions and [[pointwise continuous functions]] on the [[real numbers]] under the [[derivative]]/[[Newton-Leibniz operator]] respectively.

## Infinitely iterated preimages

The above definition of an iterated preimage is [[inductive]]; one could also consider the [[coinductive]] version of above. This leads to infinitely iterated preimages:

For every set $T$ and subset $S \subseteq T$, let $f:S \to T$ be a function from the subset $S$ to $T$. Given any subset $R \subseteq T$, the infinitely iterated preimage is defined as the largest subset $f^{-\infty}(R) \subseteq T$ such that the preimage of $f^{-\infty}(R)$ under $f$ is $f^{-\infty}(R)$ itself.

One example of an infinitely iterated preimage is the set of [[smooth functions]]  $C^\infty(\mathbb{R})$, which is the infinitely iterated preimage of [[pointwise continuous functions]] under the [[derivative]]/[[Newton-Leibniz operator]].

## Related concepts

* [[level set]], [[zero locus]]

* [[image]], [[coimage]]

For a generalisation to [[sheaf|sheaves]], see [[inverse image]].

## References

*  {#Lawvere73}  [[F. William Lawvere]], _Metric spaces, generalized logic and closed categories_, [[TAC]] Reprint, 1986.  [Web](http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html).

[[!redirects preimage]]
[[!redirects preimages]]
[[!redirects pre-image]]
[[!redirects pre-images]]

[[!redirects iterated preimage]]
[[!redirects iterated preimages]]

[[!redirects infinitely iterated preimage]]
[[!redirects infinitely iterated preimages]]


prelattice

\tableofcontents

## Idea

Prelattices are lattices which do not satisfy [[antisymmetry]]. Or equivalently, they are the bicartesian monoidal preorders, [[thin]] [[cartesian monoidal categories]] which are also [[cocartesian monoidal categories]].

One example of prelattices include [[Heyting prealgebras]]. Two other examples are the [[integers]] $\mathbb{Z}$ and the [[polynomial ring]] $K[x]$ of a [[discrete field]] $K$, with respect to the divisibility [[preorder]] $a \vert b$, the [[greatest common divisor]] $\gcd$, and the [[least common multiple]] $\lcm$; unlike the natural numbers, these only form a prelattice because there are more than one element in the [[group of units]] of both $\mathbb{Z}$ and $K[x]$, where $\mathbb{Z}^\times \coloneqq \{1, -1\}$ and $K[x]^\times \coloneqq K^\times$.

In the same way as lattices, one could either assume that prelattices have top and bottom elements, in which those without top and bottom elements are **pseudoprelattices** or **unboounded prelattices**, or prelattices do not have top and bottom elements, in which those with top and bottom elements are **bounded prelattices**. A pseudoprelattices is equivalently a thin [[locally cartesian category]] whose [[opposite category]] is also locally cartesian. Pseudoprelattices are important because given any ordered field $K$ with [[pseudolattice]] structure, every [[ordered Artinian local ring|ordered Artinian local $K$-algebra]], found in some approaches to [[analysis]], is a pseudoprelattice.

## See also

* [[preorder]]

* [[total preorder]]

* [[Heyting prealgebra]]

[[!redirects prelattice]]
[[!redirects prelattices]]

[[!redirects bounded prelattice]]
[[!redirects bounded prelattices]]

[[!redirects unbounded prelattice]]
[[!redirects unbounded prelattices]]

[[!redirects pseudoprelattice]]
[[!redirects pseudoprelattices]]

[[!redirects bicartesian monoidal preorder]]
[[!redirects bicartesian monoidal preorders]]


premetric locale

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#### Topology
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# Contents
* table of contents
{: toc}

## Definition

A **premetric locale** is a [[locally positive locale]] that is also a [[symmetric proset]] [[enriched set|enriched in]] $(\overleftarrow{\mathbb{R}_+^\infty}, \geq, +, 0)$, the [[locale]] of [[non-negative number|non-negative]] [[extended real numbers]] whose [[topology]] is upper [[semi-continuity]].

## See also

* [[locale]]

* [[locally positive locale]]

* [[premetric space]]

* [[metric locale]]

## References

* [[Simon Henry]], Localic Metric spaces and the localic Gelfand duality ([arXiv:1411.0898v1](https://arxiv.org/abs/1411.0898v1))

[[!redirects premetric locales]]


premetric space

There are multiple notions of **premetrics** and **premetric spaces** in the mathematical literature:

* [[Booij premetric space]]
* [[Richman premetric space]]
* [[premetric locale]]

## References

* Auke B. Booij, Analysis in univalent type theory ([pdf](https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf))

* [[Fred Richman]], Real numbers and other completions ([pdf](http://math.fau.edu/richman/docs/RealNums.pdf))

* [[Simon Henry]], Localic Metric spaces and the localic Gelfand duality ([arXiv:1411.0898v1](https://arxiv.org/abs/1411.0898v1))

[[!redirects premetric]]
[[!redirects premetrics]]
[[!redirects premetric space]]
[[!redirects premetric spaces]]


premodel category

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#Contents#
* table of contents
{:toc}

## Idea

The notion of a _premodel category_ is a relaxation of the notion of a [[model category]].
[[combinatorial model category|Combinatorial]] premodel categories form a [[2-category]]
that has all ([[small diagram|small]]) [[limits]] and [[colimits]] and has [[representing objects]] for [[Quillen bifunctors]].

The [[2-category]] of combinatorial $V$-[[enriched category|enriched]] premodel categories is the category of [[modules over a monoid]] $V$ in this [[2-category]].
It inherits the same set of properties
and additionally admits a [[model 2-category]] structure.
In this model structure, a [[left Quillen functor]] is a [[weak equivalence]]
if and only if it is a [[Quillen equivalence]].

## Definition

A __premodel category__ is a [[bicomplete category]]
equipped with a [[pair]] of [[weak factorization systems]]
$(C,AF)$ and $(AC,F)$ such that $AC\subset C$ (equivalently, $AF\subset F$).

[[model category|Model categories]] can be singled out among premodel categories by imposing the additional requirement that the [[class]]

$$
  W
  \coloneqq
  AF \circ AC
  \,,
$$

obtained by [[composition|composing]] [[elements]] of $AC$ with those of $AF$, is closed under the [[2-out-of-3 property]].

The members of AC are called __anodyne cofibrations__ and the members of AF are called __anodyne fibrations__ (as in _[[anodyne morphism]]_).

## Anodyne and trivial (co)fibrations

The notion of premodel category doesn't come with a good general notion of weak equivalence. But if a particular premodel category has a good notion of weak equivalence, such as one of [Barton](#Barton)'s _relaxed premodel categories_, one needs to distinguish between two types of cofibrations (and analogously between two types of fibrations):

* An __anodyne cofibration__ is a member of the class AC
* A __trivial cofibration__ is a cofibration that is also a weak equivalence

In principle one must also distinguish a third class of cofibrations that have the left lifting property with respect to fibrations between fibrant objects. However, in a relaxed premodel category, these are trivial cofibrations. ([Barton, Prop 3.5.2](#Barton))

## Related concepts

* [[model category]]

* [[weak model category]]

## References

* {#Barton} [[Reid William Barton]], _A model 2-category of enriched combinatorial premodel categories_ ([arXiv:2004.12937](https://arxiv.org/abs/2004.12937))

[[!redirects premodel categories]]


premonoidal category

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#### Monoidal categories
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# Premonoidal categories
* table of contents
{: toc}

## Idea

A premonoidal category is a generalisation of a [[monoidal category]], applied by [[John Power]] and his collaborators to [[denotational semantics]] in [[computer science]]. There, the [[Kleisli category]] of a [[strong monad]] provides a model of [[call-by-value]] programming languages. In general, if the original category is monoidal, the Kleisli category will only be premonoidal.

Recall that a [[bifunctor]] from $C$ and $D$ to $E$ (for $C,D,E$ [[categories]]) is simply a [[functor]] to $E$ from the [[product category]] $C \times D$.  We can think of this as an operation which is 'jointly functorial'.  But just as a [[function]] to $X$ from $Y$ and $Z$ (for $X,Y,Z$ [[topological spaces]]) may be [[continuous map|continuous]] in each variable yet not [[jointly continuous function|jointly continuous]] (continuous from the [[Tychonoff product]] $Y \times Z$), so an operation between categories can be functorial in each variable separately yet not jointly functorial.

Recall that a [[monoidal category]] is a [[category]] $C$ equipped with a bifunctor $C \times C \to C$ (equipped with [[extra structure]] such as the [[associator]]).  Similarly, a premonoidal category is a category equipped with an operation $C \times C \to C$, which is (at least) a [[function]] on [[objects]] as shown, but one which is functorial only in each variable separately.

## Definition

A __binoidal category__ is a [[category]] $C$ equipped with

*  for each pair $x,y$ of [[objects]] of $C$, an object $x \otimes y$;
*  for each object $x$ a [[functor]] $x \rtimes -$ whose action on objects sends $y$ to $x \otimes y$
*  for each object $x$ a [[functor]] $- \ltimes x$ whose action on objects sends $y$ to $y \otimes x$

A [[morphism]] $f\colon x \to y$ in a binoidal category is __central__ if, for every morphism $f'\colon x' \to y'$, the diagrams
$$ \array {
   x \otimes x'                      & \overset{x \rtimes f'}\to  & x \otimes y' \\
   \mathllap{f \ltimes x'}\downarrow &                            & \downarrow\mathrlap{f \ltimes y'} \\
   y \otimes x'                      & \underset{y \rtimes f'}\to & y \otimes y' \\
   } $$
and
$$ \array {
   x' \otimes x                      & \overset{x' \rtimes f}\to  & x' \otimes y \\
   \mathllap{f' \ltimes x}\downarrow &                            & \downarrow\mathrlap{f' \ltimes y} \\
   y' \otimes x                      & \underset{y' \rtimes f}\to & y' \otimes y \\
   } $$
commute.  In this case, we denote the common composites $f \otimes f'\colon x \otimes x' \to y \otimes y'$ and $f' \otimes f\colon x' \otimes x \to y' \otimes y$.

A __premonoidal category__ is a binoidal category equipped with:

*  an object $I$;
*  for each triple $x,y,z$ of objects, a central [[isomorphism]] $\alpha_{x,y,z}\colon (x \otimes y) \otimes z \to x \otimes (y \otimes z)$; and
*  for each object $x$, central isomorphisms $\lambda_x\colon x \otimes I \to x$ and $\rho_x\colon I \otimes x \to x$;

such that the following conditions hold.

*  all possible [[natural transformation|naturality squares]] for $\alpha$, $\lambda$, and $\rho$ (which make sense since we have central morphisms) commute.  Note that when written out explicitly in terms of the functors $x\rtimes -$ and $-\ltimes x$, we need three different naturality squares for $\alpha$.  (But it is possible to rephrase $\alpha$ as a single natural transformation using the slick version below.)
*  the pentagon law holds for $\alpha$, as in a [[monoidal category]].
*  the triangle law holds for $\alpha$, $\lambda$, and $\rho$, as in a monoidal category.

A __strict premonoidal category__ is a premonoidal category in which $(x \otimes y) \otimes z = x \otimes (y \otimes z)$, $x \otimes I = x$, and $I \otimes x = x$, and in which $\alpha_{x,y,z}$, $\lambda_x$, and $\rho_x$ are all [[identity morphisms]].  (We need the underlying category $C$ to be a [[strict category]] for this to make sense.)

Similarly, a **symmetric premonoidal category** is a premonoidal category equipped with a central natural isomorphism $x\otimes y \cong y\otimes x$ (as for $\alpha$, there are two naturality squares unless we use the slick approach), satisfying the usual axioms of a symmetry.

## Slick version

As a [[strict monoidal category]] is a [[monoid]] in the [[cartesian monoidal category]] [[Cat]], so a strict premonoidal category is a monoid in the [[symmetric monoidal category]] $(Cat,\Box)$, where $\Box$ is the [[funny tensor product]].

From this point of view, a binoidal category is just a category $C$ with a functor $C \Box C \to C$

We might hope that non-strict premonoidal categories are [[pseudomonoids]] in $(Cat,\Box)$ seen as a [[monoidal 2-category]], by taking either natural or [[unnatural transformations]] as 2-cells. For the requirement that coherence maps are natural, suggests taking [[natural transformations]]. However, this is not enough to ensure the coherence maps are central.

[Román 22](#R22) provides the solution, showing that [[Freyd categories]] (or *effectful categories*), which include premonoidal categories as a special case, are precisely pseudomonoids in a monoidal bicategory of [[promonads]] equipped with their *pure tensor product* (described in Section 4 of [Román 22](#R22)).

## Examples

* Every [[monoidal category]] is a premonoidal category.

* If $T$ is a [[strong monad|left- and right-strong monad]] on a monoidal category $C$ (e.g. a strong monad on a braided monoidal category), then the [[Kleisli category]] $C_T$ of $T$ inherits a premonoidal structure, such that the functor $C\to C_T$ is a strict premonoidal functor.  This premonoidal structure is only a monoidal structure if $T$ is a [[commutative monad]].

* A strict premonoidal category is the same as a [[sesquicategory]] with one object, so any object of a sesquicategory has a corresponding premonoidal category whose objects are endomorphisms and arrows are 2-cells.

## Properties

The central morphisms of a premonoidal category $C$ form a [[subcategory]] $Z(C)$, called the __centre__ of $C$, which is a [[monoidal category]].  This defines a [[right adjoint functor]] to the inclusion $MonCat \hookrightarrow PreMonCat$
 using the definition of functor of premonoidal categories in [Power-Robinson 97](#PR97).

In the same way that a (strict) monoidal category can be identified with a (strict) [[2-category]] with one object, a strict premonoidal category can be identified with a [[sesquicategory]] with one object.  In fact, a sesquicategory is precisely a category [[enriched category|enriched]] over the monoidal category $(Cat,\otimes)$ described above.

## Morphisms

A notion of (non-strict) *premonoidal functor* is somewhat tricky to define.  Part of the definition is clear: it should be a functor that preserves the tensor product up to specified coherent *central* isomorphism.  However, the tricky part is whether it should also be required to preserve centrality of morphisms (or even just isomorphisms).  Desiderata pulling in opposite directions include:

1. If a premonoidal functor is not required to preserve centrality at least of isomorphisms, then premonoidal functors may not be closed under composition, since we may not be able to define central coherence isomorphisms for $G \circ F$ if $G$ does not preserve the centrality of the coherence isomorphisms for $F$.

1. A morphism $T_1 \to T_2$ of bistrong monads on a monoidal category $C$ induces a functor $C_{T_1} \to C_{T_2}$ which preserves the premonoidal structures *strictly*, hence clearly up to coherent central isomorphism.  However, such a functor does not in general preserve centrality even of isomorphisms; a counterexample can be found in [SL13](#SL13), section 5.2.

It seems unlikely that these desiderata can be reconciled purely in the world of premonoidal categories as usually defined.  One solution is to pass to [[Freyd-categories]], which are essentially premonoidal categories equipped with a family of "special" central morphisms forming a cartesian monoidal category with the same tensor product.  Morphisms of Freyd-categories are easy to define, and include all the morphisms $C_{T_1} \to C_{T_2}$ if $C$ is cartesian (and if it isn't, then there should be a suitable non-cartesian generalization of Freyd-categories).  Other solutions that use different ways of representing a "special subfamily of central (iso)morphisms" are to consider the tensor product functor of a premonoidal category to be a not-necessarily-saturated [[anafunctor]], or (in [[homotopy type theory]]) to allow the underlying category of a premonoidal category to be merely a [[homotopytypetheory:precategory]].

## Related pages

* [[premulticategory]]
* [[Freyd category]]

## References

* {#PR97} [[John Power]] and [[Edmund Robinson]], _Premonoidal categories and notions of computation_, Math. Structures Comput. Sci., 7(5):453–468, 1997. Logic, domains, and programming languages (Darmstadt, 1995).
 [PostScript](http://www.eecs.qmul.ac.uk/~edmundr/pubs/mscs97/premoncat.ps)

* Alan Jeffrey, _Premonoidal categories and a graphical view of programs,_ [pdf file](https://web.archive.org/web/20120303033606if_/http://fpl.cs.depaul.edu/ajeffrey/papers/premonA.pdf)

* {#SL13} [[Sam Staton]] and Paul Blain Levy, *Universal Properties of Impure Programming Languages*.  ACM Sigplan Notices, 2013, [doi](https://doi.org/10.1145/2480359.2429091).

* {#R22} [[Mario Román]], *Promonads and String Diagrams for Effectful Categories*.  Proceedings of Fifth International Conference on Applied Category Theory, 2022, [doi](https://dx.doi.org/10.4204/EPTCS.380.20).

[[!redirects premonoidal category]]
[[!redirects premonoidal categories]]
[[!redirects binoidal category]]
[[!redirects binoidal categories]]


premulticategory

# Premulticategories

* toc
{: toc}

## Idea

A **premulticategory** is to a [[multicategory]] as a [[premonoidal category]] is to a [[monoidal category]].

## Definition

A definition of premulticategory is obtained by starting from a definition of [[multicategory]] in terms of the binary composition operations

$$ \circ_i : C(y_1,\dots,y_i, \dots,y_n; z) \times C(x_1,\dots,x_m; y_i) \to C(y_1,\dots,y_{i-1},x_1,\dots,x_n,y_{i+1},\dots,y_n;z) $$

and removing the "parallel associativity" or "commutativity" which says that composing $h\in C(y_1,\dots,y_n;z)$ with two morphisms $f\in C(x_1,\dots,x_n ; y_i)$ and $g\in C(w_1,\dots,w_p; y_j)$ can be done in either order.

A morphism $f$ is _central_ if for any $g$ and $h$ as above, the two methods of composition commute: $(h \circ f) \circ g = (h \circ g) \circ f$. So a premulticategory is a multicategory precisely if all morphisms are central.

## Relation to premonoidal categories

A premulticategory that has all tensor products and units, in a usual multicategorical sense, is equivalent to a [[premonoidal category]].

## References

* [[Sam Staton]] and [[Paul Blain Levy]], *Universal properties of impure programming languages*.  POPL '13: Proceedings of the 40th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages, 2013, [doi](https://doi.org/10.1145/2429069.2429091)

[[!redirects premulticategories]]
[[!redirects pre-multicategory]]
[[!redirects pre-multicategories]]


preon

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

#Contents#
* table of contents
{:toc}

## Idea

While in the [[standard model of particle physics]] all [[hadrons]] and [[mesons]] are [[bound states]] of [[quarks]], the [[quarks]] themselves, together with the [[leptons]], are assumed to be "elementary" [[particles]], hence not themselves [[bound states]] of other particles.

While this does fit very well with current [[experimental observation]], one may speculate if at a higher energy scale [[quarks]] and/or [[leptons]] are themselves bound states of more fundamental particles. These hypothetical more fundamental particles are then generically called _preons_.

## Related concepts

[[!include fields and quanta - table]]

## References

Textbook account:

* [[Robert E. Marshak]], Chapter 9 of: *Conceptual Foundations of Modern Particle Physics*, World Scientific 1993 ([doi:10.1142/1767](https://doi.org/10.1142/1767))

See also:

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

A proposal for an embedding of preon models into [[string theory]] is in

* Noriaki Kitazawa, _Dynamical Generation of Yukawa Couplings in Intersecting D-brane Models_, JHEP 0411 (2004) 044 ([arXiv:hep-th/0409146](http://arxiv.org/abs/hep-th/0409146))

[[!redirects preons]]


PreOrd

[[!redirects Preord]]

$PreOrd$ denotes the category of [[preorders]] and order-preserving maps.

The category $PreOrd$ is [[topological functor|topological]] over $Set$ and therefore enjoys strong completeness and cocompleteness properties; see [[total category]]. It is also a [[locally presentable category]].

$PreOrd$ is a [[cartesian closed category]]. In fact it is an [[exponential ideal]] in the cartesian closed category [[Cat]].


Preord > history

$Preord$ (or sometimes $PreOrd$) denotes the category of [[preorders]].