principle of equivalence

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## Idea
 {#Idea}

Sensible reasoning in [[mathematics]] in general, and particularly in [[higher structures]], such as ([[higher category theory|higher]]) [[category theory]], [[homotopy theory]],  [[homotopy type theory]], etc., should be suitably [[invariant]] under the respective notion of _[[equivalence]]_. This is a common issue whenever some structure is _[[generators and relations|presented]]_ by other structures, since the notion of equivalence of the presentation can be finer than that of the notion being presented.

For instance, a [[category]] can be presented by a [[simplicial set]] (the [[nerve]] of a corresponding [[strict category]], see [there](nerve#NerveOfACategory)), but [[isomorphism]] of simplicial sets is much finer than [[equivalence of categories|equivalences]] of their corresponding categories. It is generally a _mistake_ to mix up these two levels and, for instance, assign to a category properties that are shared only by _some_ of the simplicial sets representing it, say, by distinguishing between [[isomorphism|isomorphic]] [[objects]]. This _breaks the equivalence invariance_.  (However, see [below](#Breaking).)

The ideas here generalize in many directions.  For example, not only properties, but also constructions involving categories and functors, can fail to preserve equivalences.

## Terminology
{#terminology}

* [[Michael Makkai]] proposed  the _Principle of Isomorphism_, "all grammatically correct properties of objects of a fixed category are to be invariant under isomorphism", in his [paper](https://projecteuclid.org/ebooks/lecture-notes-in-logic/Logic%20Colloquium%20'95:%20Proceedings%20of%20the%20Annual%20European%20Summer%20Meeting%20of%20the%20Association%20of%20Symbolic%20Logic,%20held%20in%20Haifa,%20Israel,%20August%209-18,%201995/chapter/Towards%20a%20Categorical%20Foundation%20of%20Mathematics/lnl/1235415906)
"Towards a categorical foundation of mathematics", in Johann A. Makowsky, Elena V. Ravve, eds., Logic Colloquium '95: Proceedings of the Annual European Summer Meeting of the Association of Symbolic Logic, held in Haifa, Israel, August 9-18, 1995 (Berlin: Springer-Verlag, 1998), 153-190.

* [[Peter Aczel]] mentioned a similar concept, the _[[structure identity principle|Structure Identity Principle]]_, "isomorphic structures have the same structural properties" in Oberwolfach [report 52/2011](http://www.mfo.de/document/1145/OWR_2011_52.pdf). This seems a priori a little weaker to [[David Roberts|me]], but if we demand that objects should be seen as only having structural properties (as opposed to the category of [[ZF]]-[[sets]]), then we look like we get back the Principle of Isomorphism.

* A very precise way of stating this idea is encapsulated in [[Vladimir Voevodsky]]'s [[univalence axiom]], which is a fundamental part of [[homotopy type theory]] as a [[foundation]] for mathematics.  By identifying equivalences/isomorphisms with inhabitants of an [[identity type]], it ensures that all properties and structure which can be expressed within the formal [[type theory]] are invariant under such (see [AhrensNorth18](#AhrensNorth18)).

* {#evil} Floating around the web (and maybe the $n$Lab) is the idea of half-jokingly referring to a breaking of equivalence invariance as "evil". This is probably meant as a pedagogical way of amplifying that it is to be avoided.

## Motivation

### From mathematical practice

...

>Draw an analogy with vector spaces (maybe just finite-dimensional ones?).  We can *use* a basis to study a vector space, but the basis is not *part of* the vector space.  Even if we use a foundation of mathematics in which the basis literally is part of the vector space (which arguably was the case for 18th-century linear algebra, which studied only Cartesian spaces; and is still the case today if we study finite-dimensional vector spaces using a foundational type theory with propositions as types, but probably this is too obscure), anything that we consider to be a property *of the vector space* should be independent of the basis.  We can make this precise by comparing the groupoids $Vect_\sim$ and $Vect_b$.

...

### From foundations and logic

In [[material set theory]] based on [[ZFC]], any two objects are [[pure sets]] and can be compared for [[equality]].  Yet this is often mathematical nonsense which depends on precisely how one writes definitions; for example, the [[ordered pair]] $(0,0)$ is equal to the [[natural number]] $1$ by some standards and equal to $2$ by other standards (while $(1,1) = 2$ by yet other standards), but this makes no difference mathematically.  In contrast, [[structural set theory]] either implicitly (as in [[ETCS]]) or explicitly (as in [[SEAR]] or [[ETCS with elements]]) considers only equality between [[elements]] of a given [[set]] (or [[functions]] or [[binary relations]] between two given sets, etc).  This can be built into the [[logic]] using [[dependent type theory]] (which can serve as a foundation all by itself). The result is that, depending on precisely how one writes definitions, it may or may not be possible to compare two things for equality. For example, the ordered pair $(1, 1)$ is equal to $2$ under triangular encoding, equal to $3$ under binary encoding, and equal to $1$ under Chinese Remainder encoding. But if we take ordered pairs as primitive, it makes no sense to compare $(1, 1)$ to $2$. (Of course, we could do this in material set theory as well; but allowing non-set primitives there would defeat the traditional attraction of the theory.)

Does it make sense, then, to ask whether two arbitrary *sets* are equal?  In material set theory, of course it does; two sets are equal if they have the same elements (by the [[axiom of extensionality]]).  But structurally, this definition is meaningless.  All the same, a structural set theory built on [[first-order logic]] with equality automatically gives this question (whether two arbitrary sets are equal) a meaning, as do the most common forms of foundational type theory, but this is again never used in mathematics.  ($SEAR$, which is based on first-order logic without equality, does not give this question a meaning and does not suffer for it.)  So just as we don\'t need to ask whether $(0,0) = 2$, we don\'t need to ask whether $\{(0,0)\} = \{2\}$ as sets.  (Note that this is a completely different question, structurally, from whether, say, $\{(0,0)\} = \{2\}$ as *[[subsets]]* of $\mathbb{N} \coprod (\mathbb{N} \times \mathbb{N})$.)

Now let us move from the foundations of [[set theory]] to those of [[category theory]].  Usually, category theory is founded on set theory, or something very much like it; if the [[objects]] of a category don\'t form a set, then this is only because of [[size issues]], and they still form a [[proper class]].  Accordingly, it makes sense to compare them for equality.  However, if the [[category of sets]] is itself an example of a category, then by the previous paragraph, it does *not* have to make sense (and is mathematically meaningless) to test the objects of this category for equality.  So we get the idea that we cannot compare objects of a given category for equality in general (although this may make sense for particular categories).  In other words, a [[category]] is not by default a [[strict category]] (a category equipped with the [[extra stuff]] of an explicit equality predicate on its objects).

We can found mathematics on type theory without identity types, and then it is literally true that (in general) it makes no sense to compare objects of a given type for equality; one has to *define* a relevant equality predicate when there is one.  From the other direction, we can found mathematics on (weak) $\infty$-[[infinity-groupoid]] theory, i.e. [[homotopy theory]] (as in [[homotopy type theory]]), where the automatic notion of equality between object of a category is actually only isomorphism.

However, these are not commonly used foundations.  Therefore, category theory is usually written in a language in which it does literally make sense to ask whether two objects of a given category are equal (meaning something stronger than that they are isomorphic), whether two functors between two given categories are equal, etc.  If we want these definitions to make sense in the general mathematical situation (where $Set$ is an example of a category, even though comparing two arbitrary sets for equality makes no mathematical sense), then we need to check that our definitions are compatible with the principle of equivalence.

(Strictly speaking, it does not even make sense to ask if two arbitrary sets are isomorphic; going back to our earlier example, $\{(0,0)\}$ is isomorphic to $\{2\}$ as objects of $Set$ because they are both singletons, but they are not isomorphic as objects of the poset of subsets of $\mathbb{N} + \mathbb{N}\times \mathbb{N}$, because neither is a subset of the other.)

### From philosophy

In [[Tractatus Logico-Philosophicus]], 2.02331, [[Wittgenstein]] writes

> Either a thing has properties that nothing else has, in which case we can immediately use a description to distinguish it from the others and refer to it; or, on the other hand, there are several things that have the whole set of their properties in common, in which case it is quite impossible to indicate one of them. For if there is nothing to distinguish a thing, I cannot distinguish it, since otherwise it would be distinguished after all.

Given that a main theme of the Tractatus is on the limits of language, it is tempting to consider this as a description not just for 'the world' (as in _Tractatus_ 1.), but for any given language and the objects about which one reasons in that language. This seems to encapsulate Makkai's motivation alluded to above.

## General definition

If $X$ is an $\infty$-[[infinity-groupoid|groupoid]], then a property $P$ of objects of $X$ is _compatible with equivalence_ if, whenever $P(a)$ holds for an object $a$ of $X$ and $b$ is [[equivalence|equivalent]] (as an object of $X$) to $a$, then $P(b)$ holds. Alternatively, an operation $f$ from objects of $X$ to objects of (another $\infty$-groupoid) $Y$ is __compatible with equivalence__ if, whenever $a$ and $b$ are equivalent objects of $X$, $f(a)$ and $f(b)$ are equivalent objects of $Y$.

(Note that everything is an object of some $\infty$-groupoid. For example, a $2$-morphism in some random $5$-category is an object of a hom-$3$-category, which has an [[core|underlying]] $3$-groupoid, which is a special kind of $\infty$-groupoid. For some purposes, this $2$-morphism may instead need to be seen as an object in a $2$-arrow $5$-category, which has an underlying $\infty$-groupoid as well; exactly which $\infty$-groupoid is relevant depends on the [[context]]. Also note that we really do need $\infty$-groupoids here, rather than $\infty$-categories, since dualities do respect equivalences.)

The operation-version of the principle of equivalence gives the property-version if you think of a property as an operation taking values in the $\infty$-groupoid (in fact a $0$-groupoid, or [[set]]) of [[truth value|truth values]]. (The property-version also gives the operation-version, although that is a little more involved.) An operation will obey the principle of equivalence if it is given (as the object-operation) by a [[functor]] ('$\infty$-functor', if you prefer). A property adheres to the principle of equivalence precisely if it\'s a functor to the $\infty$-groupoid of truth values.

This definition should be the conclusion of a theorem that using certain language (including avoiding equations between objects of a category) makes it impossible to say anything that violates the principle of equivalence. [[Michael Makkai]] works on such a language, [[FOLDS]] ('first-order logic with dependent sorts'), which does not include an axiomatic notion of [[equality]] at all. This pertains to the [mathematical foundations of category theory](foundations#MFCT). To learn more, see his paper _[First Order Logic with Dependent Sorts, with Applications to Category Theory](http://www.math.mcgill.ca/makkai/)_.

## Examples

### In category theory

A common misconception in category theory is that [[strict categories]] violate the principle of equivalence. However, this is not true; strict categories are the mathematical structures in which the equivalence of objects are precisely equality on objects: important examples of strict categories which do not violate the principle of equivalence include [[posets]] and the [[walking parallel pair]]. In fact, every strict category is equivalent to a [[setoid]]-valued [[presheaf]] on the [[simplex category]] or the full subcategory of the [[arrow category]] of the [[(2,1)-category]] of [[weak categories]], consisting of [[essentially surjective functors]] whose [[domain]] is a [[setoid]].

Instead, what does break the principle of equivalence is to say that certain [[concrete categories]] with non-trivial [[automorphisms]] such as [[Set]] or [[Grp]] are strict categories, as the objects of the categories are [[sets]], and equality of objects is equality of sets. In this case, one has to instead say that they are [[isomorphism|isomorphic]] and then (usually) impose some [[coherence relation]] on the relevant family of isomorphisms.  But of course, this is more complicated!

#### Identity-on-objects functors

The concept of an [[identity-on-objects functor]] appears in category theory as well, particularly when defining [[Freyd categories]] and [[dagger categories]]. The traditional definition goes as follows:

\begin{definition}
An **identity-on-objects [[functor]]** $F: A\to B$ between [[categories]] $A$ and $B$ is a functor between categories with the same collection of [[objects]], and has as its underlying object function $F_{ob}$ the [[identity function]] on the collection of objects.
\end{definition}

However, in [[material set theories]] or material class theories, "having the same collection of objects" is sometimes defined as $Ob = ob(A) = ob(B)$. This definition involves equality of sets, which violates the principle of equality.

Instead, what category theorists usually mean by two categories having the same objects is actually having a single $(Set \times Set)$-[[enriched category]], where morphisms are in families of sets of pairs $(f,g):A(x,y) \times B(x,y))$ for objects $x$ and $y$, and composition is composition of pairs of morphisms $(h,k)\circ(f,g) = (h \circ f,k \circ g)$ for $f:A(x,y)$, $g:B(x,y)$, $h:A(y,z)$, and $k:B(y,z)$.

Another problem occurs in [[set theories]] with certain [[concrete categories]] like [[Rel]], [[Hilb]], and the [[permutation category]] $\mathbb{P}$: the objects of the categories are sets, and so saying that the underlying function $F_ob$ on the objects is the identity function violates the principle of equality, because by definition $F_ob(x) = x$ for all objects $x$, and because objects are sets, one is talking about equality of sets.

Instead, when category theorists are talking about two categories with the same collection of objects and and identity-on-objects functor, what they really are talking about is an $Set^I$-enriched category, where $Set^I$ is the [[arrow category]] of $Set$.

#### In the concept of $\dagger$-categories
  {#daggers}

A **[[†-category]]** is a [[category]] $C$ with a function $(-)^\dagger: Hom_C(A,B) \to hom_C(B,A)$ for every object $A,B \in Ob(C)$, such that

* For every $A \in Ob(C)$, $(1_A)^\dagger = 1_A$
* For every $A,B \in Ob(C)$ and every $f \in Hom_C(A,B)$ and $g \in Hom_C(B,C)$, $(g \circ f)^\dagger = f^\dagger \circ g^\dagger$
* For every $A,B \in Ob(C)$ and every $f \in Hom_C(A,B)$, $((f)^\dagger)^\dagger = f$.

However, there is another definition of a †-category, as a category $C$ with a functor

$$F\colon C \to C^{op} $$

which is the identity on objects and has $F^2 = 1$.

This second definition break equivalence-invariance: it imposes equations between objects, so we cannot transport a dagger-category structure along an equivalence of categories.

##### Historical comments

It was once believed that there was no known way to express the idea without equations between objects. This stemmed from the fact that category theorists were using functors to define the dagger. It was only later that it was recognized that one could include the dagger in the definition of dagger categories in the same way that one includes composition of morphisms in the definition of category, which resulted in a definition of dagger category that doesn't violate the principle of equivalence.

A discussion about this is [archived on the nForum](https://nforum.ncatlab.org/discussion/4201/principle-of-equivalence/?Focus=101442#Comment_101442).

#### In higher category theory

It violates the principle of equivalence to state that two morphisms in a $2$-[[2-category|category]] are equal, because these morphisms are objects in a [[hom-category]], but does not violate the principle of equivalence to state that that two $2$-morphisms are equal, given a common source and target.  And so on.  In an $\infty$-[[infinity-category|category]], *every* claim of equality break equivalence-invariance.

Defining higher categorial structures using such equalities tends to lead to *strict* concepts; avoiding them and imposing coherence relations leads to *weak* concepts.  Sometimes there is a [[coherence theorem]] showing that every weak concept can be strictified, which justifies using equality as a figure of speech.  See [[bicategory]], [[Gray-category]], and [[model category]] for examples of this in action.

### In physics

Since [[Hilb]] and [[nCob]] are [[dagger-categories]], the discussion [above](#daggers) is relevant, particularly for [[TQFT]].

#### In gauge theory

See at

* [[gauge transformation]], [[gauge equivalence]], [[gauge invariance]]

#### In gravity

The principle of equivalence in [[general relativity]] (see [[equivalence principle (physics)]]) is a special case of this principle of equivalence; the objects of the $\infty$-groupoid are physical "fields" (not to be confused with algebraic fields!), and the isomorphisms are [[coordinate transformation]]s. So physical properties and operations should not depend on the choice of coordinate system, and this is enforced by allowing only functorial operations, which are specifically taken to be generated by [[trace]]s, [[tensor product]]s, and linear operations acting on the [[pseudo-Riemannian metric]], and the [[covariant derivative]].

See at

* [[general covariance]]

## How to break equivalence-invariance
{#Breaking}

Just as we can make use of [[basis|bases]] in [[linear algebra]], so we may make use of [[strict categories]] to discuss [[category theory]].  Philosophically, the concept of strict category is not in itself a breaking of equivalence-invariance; what does break equivalence-invariance is to say that $Set$ (and other well-known categories such as [[Grp]], etc) *are* strict categories.  Mathematically, strict categories form a $1$-[[1-groupoid|groupoid]] $Str Cat_\sim$ that is different from the $2$-[[2-groupoid|groupoid]] $Cat_\sim$, but there is still a canonical [[pseudo functor]] from $Str Cat_\sim$ to $Cat_\sim$ that we may find useful.

Much as the [[axiom of choice]] tells us that every vector space has a basis, so the global axiom of choice tells us that every category may be given the structure of a strict category.  Then we can use this extra structure (in either case), as long as we prove that the result is independent of the structure chosen.  Even if we don\'t wish to accept the axiom of choice, we can still prove a theorem about those vector spaces that have bases or those categories that can be given a strict structure.

Perhaps the extreme case of this is using [[model categories]] to study [[homotopy theory]], which is (from the [[nPOV]]) really about $(\infty,1)$-[[(infinity,1)-category|categories]].  Even if model categories are not taken to be strict categories, they still form a $2$-groupoid and thus are still far more strict than $(\infty,1)$-categories, which only form an $\infty$-[[infinity-groupoid|groupoid]].  Nevertheless, they are quite useful (at least assuming the axiom of choice?).

## References
 {#References}

A discussion of the principle of equivalence in the very [[foundations]] of mathematics by replacing [[ZFC]] by [[homotopy type theory]] is in

* [[Vladimir Voevodsky]], _Foundations/formalization of mathematics and homotopy theory_, talk at IAS ([video](http://video.ias.edu/node/68))

* {#CoquandDanielsson} [[Thierry Coquand]], Nils Anders Danielsson, _Isomorphism is equality_ ([pdf](http://www.cse.chalmers.se/~nad/publications/coquand-danielsson-isomorphism-is-equality.pdf))

* {#Awodey14} [[Steve Awodey]], _Structuralism, Invariance, and Univalence_ (2014) ([pdf](https://www.andrew.cmu.edu/user/awodey/preprints/siu.pdf))

* {#AhrensNorth18} [[Benedikt Ahrens]], [[Paige Randall North]], _Univalent foundations and the equivalence principle_, ([pdf](https://paigenorth.github.io/ep.pdf))

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principle of extremal action

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## Idea

The _principle of extremal action_ in [[physics]] is a historical precursor of the modern understanding that [[trajectories]] of [[particles]] and configurations of [[field (physics)|fields]] realized in [[classical mechanics]] are characterized as being the _[[variational calculus|variational]] [[extrema]]_ or _[[critical loci]]_ of a [[nonlinear functional|functional]] on the space of all possible configurations, called the _[[action functional]]_.
For typical examples of this principle -- some of which were known to the ancient Greeks and which were then formalized in some generality in the 18th century -- these extrema are indeed minima, as with, for instance, the propagation of a ray of light through optical media in [[optics]]. More generally one speaks of the _principle of extremal action_ or the _principle of stationary action_. But in fact mostly today one just speaks of the [[Euler-Lagrange equations|Euler-Lagrange]] [[equations of motion]], which are the equations that characterize these extremal trajectories.

With the advent of [[quantum mechanics]], the _principle of extremal action_ found an explanation in more fundamental terms as the [[classical limit]] of quantum [[dynamics]]. A simple, though somewhat heuristic, way to see this is via [[path integral]] [[quantization]]: whereas propagation in [[quantum mechanics]] is given by the [[path integral]] by which _every_ trajectory $\phi$ between two prescribed configurations contributes with a [[probability amplitude]] given by the  exponentiated value $\exp(i S(\phi))$ of the [[action functional]] $S$ on this trajectory, at least under some conditions the main contribution to this integral is from trajectories close to the critical points of the action functional.

## Examples

* [[Einstein-Hilbert action]] $\implies$ [[Einstein equation of motion]] for [[gravity]]/[[general relativity]]

## References

Historical and expository accounts of the "principle of extremal action" or "action principle", for short:

* Agamenon R. E. Oliveira, *History of Two Fundamental Principles of Physics: Least Action and Conservation of Energy*, Advances in Historical Studies **3** 2 (2014) [[doi:10.4236/ahs.2014.32008](http://dx.doi.org/10.4236/ahs.2014.32008)]

* Walter Dittrich, *The Development of the Action Principle -- A Didactic History from Euler-Lagrange to Schwinger*, SpringerBriefs in Physics, Springer (2021) [[doi:10.1007/978-3-030-69105-9](https://doi.org/10.1007/978-3-030-69105-9)]

* [[Douglas Cline]], *Variational Principles in Classical Mechanics*, University of Rochester (2021) [[pdf](http://classicalmechanics.lib.rochester.edu/pdf/Variational_Principles_in_Classical_Mechanics_3e_final_fbcover.pdf), online version]

See also:

* Wikipedia, _[Principle of least action](http://en.wikipedia.org/wiki/Principle_of_least_action)_

* [[geometry of physics]] -- _[Equations of motion](geometry%20of%20physics#EquationsOfMotion)_

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principle of omniscience

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## Idea

In [[logic and foundations]], a __principle of omniscience__ is any principle of [[classical mathematics]] that is not valid in [[constructive mathematics]].  The idea behind the name (which is due to [Bishop (1967)](#Bishop67)) is that, if we attempt to extend the [[computation|computational]] interpretation of constructive mathematics to incorporate one of these principles, we would have to know something that we cannot compute.  The main example is the law of [[excluded middle]] ($EM$); to apply $p \vee \neg{p}$ computationally, we must know which of these [[disjunction|disjuncts]] hold; to apply this in all situations, we would have to know everything (hence 'omniscience').

Bishop\'s principles of omniscience (stated below) may be seen as principles that extend classical logic from [[predicates]] (where $EM$ may happen to be valid, even constructively, for certain predicates) to their [[quantifications]] over infinite domains (where $EM$ is typically not constructively valid).

## Definition

### The limited principle of omniscience

The __limited principle of omniscience__ ($LPO$) states that the [[existential quantification]] of any [[decidable proposition]] is again decidable.  That is,
$$ (\forall x, P(x) \vee \neg{P(x)}) \Rightarrow (\exists x, P(x)) \vee \neg(\exists x, P(x)) ,$$
or equivalently
$$ (\forall x, P(x) \vee \neg{P(x)}) \Rightarrow (\exists x, P(x)) \vee (\forall x, \neg{P(x)}) .$$

We have not stated the domain of quantification of the variable $x$.  If you take it to be the [[subsingleton]] corresponding to a given [[truth value]] and apply this principle to the constantly [[true]] proposition, then $EM$ follows; conversely, $EM$ implies $LPO$ (over any domain).  However, Bishop\'s $LPO$ takes the domain to be the set of [[natural numbers]], giving a weaker principle than $EM$.  (It appears that a [[realizability topos]] based on infinite-time [[Turing machine]]s validates $LPO$ but not $EM$; see [Bauer (2011)](#Bauer11).)  Note that propositions of the form $\exists x, P(x)$ when $P$ is decidable are the [[open truth values]] in the [[Rosolini dominance]].

We can also state the principle set-theoretically, with explicit reference to the domain of quantification.  Given a [[set]] $A$, the __limited principle of omniscience for $A$__ ($LPO_A$) states that, given any [[function]] $f$ from $A$ to the [[boolean domain]] $\{0,1\}$, either $f$ is the [[constant map]] to $0$ or $1$ belongs to the [[image]] of $f$.  Then Bishop\'s $LPO$ is $LPO_{\mathbb{N}}$, which applies to any [[infinite sequence]] of [[bits]].

While $LPO$ for $\mathbb{N}$ is a classic example of the difference between constructive and classical mathematics, $LPO$ holds for the set $\overline{\mathbb{N}}$ of [[extended natural number]]s; this is related to the fact that $\overline{\mathbb{N}}$ may constructively be given a [[compact space|compact]] topology.  See [Escardó (2011)](#Escardo11) for this and much more.

### The weak limited principle of omniscience

The __weak limited principle of omniscience__ ($WLPO$) states that the [[universal quantification]] of any [[decidable proposition]] is again decidable. That is,
$$(\forall x, P(x) \vee \neg{P(x)}) \Rightarrow (\forall x, P(x)) \vee \neg(\forall x, P(x)).$$

### The lesser limited principle of omniscience

The __lesser limited principle of omniscience__ ($LLPO$) states that, if the existential quantification of the [[conjunction]] of two [[decidable propositions]] is false, then one of their separate existential quantifications is false.  That is,
$$ (\forall x, P(x) \vee \neg{P(x)}) \Rightarrow (\forall y, Q(y) \vee \neg{Q(y)}) \Rightarrow \neg(\exists x, P(x) \wedge \exists y, Q(y)) \Rightarrow \neg(\exists x, P(x)) \vee \neg(\exists y, Q(y)) ,$$
or equivalently
$$ (\forall x, P(x) \vee \neg{P(x)}) \Rightarrow (\forall y, Q(y) \vee \neg{Q(y)}) \Rightarrow \neg(\exists x, P(x) \wedge \exists y, Q(y)) \Rightarrow (\forall x, \neg{P(x)}) \vee (\forall y, \neg{Q(y)}) .$$

If, as before, you take the domains of quantification to be subsingletons, you get [[de Morgan's law]] $\neg(p \wedge q) \Rightarrow \neg{p} \vee \neg{q}$ ($DM$), which is weaker than $EM$; conversely, $DM$ implies $LLPO$ (over any domain). Again, Bishop\'s $LLPO$ takes the domain to be $\mathbb{N}$, giving a principle weaker than $DM$ (and also weaker than $LPO_{\mathbb{N}}$).

Stated set-theoretically, the __lesser limited principle of omniscience for $A$__ ($LLPO_A$) states that, given functions $f, g\colon A \to \{0,1\}$, if $1 \notin \im f \cap \im g$, then $1 \notin \im f$ or $1 \notin \im g$.  So Bishop\'s $LLPO$ is $LLPO_{\mathbb{N}}$.

### Relation between the principles of omniscience

We have the following relations between the three principles of omniscience:

* $WLPO$ follows from $LPO$, but not conversely. If $P(x)$ is a decidable proposition, then so is $\neg{P(x)}$, and so $LPO$ gives
$$(\exists x, \neg{P(x)}) \vee (\forall x, \neg{\neg{P(x)}}), $$
which implies
$$\neg(\forall x, P(x)) \vee (\forall x, P(x))$$
as $P$ is decidable.

* $LLPO$ follows from $LPO$, but not conversely.

## Analytic versions {#analytic}

Bishop introduced the above principles of omniscience to show that certain results in pointwise [[analysis]] could not be constructive, by showing that these results implied a principle of omniscience.  The principles of omniscience that appear directly in analysis are these:

*  The __analytic LPO__ states that the usual [[apartness relation]] on the set $\mathbb{R}$ of [[real numbers]] is [[decidable relation|decidable]] ($x \neq y$ or $x = y$), or equivalently __[[trichotomy]]__ for the real numbers ($x \lt y$ or $x = y$ or $x \gt y$), or equivalently, that the real numbers form a [[discrete field]].

* The __analytic WLPO__ states that $\mathbb{R}$ has [[decidable equality]].

*  The __analytic LLPO__ states that the usual order on $\mathbb{R}$ is a [[total order]] ($x \leq y$ or $x \geq y$), which (by analogy with trichotomy) may be called __dichotomy__ for the real numbers.

The analytic principles of omniscience imply the corresponding ones for natural numbers; the converses hold if we assume [[weak countable choice]] (as Bishop did).  In any case, if we use the [[modulated Cantor real numbers]] (sequential real numbers), then the sequential-analytic principles of omniscience are the same as those for natural numbers.

(Note that we need not accept $WCC$ to see that an analytic result implies a principle of omniscience and so cannot be constructively valid.)

## In the internal logic

In [[set theory]], there are actually two different notions of logic: there is the external predicate logic used to define the set theory itself, and there is the internal predicate logic induced by the set operations on [[subsingletons]] and [[injections]]. In particular,

* An internal *[[proposition]]* is a set $P$ such that for all elements $x \in A$ and $y \in A$, $x = y$.

* The internal proposition *[[true]]* is a [[singleton]] $\top$.

* The internal proposition *[[false]]* is the [[empty set]]

* The internal conjunction of two internal propositions $P$ and $Q$ is the [[cartesian product]] $P \times Q$ of $P$ and $Q$.

* The internal disjunction of two internal propositions $P$ and $Q$ is the [[image]] of the [[uniqueness quantifier|unique]] function $!_{P \uplus Q}:P \uplus Q \to 1$ from the [[disjoint union]] of $P$ and $Q$ to the [[singleton]] $\top$.

$$P \vee Q = \mathrm{im}(!_{P \uplus Q})$$

* The internal implication of two internal propositions $P$ and $Q$ is the [[function set]] $P \to Q$ between $P$ and $Q$.

* The internal negation of an internal proposition $P$ is the function set from $P$ to the [[empty set]]

$$\neg P = P \to \emptyset$$

* An internal proposition $P$ is a [[decidable proposition]] if it comes with a function $\chi_P:P \to 2$ from $P$ to the [[boolean domain]] $2$.

* An *internal predicate* on a set $A$ is a [[set]] $P$ with [[injection]] $i:P \hookrightarrow A$, whose family of propositions indexed by $x \in A$ is represented by the [[preimages]] $i^*(x)$.

* The internal [[existential quantifier]] of an internal predicate $i:P \hookrightarrow A$ is the [[image]] of the [[uniqueness quantifier|unique]] function $!_P:P \to \top$ into the singleton $\top$.

$$\exists_A P = \im(!_P)$$

* The internal [[universal quantifier]] of an internal predicate $i:P \hookrightarrow A$ is the [[dependent product]] of the preimages

$$\forall_A P = \{f \in P^A \vert \forall x \in A, f(x) \in i^*(x) \}$$

* An internal predicate $i:P \hookrightarrow A$ is a [[decidable proposition]] if it comes with a function $\chi_P(x):i^*(x) \to 2$ into the [[boolean domain]] for all elements $x \in A$, or equivalently if it comes with a function $\chi_P:A \to 2$ from $A$ to the [[boolean domain]] $2$.

Then the **internal LPO** for a [[family of sets]] $(A_z)_{z \in I}$ is the LPO for each $A_z$ stated in the internal logic of the set theory:

* For any internal predicate $i:P \hookrightarrow A_z$, if there is a function $\chi_P:A_z \to 2$, then the internal existential quantification of $P$, $\exists_{A_z} P = \im(!_P)$ has a function $(\exists_{A_z} P) \to 2$ into the boolean domain.

or equivalently, as

* For any function $a:A_z \to 2$, the internal existential quantification of $P = \{x \in A_z \vert a(x) = 1\}$, $\exists_{A_z} P = \im(!_P)$ has a function $(\exists_{A_z} P) \to 2$ into the boolean domain.

Similarly, the **internal WLPO** for a family of sets $(A_z)_{z \in I}$ is the WLPO for each $A_z$ stated in the internal logic of the set theory:

* For any internal predicate $i:P \hookrightarrow A_z$, if there is a function $\chi_P:A_z \to 2$, then the internal universal quantification of $P$, $\forall_{A_z} P = \{f \in P^{A_z} \vert \forall x \in A_z, f(x) \in i^*(x) \}$ has a function $(\forall_{A_z} P) \to 2$ into the boolean domain.

or equivalently

* For any function $a:A_z \to 2$, the internal universal quantification of $P = \{x \in A_z \vert a(x) = 1\}$, $\forall_{A_z} P = \{f \in P^{A_z} \vert \forall x \in A_z, f(x) \in i^*(x) \}$ has a function $(\forall_{A_z} P) \to 2$ into the boolean domain.

And finally, the **internal LLPO** for a family of sets $(A_z)_{z \in I}$ is the LLPO for each $A_z$ stated in the internal logic of the set theory:

* The internal LPO for a family of sets $(A_z)_{z \in I}$ holds only for the internal predicates $i:P \hookrightarrow A_z$ which comes with an internal predicate $j:Q \hookrightarrow A_z$ such that $i^*(x) = \neg j^*(x)$ for all $x \in A_z$.

In particular, the internal LPO for the family of all [[subsingletons]] is internal excluded middle and the internal LLPO for the family of all [[subsingletons]] is internal [[weak excluded middle]]. Similarly, given a [[universe]] $U$, the internal LPO for the family of all sets in $U$ is excluded middle in $U$, and the internal LLPO for the family of all sets in $U$ is weak excluded middle in $U$.

The internal versions of the principles of omniscience, like all internal versions of axioms, are weaker than the external version of the principle of omniscience, since while [[bounded separation]] implies that one can convert any external predicate $x \in A \vdash P(x)$ on a set $A$ to an internal predicate $\{x \in A \vert P(x)\} \hookrightarrow A$, it is generally not possible to convert an internal predicate to an external predicate without a reflection rule which turns subsingletons in the set theory into propositions in the external logic.

## Truncated and untruncated versions in homotopy type theory

In the context of [[homotopy type theory]], the various principles of omniscience can be translated in two ways, by interpreting "or" as [[propositional truncation|propositionally truncated]] ("merely or") or untruncated ("purely or"). The relationships between the truncated and untruncated principles of omniscience are as follows are:

* Truncated LPO and untruncated LPO are equivalent (due to [[Martin Escardo]], see [UFP](#UFP))

* Similarly, truncated WLPO and untruncated WLPO are equivalent.

* Untruncated LLPO is equivalent to WLPO (also due to Martin Escardo).

* In <http://www1.maths.leeds.ac.uk/~rathjen/Lifschitz.pdf> is a model by Michael Rathjen that separates WLPO from LLPO.

## Equivalent statements

There are various other results that are equivalent to the principles of omniscience. Here are a few:

* Let $[0,1]/(0 \sim 1)$ be the [[quotient space|quotient]] of the unit [[interval]] that identifies the endpoints, and let $\mathbb{R}/\mathbb{Z}$ be the [[quotient ring]]; both are classically isomorphic to the [[circle]] $\mathbb{S}^1$. (Constructively, we take $\mathbb{S}^1$ to be $\mathbb{R}/\mathbb{Z}$, although $S^1$ can also be constructed as a [[uniform completion|completion]] of $[0,1]/(0 \sim 1)$.)  Constructively, there is an [[injection]] $[0,1]/(0 \sim 1) \hookrightarrow \mathbb{R}/\mathbb{Z}$, which is a [[bijection]] if and only if the $LLPO$ holds (for the appropriate kind of real number).

* Every [[semi-decidable proposition]] is a [[decidable proposition]] iff $\mathrm{LPO}_\mathbb{N}$ holds.

## Related statements

There are various other results that are related to the principles of omniscience. Here are a few:

* In the presence of [[weak countable choice]], [[existential quantifier|there exists]] a [[radix notation|radix expansion]] in any base (e.g., a decimal expansion or binary expansion) for every [[Cauchy real number]] iff $\mathrm{LLPO}_\mathbb{N}$ holds; there exists a radix expansion for every [[Dedekind real number]] has iff the analytic $\mathrm{LLPO}$ holds. Without [[weak countable choice]], [[Lifschitz realizability]] gives a model in which $\mathrm{LLPO}_\mathbb{N}$ holds but it is not true that there exists a radix expansion in any base for every [[Cauchy real number]], which implies that there are models in which the analytic LLPO holds but it is not true that there exists a radix expansion in any base for every [[Dedekind real number]]. See [[Andrew Swan]]'s answer to [Birchfield (2024)](#Swan24). In addition, the analytic $\mathrm{LLPO}$ holds for the [[Dedekind real numbers]] in [[condensed sets]], but every function from the Dedekind real numbers to the [[boolean domain]] $\mathbb{2}$ is a [[constant function]], which implies that it is not true that there exists a radix expansion in any base for every [[Dedekind real number]].

* In the presence of [[countable choice]], $\mathrm{LLPO}_\mathbb{N}$ is equivalent to the claim that the rings of radix expansions in any two bases are isomorphic. See Daniel Mehkeri\'s answer to [Feldman (2010)](#Mehkeri10).

* That every Cauchy real number has a choice of [[radix notation|radix expansion]] in any base (e.g., a decimal expansion or binary expansion) implies that the $\mathrm{WLPO}_\mathbb{N}$ holds; that every Dedekind real number has a choice of radix expansion implies that the analytic $\mathrm{WLPO}$ holds.

* [[countable set|Subcountability]] of the [[real numbers]] implies the analytic $\mathrm{WLPO}$ because [[natural numbers]] have [[decidable equality]] and injections preserve and reflect decidable equality.

* The $\mathrm{WLPO}$ implies that the [[real numbers]] are [[uncountably indexed]].

## Models

* Assuming that [[Set]] is a [[Boolean topos]], then $LPO_{\mathbb{N}}$ (the LPO for natural numbers) holds in any [[presheaf topos]] over $Set$ and indeed in any [[locally connected topos]] over $Set$, essentially since then $2^N$ is a constant object.

* The LPO for natural numbers fails in Johnstone's [[topological topos]], due to its internal continuity principle.  Hence, the analytic LPO also fails, since the modulated Cantor reals and Dedekind reals coincide in this topos.  However, the (analytic) LLPO holds, as a consequence of the preservation of finite closed unions by the inclusion of sequential spaces.

## References

* {#Bauer11}  [[Andrej Bauer]] (2011) via constructivenews, [An Injection from $\mathbb{N}^{\mathbb{N}}$ to $\mathbb{N}$](http://math.andrej.com/wp-content/uploads/2011/06/injection.pdf) (pdf)

* {#Bishop67} [[Errett Bishop]] (1967), _[[Foundations of Constructive Analysis]]_ (in the introduction or chapter 1, I forget)

* {#Escardo11} [[Martín Escardó]] (2011) via constructivenews, [Infinite sets that satisfy the principle of omniscience in all varieties of constructive mathematics](http://www.cs.bham.ac.uk/~mhe/papers/omniscient-2011-07-06.pdf) (pdf)

* {#Mehkeri10} David Feldman (2010) on Math.Overflow, [Radix notation and toposes](http://mathoverflow.net/questions/49775/radix-notation-and-toposes/)

* {#Swan24} Madeleine Birchfield (2024) on Category Theory Zulip, [Radix expansions in constructive mathematics](https://categorytheory.zulipchat.com/#narrow/stream/229199-learning.3A-questions/topic/Radix.20expansions.20in.20constructive.20mathematics)

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

* {#Booij20} [[Auke Booij]], *Analysis in univalent type theory* (2020) [[etheses:10411]( 	http://etheses.bham.ac.uk/id/eprint/10411), [pdf](https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf)]

* {#RathjenSwan20} [[Michael Rathjen]], [[Andrew Swan]], *Lifschitz Realizability as a Topological Construction*. The Journal of Symbolic Logic, Volume 85, Issue 4, December 2020, pp. 1342 - 1375. [[doi:10.1017/jsl.2021.1](https://doi.org/10.1017/jsl.2021.1), [arXiv:1806.10047](https://arxiv.org/abs/1806.10047)]

* {#Rathjen13} [[Michael Rathjen]], *Constructive Zermelo-Fraenkel set theory and the limited principle of omniscience*. [[arXiv:1302.3037](https://arxiv.org/abs/1302.3037)]

* {#Richman90} [[Fred Richman]], *Polynomials and linear transformations*. Linear Algebra and its Applications, Volume 131, 1 April 1990, Pages 131-137. [doi:10.1016/0024-3795(90)90379-Q]

* {#Ishihara06} [[Hajime Ishihara]], _Reverse Mathematics in Bishop's Constructive Mathematics_, Philosophia Scientiæ, CS 6 (2006) ([doi:10.4000/philosophiascientiae.406](https://doi.org/10.4000/philosophiascientiae.406),  [pdf](https://philosophiascientiae.revues.org/pdf/406))

The analytic WLPO and LLPO are mentioned in the following paper, although unfortunately it uses the phrase "analytic LPO" for what should really be the analytic *WLPO* (the reals have decidable equality):

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

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principle of unique choice

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# Principle of Unique Choice
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## Idea

The __principle of unique choice__, also known as __function comprehension__, is the principle that every [[total relation|total]] [[functional relation|functional]] [[relation]] determines a (necessarily unique) function.

This is true in [[classical logic]] and most forms of [[intuitionistic logic]] (for example the [[internal language]] of any [[topos]]), but fails in weaker systems such as the internal language of a [[tripos]], or one of the internal languages of a [[quasitopos]].

## 2-Categorical Formulation

A total, functional relation is precisely an [[adjunction]] in the [[bicategory]] of relations [[Rel]]. So the principle of unique choice can be seen as a functor $Adj(Rel) \to Set$, in fact, an [[equivalence of categories]]. For this reason, bicategories modeling relations such as [[cartesian bicategories]] or [[allegories]] can be used to model functions as well by using the adjoint pairs of relations.

However, note that a generalization from [[Rel]] to the bicategory [[Prof]] of categories and [[profunctors]] is not possible: in general an adjoint pair of profunctors determines only a [[semifunctor]], which can be improved to a [[functor]] if the codomain category is [[Cauchy complete]]. For this reason, sometimes people use algebraic structures modeling profunctors such as [[proarrow equipments]] and [[virtual double categories]] that include notions of functor in addition to profunctor.

## In dependent type theory

In dependent type theory, the **type-theoretic principle of unique choice** states that the dependent product type of a family of contractible types is pointed.

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash p(x):\mathrm{isContr}(B(x))}{\Gamma\vdash \mathrm{uniquechoice}(\lambda x.p(x)):\prod_{x:A} B(x)}$$

The [[principle of unique choice]] is not equivalent to [[weak function extensionality]]. Weak function extensionality states that the dependent product type of a family of contractible types is contractible

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash p(x):\mathrm{isContr}(B(x))}{\Gamma\vdash \mathrm{weakfunext}(\lambda x.p(x)):\mathrm{isContr}\left(\prod_{x:A} B(x)\right)}$$

However, this is not the same principle of unique choice as in set theory. The **set-theoretic principle of unique choice** in states that every *proposition-valued* binary type family $x:A, y:B \vdash C(x, y)$ which is functional and total determines a necessarily unique function, and is only true for propositional-valued binary type families in [[dependent type theory]] for which the second index type $B$ is a set. Making the set-theoretic principle of unique choice hold for all types is tantamount to saying that all types are sets; thus the set-theoretic principle of unique choice makes the type theory into a [[set-level type theory]].

## Related concepts

* [[anafunction]]

## References

* *Homotopy Type Theory: Univalent Foundations of Mathematics*, The [[Univalent Foundations Project]], Institute for Advanced Study, 2013. ([web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf))

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prism

_This article is on prisms as a geometric shape. A different sense of 'prism' is studied in [[prismatic cohomology]]._

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## Overview

The category of __prisms__ is one of the [[geometric shapes for higher structures]].
The resulting [[presheaves]] of sets are known as [[multisimplicial sets]].
These were used by Laures and McClure in their work on [[Quinn spectra]].

## Definition

The category of $n$-fold prisms is the $n$-fold [[product]]
of the [[category of simplices]] $\Delta$.

[[Presheaves]] on this category are known as $n$-fold multisimplicial sets.

Restricting to injective maps of simplices,
we get $n$-fold multisemisimplicial sets.

## Related notions

* [[geometric shapes for higher categories]]

## References

* [[Gerd Laures]], [[James E. McClure]],
_Commutativity properties of Quinn spectra_,
([arXiv:1304.4759](https://arxiv.org/abs/1304.4759))

* [[Gerd Laures]], [[James E. McClure]],
_Multiplicative properties of Quinn spectra_,
([arXiv:0907.2367](https://arxiv.org/abs/0907.2367))

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prismatic cohomology

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## Idea

Prismatic cohomology is a cohomology theory for p-adic [[formal schemes]] which can specialize into various $p$-adic cohomology theories, including [[étale cohomology]], [[de Rham cohomology]] and [[crystalline cohomology]], as well as the so far conjectural $q$-de Rham cohomology of [[Peter Scholze]]. It is a geometric approach to integral [[p-adic Hodge theory]].

## Prisms

A prism is a pair $(A,I)$ where $A$ is a $\delta$-ring and $I$ is an ideal defining a Cartier divisor on $\mathrm{Spec}(A)$, such that $A$ is derived $(p,I)$-complete, and $p \in I + \phi(I)A$.

## Examples of Prisms

* $A=\mathbb{Z}_{p}[[u]]$ and $I=(u-p)$
* $A=A_{\mathrm{inf}}(R)$ and $I=\mathrm{ker}(\theta)$, where $\theta:A_{\mathrm{inf}}(R)\to R$ is the canonical surjection

## Relation to Perfectoid Rings

A prism $(A,I)$ is _perfect_ if $A$ is perfect. The category of perfect prisms is equivalent to the category of _integral perfectoid rings_ which are related to, but not the same as, the perfectoid rings in [[perfectoid spaces]]. The definition of integral perfectoid ring can be found in Definition 3.5 of  [BhattScholze19](#BhattScholze19) and the relation between the two notions of perfectoid can be found in Lemma 3.20 of the same paper.

## Definition of Prismatic Cohomology

Let $(A,I)$ be a prism as defined above. Let $R$ be a formally smooth $A/I$-algebra. The _prismatic site_ $(R/A)_{\Delta}$ has objects which are prisms $(B,I B)$ over $(A,I)$ together with a map $R\to B/I B$ over $A/I$. Such an object is written $(R\to B/I B\leftarrow B)$.

We have functors $\mathcal{O}_{\Delta}$ and $\overline{\mathcal{O}}_{\Delta}$ which sends $(R\to B/I B\leftarrow B)$ to $B$ and $B/I B$ respectively. The _prismatic cohomology_ of $R$ is defined to be $\Delta_{R/A}:=R\Gamma((R/A)_{\Delta},\mathcal{O}_{\Delta})$.

## Comparison theorems

We have the following comparison theorems relating prismatic cohomology to the crystalline and étale cohomology:

\begin{theorem}
Let $A,(p)$ be a bounded prism and let $R$ be a smooth $A/p$-algebra. There exists a canonical isomorphism
$$\phi_{A}^{*}\Delta_{R/A}^{\wedge}\cong R\Gamma_{crys}(R/A)$$
in $D(A)$, compatible with the action of Frobenius on both sides.
\end{theorem}

\begin{theorem}
Let $(A,I)$ be a perfect prism and let $R$ be a $p$-complete $A/I$-algebra. For all $n\geq 1$, there exists a canonical isomorphism
$$R\Gamma_{\et}(\mathrm{Spec}(R[1/p]),\mathbb{Z}/p^{n})\cong (\Delta_{R/A}[1/d]/p^{n})^{\phi=1}$$
where $d$ is a generator of $I$.
\end{theorem}

## Absolute Prismatic Cohomology

By forgetting the choice of base prism, one obtains the absolute prismatic cohomology. Given a p-adic formal scheme $X$, its absolute prismatic site ${X}_{\Delta}$ is the category of all bounded  prisms $(B,J)$ equipped with a map $\mathrm{Spf}(B/J)\to X$, topologized with the flat topology. We have sheaves $\mathcal{O}_{\Delta}$, $I_{\Delta}$, and $\overline{\mathcal{O}}_{\Delta}$ obtained by remembering $B$, $J$, and $B/J$ respectively.

The absolute prismatic cohomology $R\Gamma(X_{\Delta},\mathcal{O}_{\Delta})$ is equipped with a filtration called the _Nygaard filtration_, playing a role analogous to that of the [[Hodge filtration]] for [[de Rham cohomology]].

## Prismatic Crystals

Let $X$ be a p-adic formal scheme and let $X_{\Delta}$ be its absolute prismatic site as above. A _prismatic crystal_ is an assignment

$$(B,J)\in X_{\Delta}\mapsto \mathcal{E}(B)\in\mathrm{Vect}_{B}$$

where $\mathrm{Vect}_{B}$ is the set of finite projective $B$-modules.

A stacky approach to the study of prismatic crystals has been developed independently by Drinfeld ([Drinfeld20](#Drinfeld20)) and Bhatt-Lurie ([BhattLurie22](#BhattLurie22)).

Given p-adic formal scheme $X$, one can attach a stack, called the _Cartier-Witt stack_ and denoted $\mathrm{WCart}_{X}$ by Bhatt-Lurie, and called the _[[prismatization]]_ of $X$ and denoted $X^{\Delta}$ by Drinfeld. When $X$ is quasi-syntomic, one can identify the $\infty$-category of crystals of $(p,I_{\Delta})$-complete complexes on $(X_{\Delta},\mathcal{O}_{\Delta})$ with the derived $\infty$-category of quasi-coherent sheaves on $\mathrm{WCart}_{X}$ ([Bhatt21](#Bhatt21), Remark 2.6).

## Applications

Prismatic cohomology can be used to obtain the following inequality for $X$ a proper smooth [[formal scheme]] over $\mathbb{C}_{p}$ (here $k$ is the residue field of $\mathcal{O}_{\mathbb{C}_{p}}$):

$$dim_{\mathbb{F}_{p}}H_{et}^{i}(X_{\mathbb{C}_{p}},\mathbb{F}_{p})\leq dim_{k}H_{dR}^{i}(X_{k})$$

Prismatic cohomology has also been used in [Bhatt20](#Bhatt20) to prove that modulo a prime power the absolute integral closure of an excellent Noetherian domain is Cohen-Macaulay. The proof also uses a p-adic version of the [[Riemann-Hilbert correspondence]] being developed in yet-unpublished work of [[Bhargav Bhatt]] and [[Jacob Lurie]].

Prismatic cohomology has also been used to construct a version of [[syntomic cohomology]], which describes the graded pieces of a "motivic" filtration on p-adic etale K-theory. This is analogous to the filtration on [[topological K-theory]] whose graded pieces are described by the shifted singular cohomology complex, and is also analogous to the filtration on [[algebraic K-theory]] by [[motivic cohomology]]. This is used by Antieau, Krause and Nikolaus in [AKN22](#AKN22) to determine the K-groups of $\mathbb{Z}/p^{n}$.

## Historical Precursors

Prismatic cohomology was developed by [[Bhargav Bhatt]] and [[Peter Scholze]] following earlier work on developing an integral version of [[p-adic Hodge theory]]. Some of this earlier work includes $A_inf$-cohomology in [BhattMorrowScholze16](#BhattMorrowScholze16) and integral p-adic Hodge theory via [[topological Hochschild homology]] in [BhattMorrowScholze16](#BhattMorrowScholze16), both of which were developed together with [[Matthew Morrow]].

## Related entries

* [[p-adic Hodge theory]]
* [[perfectoid spaces]]
* [[topological Hochschild homology]]
* [[arithmetic differential geometry]]
* [[Borger's absolute geometry]]

## References

Prismatic cohomology was introduced in

* {#BhattScholze19}[[Bhargav Bhatt]], [[Peter Scholze]], _Prisms and Prismatic Cohomology_, preprint (2019) arXiv:[1905.08229](https://arxiv.org/abs/1905.08229)

A survey of recent developments is given in

* {#Bhatt21}[[Bhargav Bhatt]], _Algebraic Geometry in Mixed Characteristic_, preprint (2021) arXiv:[2112.12010](https://arxiv.org/abs/2112.12010)

Lecture notes include

* [[Bhargav Bhatt]], _Prismatic cohomology_, course at Columbia University, 2018 [lecture notes](http://www-personal.umich.edu/~bhattb/teaching/prismatic-columbia/)

Lecture notes taken by Chao Li, from the same lectures (contains some material not included in the above)

* [[Bhargav Bhatt]], _Prismatic cohomology_, course at Columbia University, 2018 (lecture notes taken by Chao Li) [lecture notes](http://www.math.columbia.edu/~chaoli/doc/BhattEilenberg.html)

Some other lectures on prismatic cohomology:

* [[Kiran Kedlaya]], _Notes on prismatic cohomology_, 2020 [lecture notes](https://kskedlaya.org/prismatic/prismatic.html)

* [[Matthew Emerton]], _Prismatic cohomology_, lecture notes (2020) [[webpage](http://www.math.uchicago.edu/~emerton/prismatic/prismatic.html)]

For some introductory comments see

* [[Terence Tao]], _Prismatic cohomology_, ([blog post](https://terrytao.wordpress.com/2019/03/19/prismatic-cohomology/))

* Riccardo Pengo ([MO answer](https://mathoverflow.net/a/333062/447))

Recent developments include

* {#Drinfeld20}[[Vladimir Drinfeld]], _Prismatization_ (2020) arXiv:[2005.04746] (https://arxiv.org/abs/2005.04746)

* {#BhattLurie22}[[Bhargav Bhatt]], [[Jacob Lurie]], _Absolute Prismatic Cohomology_, preprint (2022) arXiv:[2022.06120](https://arxiv.org/abs/2201.06120)

* [[Bhargav Bhatt]], [[Jacob Lurie]], _The Prismatization of $p$-adic Formal Schemes_, preprint (2022) arXiv:[2022.06124](https://arxiv.org/abs/2201.06124)

* {#Bhatt20}[[Bhargav Bhatt]], _Cohen-Macaulayness of Absolute Integral Closures_, preprint (2020) arXiv:[2008.08070](https://arxiv.org/abs/2008.08070)

* [[Jeremy Hahn]], [[Arpon Raksit]], [[Dylan Wilson]], _A motivic filtration on the topological cyclic homology of commutative ring spectra_, preprint (2022) arXiv:[2206.11208] (https://arxiv.org/abs/2206.11208)

An application to [[algebraic K-theory]] is

* {#AKN22} [[Benjamin Antieau]], [[Achim Krause]], [[Thomas Nikolaus]], _On the K-theory of $\mathbb{Z}/p^{n}$ -- Announcement_ [[arXiv:2204.03420](https://arxiv.org/abs/2204.03420)]

Historical precursors of prismatic cohomology are

* {#BhattMorrowScholze16}[[Bhargav Bhatt]], [[Matthew Morrow]], [[Peter Scholze]] *Integral $p$-adic Hodge Theory* [[arXiv:1602.03148](https://arxiv.org/abs/1602.03148)]

* {#BhattMorrowScholze18}[[Bhargav Bhatt]], [[Matthew Morrow]], [[Peter Scholze]], *Topological Hochschild Homology and Integral $p$-adic Hodge Theory* [[arXiv:1802.03261](https://arxiv.org/abs/1802.03261)]


prismatization

#Contents#
* table of contents
{:toc}

## Idea

Prismatization is a stacky approach to [[prismatic cohomology]] developed independently by Drinfeld in [#Drinfeld20] (#Drinfeld20) and Bhatt-Lurie in [#BhattLurie22](#BhattLurie22) and [#BhattLurie22b](#BhattLurie22b).

## Motivation: Linear algebra via stacks

The reference for this section is Section 2 of [#BhattNotes](#BhattNotes).

## Definitions

\begin{definition}(Definition 3.1.4 of [#BhattLurie22a](#BhattLurie22a), Definition 5.13 of [#BhattNotes](#BhattNotes))
Let $R$ be a p-nilpotent ring and $W(R)$ its ring of Witt vectors. A _Cartier-Witt divisor_ on $R$ is an invertible $W(R)$-module $I$ together with a morphism $\alpha:I\to W(R)$ of $W(R)$-modules such that

* The image of the map $I\to W(R)\twoheadrightarrow R$ is a nilpotent ideal of $R$.
* The image of the map $I\to W(R)\xrightarrow{\delta} W(R)$ generates the unit ideal of $W(R)$.

\end{definition}

\begin{definition}(Definition 5.1.6 of [#BhattNotes](#BhattNotes))
Let $X$ be a bounded $p$-adic formal scheme. The _prismatization_ $X^{\Delta}$ is the presheaf over $\mathrm{Spf}(\mathbb{Z}_{p})$ defined as follows. For a $p$-nilpotent ring $R$, $X^{\Delta}(R)$ is the groupoid of Cartier-Witt divisors on $R$ together with a map $\mathrm{Spec}(\mathrm{Cone}(W(R)/I))\to X$ of derived $\mathrm{Spf}(\mathbb{Z}_{p})$-schemes.
\end{definition}

## The Cartier-Witt stack

The prismatization $\mathrm{Spf}(\mathbb{Z}_{p})^{\Delta}$ of $\mathrm{Spf}(\mathbb{Z}_{p})$ is also called the _Cartier-Witt stack_, and denoted $\mathrm{WCart}$ or $\mathbb{Z}_{p}^{\Delta}$.

## Filtered prismatization

The reference for this section is Section 5.3 of [#BhattNotes](#BhattNotes).

\begin{definition}(Definition 5.2.4 of [#BhattNotes](#BhattNotes))
Let $R$ be a $p$-nilpotent ring. Let $W$ be the ring scheme of Witt vectors over $\mathrm{Spf}(\mathbb{Z}_{p})$. Let $F:W\to F_{*}W$ be the Frobenius. An _admissible $W$-module_ over $R$ is an affine $W$-module scheme $M$ which can be realized as an extension of a twisted form of $F_{*}M$ by a twisted form of the $\mathbb{G}_{a}^{#}$, where $\mathbb{G}_{a}^{#}$ is the PD-hull of the origin in $\mathbb{G}_{a}$ over $\mathbb{Z}$.
\end{definition}

\begin{definition} (Definition 5.3.1 of [#BhattNotes](#BhattNotes))
Let $R$ be a $p$-nilpotent ring. Let $W$ be the ring scheme of Witt vectors over $\mathrm{Spf}(\mathbb{Z}_{p})$. Let $F:W\to F_{*}W$ be the Frobenius. A _filtered Cartier-Witt divisor over $R$ is an admissible $W$-module $M$ over $R$ and a map $M\to W$ such that the induced map $F_{\ast}M'\to F_{\ast}W$ of twisted forms of $F_{\ast}W $ comes from a Cartier-Witt divisor.

\end{definition}

\begin{definition} (Definition 5.3.10 of [#BhattNotes](#BhattNotes))

Let $X$ be a bounded $p$-adic formal scheme. Define $\mathrm{Spf}(\mathbb{Z}_{p})^{\mathcal{N}}$ to be the stack which takes a $p$-nilpotent ring $R$ to the groupoid $\mathrm{Spf}(\mathbb{Z}_{p})^{\mathcal{N}}(R)$ of filtered Cartier-Witt divisors over $R$. The _filtered prismatization_ of $X$ is the stack $X^{\mathcal{N}}$ over $\mathbb{Z}_{p}^{\mathcal{N}}$ obtained via transmutation from $\mathbb{G}_{a}^{\mathcal{N}}\to\mathbb{Z}_{p}^{\mathcal{N}}$, where $\mathbb{G}_{a}^{\mathcal{N}}$ is the stack over $\mathbb{Z}_{p}^{\mathcal{N}}$ taking a filtered Cartier-Witt divisor $d:M\to W$ to $R\Gamma(\mathrm{Spec}(R),W/M)$.

\end{definition}

## References

* {#Drinfeld20}[[Vladimir Drinfeld]], _Prismatization_ (2020) arXiv:[2005.04746] (https://arxiv.org/abs/2005.04746)

* {#BhattLurie22a}[[Bhargav Bhatt]], [[Jacob Lurie]], _Absolute Prismatic Cohomology_, preprint (2022) arXiv:[2022.06120](https://arxiv.org/abs/2201.06120)

* {#BhattLurie22b}[[Bhargav Bhatt]], [[Jacob Lurie]], _The Prismatization of $p$-adic Formal Schemes_, preprint (2022) arXiv:[2022.06124](https://arxiv.org/abs/2201.06124)

* {#BhattNotes} [[Bhargav Bhatt]], _Prismatic F-gauges_, ([Lecture notes](https://www.math.ias.edu/~bhatt/teaching/mat549f22/lectures.pdf))


Prithvi Narayan

* [webpage](https://www.icts.res.in/people/prithvi-narayan)

## Selected writings

On [[weight systems]] on [[chord diagrams]] as [[single trace observables]] in [[SYK-model]]-like [[AdS-CFT duality|dual]] descriptions of [[black hole thermodynamics]] (see at [[weight systems on chord diagrams in physics]]):

* {#BerkoozNarayanSimon18} [[Micha Berkooz]], [[Prithvi Narayan]], [[Joan Simón]], Section 2.1 of: _Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction_, JHEP 08 (2018) 192 ([arxiv:1806.04380](https://arxiv.org/abs/1806.04380))

category: people


PRO

#Contents#
* table of contents
{:toc}

## Idea

A PRO, an abbreviation for "product category", is similar to a [[PROP]] but more general, being merely a monoidal category but not necessarily symmetric monoidal. They may be used to describe theories for finitary algebraic and coalgebraic structures which make sense in any monoidal category, for example monoids and comonoids.

## Definitions

A **PRO** is a strict monoidal category $T$ in which every object is of the form

$$x^{\otimes n} = \underset{n \: times}{\underbrace{x \otimes \ldots \otimes x}} \qquad (product \: of \: n \: copies \: of \: x)$$

for some $n \geq 0$. Somewhat more precisely, a PRO is a strict monoidal category for which the unique strict monoidal functor

$$(\mathbb{N}, +, 0) \to (T, \otimes, I)$$

is an isomorphism on objects.

+--{: .query}
[[Adam]]: Is there a variant where instead of a single object $x$ we start with some set of objects (or the objects of some other category) and require that any object of the new category is isomorphic to the tensor of zero or more objects from that set?  I suppose this would be a "multi-sorted PRO" or something like that, although you'd have to specify the base set.  Maybe "PRO generated by XYZ" or "PRO over XYZ" or some such thing.  If there's an established term for this, or something close to an established term, I'm keen on hearing about it.

RD : The term "coloured PROP" has been used.  See arXiv:1207.2773
=--

Morphisms $x^{\otimes m} \to x^{\otimes n}$ in a PRO are often thought of as operations which accept $m$ inputs and produce $n$ outputs, hence PROs are like nonpermutative operads but for the multiple outputs, which make them more general.

Let $C$ be a monoidal category. A **$T$-model** in $C$ (or a $C$-representation of $T$) is a strong monoidal functor $F: T \to C$; the _underlying object_ of the $T$-model is the value $F(x)$. A $T$-model is thus an object $c$ of $C$ equipped with operations

$$\theta_c: c^{\otimes m} \to c^{\otimes n},$$

one for each morphism $x^{\otimes m} \to x^{\otimes n}$, reflecting the monoidal category structure of $T$. A **homomorphism** $F \to G$ of $T$-models $F, G: T \to C$ is a monoidal transformation between $F$ and $G$; it may be equivalently expressed as a morphism $\phi: F(x) \to G(x)$ in $C$ which respects the modelings of the $T$-operations $\theta$.

## Examples

The [[augmented simplex category]] $\Delta_a$ is a PRO whose models are [[monoid]]s in monoidal categories. Similarly, $\Delta_a^{op}$ is a PRO whose models are [[comonoid]]s.

The [[cube]] category is a PRO whose models are objects $X$ equipped with "elements" $x_0, x_1: I \to X$ and a projection $p: X \to I$ which is a retraction of both $x_0$ and $x_1$, where $I$ denotes the monoidal unit.

The monoidal category of planar thickened 1d tangles (is this the right vocabulary?) is a PRO whose models are noncommutative Frobenius objects in monoidal categories.

## Variations

Just as the notion of [[PROP]] is a symmetric monoidal analogue of PRO, there is a braided monoidal analogue called a **PROB**. The cartesian monoidal analogue is known as a [[Lawvere theory]].

More generally, given any [[doctrine]] $D$, let $I(D)$ be an initial algebra of the doctrine. A PRO-D can then be defined as an algebra $T$ such that the algebra map $I(D) \to T$ induces an isomorphism on objects. This notion clearly violates the [[principle of equivalence]], and yet is a useful and general one in practice.

## References

An account of how the tensor product of PROs and the [[smash product]] of [[pointed spaces]] are two facets of the same construction is in

* [[Amar Hadzihasanovic]], _The smash product of monoidal theories_, ([arXiv:2101.10361](https://arxiv.org/abs/2101.10361))

[[!redirects PROs]]


pro-C-star-algebra

A __pro-C-star-algebra__ (also called locally C-star algebra) is very much like a [[pro-object]] in the category of [[C-star-algebra]]s, but the inverse limits are in fact taken in an ambient category of [[topological algebra|topological]] [[*-algebra]]s. $\sigma$-$C^*$-algebras are those pro-$C^*$-algebras which may be obtained as countable inverse limits (there are also intrinsic characterizations in terms of seminorms).

* N. C. Phillips, _Inverse limits of $C^\ast$-algebras_, J. Operator Theory 19 (1988) 159--195, [pdf](http://www.theta.ro/jot/archive/1988-019-001/1988-019-001-010.pdf); _Inverse limits of $C^\ast$-algebras_, in: Operator Algebras and Applications vol. 1, Structure Theory K-theory, Geometry and Topology (London Math. Soc. Lecture Note Series), D. E. Evans, M. Takesaki, eds.; _Representable K-theory for $\sigma$-$C^*$-algebras_, K-Theory __3__, 441–478 (1989)
* Massoud Amini, _Locally compact pro-$C^\ast$-algebras_, [math.OA/0205253](http://arxiv.org/abs/math/0205253)
* Maria Joița, _Hilbert modules over locally $C^*$-algebras_, thesis, Editura Universitatii Bucuresti 2006 [pdf](http://www.geocities.ws/m_joita/Hilbertm.pdf); _Crossed products of locally $C^*$-algebras_, Rocky Mountain J. Math. __37__:5 (2007) 1623-1644 [jstor](https://www.jstor.org/stable/44239423); _A new look at the crossed products of pro-$C^*$-algebras_; Annals of Functional Analysis, 2015; _On representations associated with completely n-positive linear maps on pro-$C^*$-algebras_, Chinese Annals of Mathematics, Series B, 2008; _Pro-$C^*$-algebras associated to tensor products of pro-$C^*$-correspondences_, J. Math. Anal. Appl.
__455__: 2 (2017) 1822-1834 [doi](https://doi.org/10.1016/j.jmaa.2017.06.065); Crossed Products of pro-$C^*$-algebras and Hilbert pro-$C^*$-modules_, M. Bull. Malays. Math. Sci. Soc. (2015) 38:1053 [doi](https://doi.org/10.1007/s40840-014-0067-z)
* Maria Joiţa, Ioannis Zarakas, _Crossed products by Hilbert pro-$C^*$-bimodules_, Studia Mathematica 215 (2013), 139-156 [doi](http://dx.doi.org/10.4064%2Fsm215-2-4); _A construction of pro-$C^*$-algebras from pro-$C^*$-correspondences_, J. Operator Theory __74__:1 (2015) 195-211, [doi](http://dx.doi.org/10.7900/jot.2014may27.2025)
* Rachid El Harti, N. Christopher Phillips, Paulo R. Pinto, _Profinite pro-$C^\ast$-algebras and pro-$C^\ast$-algebras of profinite groups_, [arxiv/1110.3411](http://arxiv.org/abs/1110.3411)
* Snigdhayan Mahanta, _Noncommutative correspondence categories, simplicial sets and pro-$C^\ast$-algebras_, [arxiv/0906.5400](http://arxiv.org/abs/0906.5400)
* Ilan Barnea, Michael Joachimand Snigdhayan Mahanta, _Model structure on projective systems of $C^*$-algebras and bivariant homology theories_, New York J. Math. __23__ (2017) 383–439 [pdf](http://emis.ams.org/journals/NYJM/j/2017/23-19v.pdf)
* _Atiyah-Jänich Theorem for σ-C*-algebras_ 2017, Applied Categorical Structures __25__:5 (2017) 893–905 [doi](https://doi.org/10.1007/s10485-016-9474-7)

There is a functor $(-)_b:pro-C^\ast\to C^\ast$ which is the dual of Stone-Čech compactification:

* Rachid El Harti, Gábor Lukács, _Bounded and unitary elements in pro-$C^\ast$-algebras_, Applied Categorical Structures  __14__:2, 2006, 151--164 [doi](http://dx.doi.org/10.1007/s10485-006-9012-0) [math.CT/0511068](https://arxiv.org/pdf/math/0511068)

A class of topological algebras which correspond to pro-$C^\ast$-algebras can be characterized in terms of properties of the topological algebra:

* S.J. Bhatt, D.J. Karia, _An intrinsic characterization of pro-$C^\ast$-algebras and its applications_, J. Math. Anal. Appl. __175__:1 (1993) 68–80, [doi](http://dx.doi.org/10.1006/jmaa.1993.1152)


pro-homotopy theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}

##Idea

Pro-homotopy theory involves the study of [[model categories]] and other abstract [[homotopy theory|homotopy theoretic]] structure on pro-categories of [[topological spaces]] or [[simplicial sets]]. (The term can also be used for any extension of homotopical structures for a category $C$ to the corresponding category $Pro(C)$ of [[pro-objects]] in $C$.)

## Uses

* In [[strong shape theory]],

* In [[etale homotopy theory|étale homotopy theory]] and motivic homotopy.

*  In [[proper homotopy theory]]

and is closely related to [[profinite homotopy theory]].

## Related concepts

* [[profinite space]]

* [[simplicial profinite space]]

* [[pro-left adjoint]]

The homotopy theory of simplicial profinite spaces has been explored by [[Fabien Morel]] and [[Gereon Quick]].

For Morel's theory see

* [[F. Morel]], _Ensembles profinis simpliciaux et interprétation géométrique du foncteur $T$_, Bull. Soc. Math. France, 124, (1996), 347–373,

The initial reference to Quick's work is :

* G. Quick, _Profinite homotopy theory_, [PDF](https://eudml.org/doc/55986)

but a correction to an error in the proof of the main result was included in

* G. Quick,  _Continuous group actions on profinite spaces_, J. Pure Appl. Algebra 215 (2011),
1024-1039.

##References

For one of the earliest model structures, namely the strict model structure on $Pro(C)$, see

* [[D.A. Edwards]] and [[H. M. Hastings]], (1976), _Čech and Steenrod homotopy theories with applications to geometric topology_, Lecture Notes in Maths. 542, Springer-Verlag, [pdf](http://www.math.uga.edu/~davide/Cech_and_Steenrod_Homotopy_Theories_with_Applications_to_Geometric_Topology.pdf)

More recent contributions include:

* [[I. Barnea]] and [[T. M. Schlank]], 2011, _A Projective Model Structure on
Pro Simplicial Sheaves, and the Relative Etale Homotopy Type_, [arXiv:1109.5477](http://arxiv.org/abs/1109.5477)

*  [[I. Barnea]] and [[T. M. Schlank]], 2013, _Functorial Factorizations in Pro
Categories_, [arXiv:1305.4607](http://arxiv.org/abs/1305.4607).

*  [[D. C. Isaksen]], _A model structure on the category of pro-simplicial sets_,
Trans. Amer. Math. Soc., 353, (2001), 2805–2841.

* [[D. C. Isaksen]], _Calculating limits and colimits in pro-categories_, Fundamenta Mathematicae, 175, (2002), 175 – 194.

* [[D. C. Isaksen]], 2004, Strict model structures for pro-categories , in Categorical decomposition techniques in algebraic topology (Isle of Skye, 2001) , volume 215 of Progr. Math., 179 – 198, Birkhauser, Basel.

* [[D. C. Isaksen]], Completions of pro-spaces , Math. Z., 250, (2005), 113
– 143.

* {#FauskIsaksen07} [[Halvard Fausk]], [[D. C. Isaksen]], _Model structures on pro–categories_, Homology, Homotopy and Applications, Vol. 9 (2007), 367–398 ([arXiv:math/0511057](https://arxiv.org/abs/math/0511057))

* Ilan Barnea, [[Yonatan Harpaz]], [[Geoffroy Horel]], _Pro-categories in homotopy theory_ ([arXiv:1507.01564](http://arxiv.org/abs/1507.01564))


pro-left adjoint

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

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

For $R \,\colon\, \mathcal{D} \to \mathcal{C}$ a [[functor]] which [[preserved limit|preserves]] [[finite limits]], the [[left adjoint]] [[profunctor]]

\[
  \label{FormulaForLeftAdjointProfunctor}
  \array{
    \mathcal{C}
    &\longrightarrow&
    Func(\mathcal{D}, Set)^{op}
    \\
    c
    &\mapsto&
    \Big(
    d
      \mapsto
      \mathcal{C}
      \big(
        c
        ,\,
        R(d)
      \big)
    \Big)
  }
\]

factors through the [[pro-objects]] $Pro(\mathcal{D})$. This $\mathcal{C}\to Pro(\mathcal{D})$ is called the _pro-left adjoint_ to $R$. Its extension $L\colon Pro(\mathcal{C})\to Pro(\mathcal{D})$ is a genuine [[left adjoint]] to $Pro(R)$, the _[[proadjoint]]_.

The analogous definition applies in the generality of [[(infinity,1)-category theory|$(\infty,1)$-category theory]] -- if [[Set]] is replaced by [[∞Grpd]] in (eq:FormulaForLeftAdjointProfunctor) -- to yield the notion of pro left-adjoint [[(∞,1)-functors]], generalizing [[left adjoint|left]] [[adjoint (∞,1)-functors]].

## Example

### Pro-étale homotopy type

Consider any [[(∞,1)-sheaf ∞-topos]] with its [[global sections]] [[geometric morphism]]

$$
  \mathbf{H}
   \stackrel{\overset{\Delta}{\longleftarrow}}{\underset{\Gamma}{\longrightarrow}}
  \infty Grpd
 \,.
$$

If here the [[constant ∞-stack]] [[inverse image]] $\Delta$ does have a further [[left adjoint|left]] [[adjoint (∞,1)-functor]] $\Pi$ (so that $\mathbf{H}$ is a [[locally ∞-connected (∞,1)-topos]]) then $\Pi$ may be understood as taking any object to its [[fundamental ∞-groupoid]] or [[geometric realization]] [[homotopy type]].

In general $\Pi$ does not exist, but the pro-left adjoint $\Pi_{pro}$ of $\Delta$ may always be formed. This sends any object to what in the case of the $\infty$-topos over an [[étale site]] is called its _[[étale homotopy type]]_.

In general $\Pi_{pro}$ sends the [[terminal object]] to the [[shape of an (∞,1)-topos|shape of the $(\infty,1)$-topos]] of $\mathbf{H}$.

##Related topics

* [[proadjoint]]

* [[prorepresentable functor]]

## References

Discussion in the context of [[shape of an (infinity,1)-topos|shape of an $(\infty,1)$-topos]]:

* {#Hoyois13} [[Marc Hoyois]], p. 3 of _Higher Galois theory_, Algebraic & Geometric Topology __17__ 1 (2017) 567-643 ([arxiv/1506.07155](https://arxiv.org/abs/1506.07155), [doi:10.2140/agt.2017.17.567](https://doi.org/10.2140/agt.2017.17.567))

See also:

* E. Giuli, Pro-reflective subcategories, J. Pure Appl. Algebra 33 (1984) 19–29.

[[!redirects pro-left adjoints]]


pro-manifold

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

#Contents#
* table of contents
{:toc}

## Idea

A _pro-manifold_ is a [[pro-object]] in a [[category]] of [[manifolds]], i.e. a formal [[projective limit]] of [[manifolds]].

Details depend on what exactly is understood by "[[manifold]]", i.e. whether [[topological manifolds]] or [[smooth manifold]], etc.

Typically one wants to mean [[pro-objects]] in manifolds of [[finite number|finite]] [[dimensions]], the point being then that a pro-manifold is like an [[infinite-dimensional manifold]] but with "mild" infinite dimensionality, expressed by the very fact that it may be presented as a formal projective limit of finite dimensional manifolds.

To amplify this specification, one should properly speak of "pro-(finite dimensional smooth manifolds)", but beware that people often abbreviate to "pro-manifold" regardless. Also "pro-finite manifold" is in use, which however, strictly speaking, is a misnomer since a "finite manifold" is one with a [[finite number]] of points.

An important example of pro-objects in finite-dimensional smooth manifolds are infinite [[jet bundles]]. These are the formal projective limits of the underlying finite-order jet bundles.

## Pro-Cartesian spaces

### Embedding into smooth loci

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

Write [[CartSp]] for the [[full subcategory]] of that of [[smooth manifolds]] on the [[Cartesian spaces]], i.e. on those of the form $\mathbb{R}^n$, for $n \in \mathbb{N}$. Write

$$
  ProCartSp \coloneqq Pro(CartSp)
$$

for its category of [[pro-objects]], the **pro-Cartesian spaces**.

=--

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

The functor which sends a formal cofiltered limit of Cartesian spaces to its actual [[cofiltered limit]] of [[smooth loci]] is a [[fully faithful functor]], hence constitutes a [[full subcategory]] inclusion of [[pro-Cartesian spaces]] (def. \ref{proCartSp}) into [[smooth loci]]:

$$
  Pro(CartSp) \hookrightarrow SmthLoc
  \,.
$$

=--

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

Since $Pro(\mathcal{C}) \simeq (Ind(\mathcal{C}^{op}))^{op}$ ([remark](pro-object#ProObjectAsFormalDualOfIndObject)) it is sufficient to show that the functor in question is on opposite categories a fully faithful functor of the form

$$
  Ind(CartSp^{op})
    \hookrightarrow
  SmthLoc^{op} = SmthAlg_{\mathbb{R}}
  \,,
$$

where $SmothAlg_{\mathbb{R}}$ is the category of [[smooth algebras]].

Now, there is the [[fully faithful functor]]

$$
  i \;\colon\; CartSp \hookrightarrow SmthLoc
$$

([prop.](smooth+locus#SmoothManifoldsFullSubcategoryOfSmoothLoci)) hence a fully faithful functor

$$
  i^{op} \colon CartSp^{op} \hookrightarrow SmthAlg_{\mathbb{R}}
  \,.
$$

Moreover, the image of the latter is in [[compact objects]] $i^{op} \colon CartSp^{op} \hookrightarrow (SmthAlg_{\mathbb{R}})_{cpt} \hookrightarrow SmthAlg$, because

$$
  C^\infty(\mathbb{R}^n) \simeq y(\mathbb{R}^n)
  \in
  SmthAlg_{\mathbb{R}} \simeq Func_\times(CartSp,Set)
$$

is [[representable functor|co-representable]], hence [[compact object|compact]] (by the [[Yoneda lemma]] and since colimits are computed objectwise [prop.](limits+and+colimits+by+example#LimitsOfPresheaves)).

This implies that the composite

$$
  Ind(CartSp^{op})
   \overset{Ind(i^{op})}{\hookrightarrow}
  Ind(SmthAlg_{\mathbb{R}})
    \overset{L}{\longrightarrow}
  SmthAlg_{\mathbb{R}}
$$

is also fully faithful ([prop.](ind-object#JFIsFullyFaithful)).

Here $Ind(i^{op})$ takes formal filtered colimits in $CartSp^{op}$ to the corresponding formal colimits in $SmthAlg_{\mathbb{R}}$ ([prop.](ind-object#FunctorialityOfInd)), while $L$ takes formal filtered colimits to actual [[filtered colimits]] ([prop.](ind-object#ReflectionToYonedaEmbedding)). Hence this is indeed the functor in question.

=--

### The site of towers of Cartesian spaces and pro-morphisms
 {#SiteOfTowersOfCartesianSpaces}

> under construction

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

Write

$$
  TowCartSp \hookrightarrow ProCartSp
$$

for the [[full subcategory]] of the category of [[pro-Cartesian spaces]] (def. \ref{proCartSp}) on those [[pro-objects]] in [[CartSp]] which are presented as formal [[sequential limits]] of [[tower]] diagrams, i.e. where the indexing category $\mathcal{K} = \mathbb{N}_{\geq}$.

=--

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

For $U \in TowCartSp$ a [[tower of Cartesian spaces]]  (def. \ref{TowersOfCartesianSpaces}), say that a **tower of [[good open covers]]** of $U$ is a sequence of morphisms $\{U_i \overset{\phi_i}{\to} U\}$ in $TowCartSp$ such that these are the formal [[sequential limit]] of a cofiltered diagram of [[good open covers]] $\{U_i^k \overset{\phi_i^k}{\to} U^k\}$.

$$
  \array{
    U_i^{k} &\overset{\underset{\longleftarrow}{\lim}^f}{\mapsto}& U_i
    \\
    {}^{\mathllap{\phi_i^k}}\downarrow && \downarrow^{\mathrlap{\phi_i}}
    \\
    U^k &\overset{\underset{\longleftarrow}{\lim}^f}{\mapsto}& U
  }
$$

=--

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

The collection of towers of [[good open covers]] on $TowCartSp$, according to def. \ref{ProGoodOpenCoverOnACartesianSpace}, constitutes a [[coverage]].

=--

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

By the definition of [[coverage]] ([def.](coverage#ConditionsOnACoverage)) we need to check that for every tower of [[good open covers]] $\{U_i \overset{\phi_i}{\to} U\}$ and for every morphism $V \overset{g}{\longrightarrow} U$ in $TowCartSp$, there exists a tower of [[good open covers]] $\{V_j \overset{\psi_j}{\longrightarrow} V\}$ of $V$ such that for each index $j$ we may find an index $i$ and a morphism $V_j \overset{}{\to} U_i$ such as to make a [[commuting diagram]] of the form

$$
  \array{
    V_j &\overset{}{\longrightarrow}& U_i
    \\
    \downarrow
      &&
    \downarrow^{\mathrlap{\phi_i}}
    \\
    V &\underset{g}{\longrightarrow}& U
  }
  \,.
$$

Now by [this prop.](tower#ProMorphismsBetweenTowerDiagrams) the bottom morphism is represented by a sequence of component morphisms

$$
  V^{n(k)} \overset{}{\longrightarrow} U^k
  \,.
$$

Since ordinary [[good open covers]] do form a [[coverage]] on [[CartSp]] ([prop.](good+open+cover#GoodOpenCoversFormACoverageOnParacompactSmooothManifolds)) each of these component diagrams may be completed

$$
  \array{
    \tilde V^{n(k)}_j &\overset{}{\longrightarrow}& U^k_i
    \\
    \downarrow
      &&
    \downarrow^{\mathrlap{\phi^k_i}}
    \\
    V^{n(k)} &\underset{g^k}{\longrightarrow}& U^k
  }
$$

by first forming the [[pullback]] [[open cover]] $(g^k)^\ast U^k_i \to V^{n(k)}$ and then refining this to a [[good open cover]] $\tilde V^{n(k)}_j \to V^{n(k)}$. By the [[universal property]] of the [[pullback]], there are morphisms

$$
  \tilde V^{n(k+1)} \longrightarrow (g^k)^\ast U^k_i
$$

that make the evident cube commute

$$
  \array{
    \tilde V^{n(k+1)}_j &\overset{}{\longrightarrow}& U^{k+1}_i
    \\
    \downarrow
      &&
    \downarrow^{\mathrlap{\phi^{k+1}_i}}
    \\
    V^{n(k+1)} &\underset{g^{k+1}}{\longrightarrow}& U^{k+1}
  }
  \;\;\;\;\;\;\;\;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;\;\;\;\;\;\;\;
  \array{
    (g^{k})^\ast U_i^k &\overset{}{\longrightarrow}& U^k_i
    \\
    \downarrow
      &&
    \downarrow^{\mathrlap{\phi^k_i}}
    \\
    V^{n(k)} &\underset{g^k}{\longrightarrow}& U^k
  }
$$

Take

$$
  V^{n(0)}_j \coloneqq \tilde V^{n(0)}_j
$$

and then inductively define

$$
  V^{n(k+1)}_j
$$

to be a refinement by a good open cover of the joint refinement of $\{\tilde V^{n(k+1)}_j\}$ with the pullback of $\{V^{n(k)}_j\}$ to $V^{n(k+1)}$.

This refines the above commuting cubes to

$$
  \array{
    V_j^{n(k+1)} &\overset{}{\longrightarrow}& U^{k+1}_i
    \\
    \downarrow
      &&
    \downarrow^{\mathrlap{\phi^{k+1}_i}}
    \\
    V^{n(k+1)} &\underset{g^{k+1}}{\longrightarrow}& U^{k+1}
  }
  \;\;\;\;\;\;\;\;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;\;\;\;\;\;\;\;
  \array{
    V_j^{n(k)} &\overset{}{\longrightarrow}& U^k_i
    \\
    \downarrow
      &&
    \downarrow^{\mathrlap{\phi^k_i}}
    \\
    V^{n(k)} &\underset{g^k}{\longrightarrow}& U^k
  }
$$

and hence provides components for the required diagram in $TowCartSp$.

=--

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