scissors congruence

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

#Contents#
* table of contents
{:toc}

## Idea

Two [[polygons]] $P,P'$ in the [[Euclidean space|Euclidean plane]] have the same [[area]] iff they are __scissors congruent__ in the sense that they can be subdivided into _finitely_ many pieces such that the pieces of $P$ are congruent to the pieces of $P'$.

__Hilbert's third problem__ (see _[[Hilbert's problems]]_) was to decide whether the analogue of this elementary fact holds for [[polyhedra]] in 3-[[dimension|dimensional]] space. Dehn solved this problem using what is now called the Dehn invariant. In modern language, it assigns to a polyhedron $P$ an element in $\mathbf{R}\otimes_{\mathbf{Z}} \mathbf{R}/\mathbf{Z}$. If both the Dehn invariant and volume of two (3-dimensional finite) polyhedra in Euclidean space are equal then they are scissors congruent.

The __generalized Hilbert's third problem__  asks whether two finite $n$-[[polytopes]] in Euclidean, spherical or hyperbolic $n$-space must be scissors equivalent if they have the same volume and generalized invariants.

The __scissors congruence group__  $\mathcal{P}(X,G)$ where $G$ is a subgroup of the group of isometries of $X$, is the free abelian group on symbols $[P]$, for all polytopes in $X$ modulo the relations

(i) $[P]-[P']-[P'']$ when $P = P'\coprod P''$,

(ii) $[g P]-[P]$.

In the book

* J.L. Dupont, Scissors congruences, group homology and characteristic classes (.ps) , Lecture notes from Nankai Institute of Mathematics, 1998, Manuscript, University of Aarhus, Department of Math., Aarhus, 1999, 122 pp., and in: Nankai Tracts in Mathematics 1, World Scientific, Singapore, 2001.

Dupont explains the relations of dilogarithms, Reidemeister torsion, Dehn invariant (which is from 1900) and their role in understanding the 19th century scissors congruence...The subject is very active now. After physicist Anatole Kirillov found the [[quantum dilogarithm]] and L. D. Fadeeev and Kashaev published a paper about it (and discovered the pentagon relation), this became an important topic in quantization (Fadeev's modular double, quantization of Teichmuller spaces, work of A. Fock, L. Takhdajan, Aldrovandi etc.) of hyperbolic spaces and generally the hyperbolic geometry and with relations, together with classical [[dilogarithm]] to many other subjects (e.g. to the work of Reznikov and theory or regulators, specially Borel regulator in higher algebraic K-theory, important for study of motives (Goncharov et al.) and, as W. Nahm shown, also relevant for understanding some phenomena in CFT).

## References

### Scissor congruence

* C. H. Sah, _Hilbert's third problem: scissors congruence_, Research Notes in Mathematics __33__, Pitman 1979.

* Inna Zakharevich, _Scissors congruence as K-theory_, [arxiv/1101.3833](http://arxiv.org/abs/1101.3833)

* J.-P. Sydler, _Conditions nécessaires et suffisantes pour l'équivalence des polyèdres de l'espace euclidean à trois dimensions_, Comment. Math. Helv. 40, 43-80, 1965.

* Wikipedia, *[Hilbert's third problem](https://en.wikipedia.org/wiki/Hilbert%27s_third_problem)*

### Cut-and-paste of manifolds
 {#CutAndPasteOfManifolds}

Generalization to cut-and-paste of [[smooth manifolds]] (SK-groups, D: *Schneiden-und-Kleben Gruppen*):

* {#KarrasKreckNeumannOssa73} U. Karras, [[Matthias Kreck]], [[Walter D. Neumann]], E. Ossa, *Cutting and Pasting of Manifolds; SK-Groups*, Publish or Perish (1973) [[pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/skbook.pdf), [[Karras-Kreck-Neumann-Ossa_CutNPaste.pdf:file]]]

* [[Walter D. Neumann]], *Manifold cutting and pasting groups*, Topology **14** 3 (1975) 237-244 [doi:10.1016/0040-9383(75)90004-X]

Relation to [[TQFTs]]:

* [[Carmen Rovi]], Matthew Schoenbauer, *Relating Cut and Paste Invariants and TQFTs*, The Quarterly Journal of Mathematics **73** 2 (2022) 579–607 [[arXiv:1803.02939](https://arxiv.org/abs/1803.02939), [doi:10.1093/qmath/haab044](https://doi.org/10.1093/qmath/haab044)]

Survey:

* [[Carmen Rovi]], *Relating cut and paste invariants and TQFTS*, [talk](Center+for+Quantum+and+Topological+Systems#RoviArp2023) at [[CQTS]] (Apr. 2023) [video: [YT](https://www.youtube.com/watch?v=lD8wQhwyrdg)]

[[!redirects scissor congruences]]

[[!redirects Hilbert's third problem]]

[[!redirects cutting and pasting of manifolds]]
[[!redirects cut-and-paste of manifolds]]

[[!redirects SK-group]]
[[!redirects SK-groups]]

[[!redirects SK invariant]]
[[!redirects SK invariants]]

[[!redirects SK-invariant]]
[[!redirects SK-invariants]]

[[!redirects SKK invariant]]
[[!redirects SKK invariants]]

[[!redirects SKK-invariant]]
[[!redirects SKK-invariants]]


Scott Aaronson

* [webpage](http://www.scottaaronson.com)

## Selected writings

On [[quantum information theory]]:

* [[Scott Aaronson]], *Introduction to Quantum Information Science* (2018) [[pdf](https://www.scottaaronson.com/qclec.pdf), [webpage](https://www.scottaaronson.com/cs378/)]

  *Introduction to Quantum Information Science II* (2022) [[pdf](https://www.scottaaronson.com/qisii.pdf)]

On [[quantum complexity theory]]:

* [[Scott Aaronson]], *Quantum computing, postselection, and probabilistic polynomial-time*, Proceedings of the Royal Society A, Volume 461, Issue 2063, pages 3473–3482, 08 November 2005. ([doi:10.1098/rspa.2005.1546](https://doi.org/10.1098%2Frspa.2005.1546), [arXiv:quant-ph/0412187](https://arxiv.org/abs/quant-ph/0412187))

On [[complexity theory|complexity theoretic]] aspects of [[quantum supremacy]]:

* [[Scott Aaronson]], [[Lijie Chen]], *Complexity-Theoretic Foundations of Quantum Supremacy Experiments*, CCC '17: Proceedings of the 32nd Computational Complexity Conference (2017) **22** 1–67 [[arXiv:1612.05903](https://arxiv.org/abs/1612.05903), [doi;10.5555/3135595.3135617](https://dl.acm.org/doi/10.5555/3135595.3135617)]

* [[Scott Aaronson]], *How Much Structure Is Needed for Huge Quantum Speedups?*, talk at: *28th Solvay Physics Conference* in Brussels on May 21, 2022 [[arXiv:2209.06930](https://arxiv.org/abs/2209.06930), slides: [ppt](https://www.scottaaronson.com/talks/aar-solvay.ppt)]

On the [[Ryu-Takayanagi formula]]:

* [[Scott Aaronson]], [[Jason Pollack]], *Discrete Bulk Reconstruction* [[arXiv:2210.15601](https://arxiv.org/abs/2210.15601)]

## Related entries

* [[quantum computation]]

category: people


Scott Axelrod

* [genealogy entry](http://genealogy.math.ndsu.nodak.edu/id.php?id=65308)

category: people


Scott Balchin

* [personal page](http://bifibrant.com/)

## Selected writings

On [[model categories]]:

* [[Scott Balchin]], *A Handbook of Model Categories*, Algebra and Applications, **27** Springer (2021) $[$[doi:10.1007/978-3-030-75035-0](https://doi.org/10.1007/978-3-030-75035-0)$]$

category: people


Scott Carnahan

* [webpage](http://member.ipmu.jp/scott.carnahan/)

category: people


Scott Carter

* [webpage](http://www.southalabama.edu/mathstat/personal_pages/carter/)

## Related entries

* [[3d TQFT]], [[4d TQFT]]

* [[quantum gravity]]

category: people

[[!redirects J. Scott Carter]]


Scott Collier

* [institute page](https://pcts.princeton.edu/people/scott-collier)

* [GoogleScholar page](https://scholar.google.com/citations?user=S8hOpd0AAAAJ&hl=en)

## Selected writings

On 3-dimensional ([[quantum gravity|quantum]]) [[gravity]] ([[general relativity]]) with [[cosmological constant]], and its (non-)relation to [[Chern-Simons theory]] with non-[[compact Lie group|compact]] [[gauge groups]]:

* [[Scott Collier]], [[Lorenz Eberhardt]], [[Mengyang Zhang]], *3d gravity and Chern-Simons theory -- references* [[arXiv:2304.13650](https://arxiv.org/abs/2304.13650)]

category: people


Scott Fleischman

* [Azimuth page](https://forum.azimuthproject.org/discussion/comment/15892)

* [Twitter account](https://twitter.com/scottfleischman)

## Selected work

Introduction to [[Agda]]:

* [[Scott Fleischman]], *Agda from Nothing*, talk at λC (2016) [[video 1](https://www.youtube.com/watch?v=-i-QQ36Nfsk&list=PLCqzQM-xq02Uv4OmcckrFsIyZ20hWEeAx&index=7), [video 2](https://www.youtube.com/watch?v=XprGyVWXwks)]

category: people


Scott Morrison

* [website](http://tqft.net/)

## Selected writings

On [[monoidal enriched categories]]:

* [[Scott Morrison]], [[David Penneys]], *Monoidal Categories Enriched in Braided Monoidal Categories*, International Mathematics Research Notices **2019** 11 June 2019 3527–3579 [[doi:10.1093/imrn/rnx217](https://doi.org/10.1093/imrn/rnx217), [arXiv:1701.00567](https://arxiv.org/abs/1701.00567)]

## Related entries

* [[blob homology]]

category: people


Scott Tanona

* [webpage](http://www.k-state.edu/philos/faculty/tanona.html)

category: people


Scott topology

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

#Contents#
* table of contents
{:toc}

## Definition

The __Scott topology__ on a [[preordered set]] is the [[topological space|topology]] in which the [[open subsets]] (called __Scott-open__) are precisely those whose [[characteristic functions]] (from the given preorder into the preorder of [[truth values]]) preserve [[directed joins]] (and this makes them necessarily [[monotone function|monotonic]]).

This in fact ensures that, in general, the [[continuous functions]] between preorders with Scott topologies are precisely those (necessarily monotonic) functions between them which preserve directed joins (called __Scott-continuous__). The [[poset of truth values]] itself, therefore, when equipped with the Scott topology, becomes the open-set [[classifying space|classifier]], [[Sierpinski space]].

## Properties

### As injective objects in $T_0$-spaces

In the category of $T_0$ [[topological spaces]] (see at _[[separation axiom]]_), the [[injective objects]] are precisely those given by Scott topologies on [[continuous lattices]]; as [[locales]] these are [[local compactum|locally compact]] and [[spatial locale|spatial]].

## Related entries

* [[domain theory]]

* [[specialization topology]]

* [[order topology]]

* [[Scott topos]]

* [[Dana Scott]]

[[!redirects Scott topology]]
[[!redirects Scott topologies]]

[[!redirects Scott open]]
[[!redirects Scott-open]]
[[!redirects Scott open subset]]
[[!redirects Scott open subsets]]
[[!redirects Scott-open subset]]
[[!redirects Scott-open subsets]]
[[!redirects Scott open set]]
[[!redirects Scott open sets]]
[[!redirects Scott-open set]]
[[!redirects Scott-open sets]]
[[!redirects Scott open subspace]]
[[!redirects Scott open subspaces]]
[[!redirects Scott-open subspace]]
[[!redirects Scott-open subspaces]]

[[!redirects Scott continuity]]
[[!redirects Scott continuous]]
[[!redirects Scott-continuous]]
[[!redirects Scott continuous function]]
[[!redirects Scott continuous functions]]
[[!redirects Scott-continuous function]]
[[!redirects Scott-continuous functions]]
[[!redirects Scott continuous map]]
[[!redirects Scott continuous maps]]
[[!redirects Scott-continuous map]]
[[!redirects Scott-continuous maps]]
[[!redirects Scott continuous mapping]]
[[!redirects Scott continuous mappings]]
[[!redirects Scott-continuous mapping]]
[[!redirects Scott-continuous mappings]]


Scott topos

[[!redirects empty 109]]

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

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

A **Scott topos** is a topos-theoretic generalization of the [[Scott topology]] on a domain.

## Definition

The following definition is due to Karazeris ([2001](#Kar01)).

+-- {: .num_defn}
###### Definition
Let $\mathcal{K}$ be a [[accessible category|finitely accessible category]] and $\mathcal{K}_f$ the full subcategory of its [[finitely presentable objects]]. The topos $Set^{\mathcal{K}_f}$ is called the **Scott topos** of $\mathcal{K}$ and denoted by $\sigma\mathcal{K}$.
=--

## Properties

...

## Example

...

## Related entries

* [[Scott topology]]

* [[domain theory]]

* [[accessible category]]

* [[exponentiable topos]]

* [[Dana Scott]]

## References

* [[Jiri Adamek|J. Adámek]], _A categorical generalization of Scott domains_ , Math. Struc. Comp. Sci. **7** pp.419-443.

* {#Kar01}P. Karazeris, _Categorical Domain Theory: Scott Topology, Powercategories, Coherent Categories_ , TAC **9** (2001) pp.106-120. ([abstract](http://www.tac.mta.ca/tac/volumes/9/n6/9-06abs.html))

* J. Velebil, _Categorical Domain Theory_ , Diss. Prague 1998. ([pdf](ftp://math.feld.cvut.cz/pub/velebil/downloads/thesis.pdf))

* J. Velebil, _Categorical Generalization of a Universal Domain_ , Appl. Cat. Struc. **7** (1999) pp.209-226.


Scott Tremaine

* [institute page](https://www.ias.edu/scholars/tremaine)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Scott_Tremaine)

## related $n$Lab entries

* [[fuzzy dark matter]]

category: people


Scott Wilson

* [webpage](http://qcpages.qc.cuny.edu/~swilson/)

category: people


Scott's trick

## Idea

*Scott's trick* (due to [[Dana Scott]]) is a technical device in a [[material set theory]] such as ZF, that is used to construe a [[quotient]] of a [[proper class]] as another class.

## Discussion

We work throughout with the theory [[ZF]], leaving it to the reader to consider how the discussion applies to other material set theories of comparable strength.

A "class" is not a formal object of ZF, but a meta-object defined by a formula of ZF such as $x \notin x$. (Formally, then, a class is a formula considered up to logical equivalence.) To some extent such meta-objects can be handled like "sets" (the formal objects), so long as constructions remain safely within the confines of first-order, not higher-order logic. For example, there are various ZF hacks for defining the product of two classes as a class, the sum of two classes as a class, etc., but there is no way of forming exponentials of classes, or the power object of a class. In technical terms, the meta-category of classes and class functions may be regarded as a [[pretopos]], but not a [[topos]].

A critical case here is forming the quotient of a class by an equivalence relation. In the usual development of axiomatic set theory, the quotient of a set $X$ by an equivalence relation $\sim$ on $X$ is located within the power set $P X$: each $\sim$-equivalence class is an element of $P X$, and the set of equivalence classes is thus rendered as a definable subset of $P X$. This naive approach to defining quotients is not available for classes $C$, since in the first place we do not have a power class $P C$ to work with.

As a workaround, Dana Scott introduced the following trick. In ZF set theory, we define the ordinal [[rank]] of a set by transfinite induction ([[recursion]]), giving a class function $V \to Ord$. This uses the axioms of [[axiom of foundation|foundation]] and of [[replacement axiom|replacement]]. The collection of sets of ordinal rank $\alpha$ or below is itself a set $V_\alpha$; this is the underpinning of the [[cumulative hierarchy]] picture of set theory.

Thus, given an equivalence relation $\sim \subseteq C \times C$ on a class $C$, we may consider in each equivalence class just those sets whose rank is the minimum $\alpha$ within that class. The collection of those sets is again a set, a subset $c$ of $V_\alpha$, by the [[axiom of separation]]. We let those sets $c$ proxy as equivalence classes; those sets form a class, and this class is the desired quotient $Q$. Indeed, there is a quotient map $C \to Q$, taking $x \in C$ to the set of smallest-rank sets $\sim$-equivalent to $x$, and one may argue that this quotient map has the requisite universal property.

## Applications

There are many applications of Scott's trick. See for instance

* [[localization]]

where the general construction for a large category involves passage to a quotient, and also

* [[ultraproduct]]

where standard discussions of ultrapowers of class-sized models take this technical trick into account.

## References

The eponymous trick was introduced in a conference:

* [[Dana Scott]], *Definitions by abstraction in axiomatic set theory*,  Meeting of the American Mathematical Society, University of British Columbia,  Vancouver, Canada (June 1955). ([web](http://www.ams.org/journals/bull/1955-61-05/S0002-9904-1955-09941-5/S0002-9904-1955-09941-5.pdf))

Scott's immediate application was to give a viable notion of cardinal number, not as a von Neumann ordinal which requires the [[well-ordering principle]], but via a proxy of the equivalence class for the equivalence relation "is in bijection with", as explained above.

A brief textbook account (not mentioning Scott's name) is in

* [[Thomas Jech]], *Set Theory*, 3rd millennium (revised) ed., Springer Monographs in Mathematics, Springer (2003).

See particularly page 65.


Scott-complete category

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

# Contents
* table of contents
{: toc}

## Idea

The notion of *Scott-complete categories* is a [[category theory|category theoretic]] generalization of the notion of *[[Scott domains]]* from [[domain theory]], and as such it provides [[categorical semantics]] for the *un-typed* [[lambda calculus]] and related [[semantics of a programming language|programming languages]].

A Scott domain is a [[directed colimit|directed-complete]] [[partial order]] ([[dcpo]]) that is

* [[algebraic lattice|algebraic]], in that every [[element]] is a [[directed colimit|directed]] [[join]] of [[compact]] elements, and there is a [[bottom]] [[element]];

* consistently complete, i.e., every [[inhabited set|nonempty]] [[subset]] with an [[upper bound]] has a [[least upper bound]].

The suitable notion of [[morphisms]] between Scott domains is that of [[directed colimit|directed]]-[[join]]-[[preserved colimit|preserving]] [[monotone functions]]. These are the same as [[continuous functions]] with respect to the [[Scott topology]].

Scott domains forms a [[cartesian closed category]] that supports the solution of recursive domain equations.

Scott domains admit a simple description in terms of "information systems", which can help to calculate Scott domains and to cement the computation / information-based intuitions.

## Definition {#Def}

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

A category is **Scott-complete** if it is

* [[accessible category|finitely accessible]];

* every [[diagram]] with a [[cocone]] has a [[colimit]].

=--

Scott-complete categories and [[directed colimit]]-[[preserved colimit|preserving]] [[functors]] form a [[category]] $SCC$.

This category $SCC$ is [[cartesian closed]] and supports the solution of recursive domain equations.

## See also

* [[Scott topos]]

## References

* [[Jiří Adámek]], *A categorical generalization of Scott domains*, Mathematical Structures in Computer Science **7** 5  (1997) 419 - 443 [[doi:10.1017/S0960129597002351](https://doi.org/10.1017/S0960129597002351)]

[[!redirects Scott-complete categories]]


scrambling of a bit sequence

\tableofcontents

\section{Idea}

When sending a [[digital signal]], long sequences of 0s or 1s in the original bit sequence can lead to a [[signal]] which in practise is difficult to decode when received. Thus a first step in the process of creating a [[waveform]] from a sequence of bits typically involves _scrambling_ the bit sequence: replacing the original bit sequence in a reversible way by one with a roughly equal number of 0s and 1s, so that whether a particular bit after scrambling is 0 or 1 is pseudo-random.


SDiff

The [[category]] $S Diff$ is the category whose objects are smooth [[supermanifold]]s and whose morphisms are the standard morphisms between them.  Compare to [[Diff]].

category: category


Sean Carroll

* [webpage](http://preposterousuniverse.com)

## Selected writings

On [[extremal black holes]] and [[black hole entropy]]:

* [[Sean Carroll]], Matthew C. Johnson, [[Lisa Randall]], _Extremal limits and black hole entropy_, JHEP 0911:109,2009 ([arXiv:0901.0931](https://arxiv.org/abs/0901.0931))

category: people


Sean Hartnoll

* [webpage](https://sitp.stanford.edu/people/sean-hartnoll)

## Selected writings

Discussion of [[superconductivity]] via [[AdS/CFT in condensed matter physics]]:

* [[Sean Hartnoll]], [[Christopher Herzog]], [[Gary
 Horowitz]], _Building an AdS/CFT superconductor_, Phys. Rev. Lett. 101:031601, 2008 ([arXiv:0803.3295](https://arxiv.org/abs/0803.3295))

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

* [[Sean Hartnoll]], _Lectures on holographic methods for condensed matter physics_, Class. Quant. Grav. 26:224002, 2009 ([arXiv:0903.3246](https://arxiv.org/abs/0903.3246))

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

* Mike Blake, Yingfei Gu, [[Sean A. Hartnoll]], Hong Liu, [[Andrew Lucas]], Krishna Rajagopal, [[Brian Swingle]], [[Beni Yoshida]], *Snowmass White Paper: New ideas for many-body quantum systems from string theory and black holes* [[arXiv:2203.04718](https://arxiv.org/abs/2203.04718)]

[[!redirects Sean A. Hartnoll]]

category: people


SEAR

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

# SEAR
* a table of contents
{: toc}

## Introduction #

**SEAR**, short for **Sets, Elements, And Relations**, is a [[structural set theory]] with the following properties:

* It is equivalent in strength to the [[material set theory]] [[ZF]].  (It can be augmented by an [[axiom of choice]] to produce *SEARC*, which is strongly equivalent to [[ZFC]].)
* It contains a subtheory called **bounded SEAR** which (when augmented by choice) is strongly equivalent to [[ETCS]].  Conversely, ETCS can be augmented by a replacement axiom to become equivalent to SEARC.

Being a structural set theory, SEAR differs from ZF in the following ways (which are also ways in which it is similar to ETCS):

* There is a type distinction between *sets* and *elements*; a set is never an element of another set.  Every element is an element *of some set*.
* The elements of a set have no "internal structure;" they are merely featureless points.
* The "elementhood" relation $x\in A$ only makes sense when $x$ is known to be an element of some ambient set $X$ and $A$ is known to be a subset of $X$.

A good description of the difference between material and structural set theory can be found at [[Trimble on ETCS I]].  However, ZFC and ETCS also differ along another axis than the material-structural.  At [[Trimble on ETCS III]] the following metaphor is proposed for this dimension:

> with ZFC it's more as if you can just hop in the car and go; with ETCS you build the car engine from smaller parts with your bare hands, but in the process you become an expert mechanic, and are not so rigidly attached to a particular make and model.

Using this metaphor, SEAR can be thought of as an ETCS-car which comes preassembled with a nice slick control panel.  Or, using an alternate metaphor, ZFC is like Windows, ETCS is like UNIX, and SEAR is like OS X (or maybe Ubuntu).  With SEAR you get a nice familiar interface with which it is easy to do standard things, there is less cruft than you get with ZFC, and behind the scenes you have all the power of ETCS (and more).  (Of course, if you like Microsoft products, then this metaphor probably does not appeal to you.)

Note that experts will probably always prefer to build their own car/OS; the goal of SEAR is to make structural set theory accessible to a wider audience.  In particular, SEAR is intended to demonstrate the complete independence of [[structural set theory]] from [[category theory]].  Thus, apart from being (by default) stronger than ETCS, SEAR differs from it in the following ways:

* It includes the notion of an *element* of a set as a primitive concept, rather than defining it as a particular sort of function.
* It does *not* include the notion of a [[function]] as a primitive concept (instead it includes the notion of a [[binary relation]], with functions defined via their [[graph of a function|graphs]] in a familiar way).
* It does not include the notions of composition or identities (of either functions or relations) as primitive concepts.  In particular, it does not require the user to know what a [[category]] is, even implicitly.
* As a corollary, its axioms do not require knowledge of categorical concepts such as [[limits]] and [[natural numbers objects]] (although these concepts can be very naturally introduced while learning SEAR).
* The property of [[axiom of separation|separation]] (one of the most important facts in the everyday use of set theory) is a direct consequence of the axioms of SEAR.  Compare how in ETCS, the property of bounded separation follows from the theorem that a [[topos]] is a [[Heyting category]], which requires [[Trimble on ETCS II|substantial work]].

## Description of SEAR

### Types

SEAR is a theory about three kinds of things: **sets**, **elements**, and **relations**.  Every element, and every variable that ranges over elements, is always associated to something denoting a set (which might be a constant or a variable); we say it is an element **of** that set.  If $x$ is an element of $A$ we write $x\in A$; note that this is not an *assertion* which may be true or false, but a *typing* declaration.  In formal terms, this means that SEAR is a [[dependent type theory|dependently typed]] theory, with a type of sets, a type of elements for each set term, and a type of relations for each pair of set terms.

One consequence of this is that whenever we [[quantification|quantify]] over elements we must always quantify over elements *of some set*; thus we can say "for all elements $x\in A$" but not "for all elements $x$."  Another consequence is that the assertion $x=y$ is only well-formed (a precondition to its being true _or_ false) if $x$ and $y$ are elements of the same set.

In a similar manner, every relation is always associated to an ordered pair of sets, the first called its **[[source]]** and the second its **[[target]]** (thus the fundamental relations in SEAR are binary relations).  If $\varphi$ is a relation from $A$ to $B$ we write $\varphi:A\looparrowright B$.  As with elements, the assertion $\varphi=\psi$ is only well-formed if $\varphi$ and $\psi$ have the same source and the same target.

Implicit in the existence of three types of things is that nothing is both a set and an element, etc., so in particular a statement such as $x=A$ is not well-formed if $x$ is an element and $A$ a set.  Furthermore, SEAR does not include an equality relation between sets: even if $A$ and $B$ are both sets we do not consider $A=B$ to be well-formed.  (Thus SEAR respects the [[principle of equivalence]].)

The final piece of data that we have is a notion of when a relation $\varphi:A\looparrowright B$ **holds** of a pair of elements $x\in A$ and $y\in B$.  We write $\varphi(x,y)$ when $\varphi$ holds of $x$ and $y$.

### Basic axioms

Now we can state the axioms of SEAR, beginning with the most basic ones.

**Axiom 0 (Nontriviality):** _There exists a set which contains an element._

**Axiom 1 (Relational comprehension):** _For any two sets $A$ and $B$, and any property $P$ that can obtain of an element of $A$ and an element of $B$, there exists a unique relation $\varphi:A\looparrowright B$ such that $\varphi(x,y)$ if and only if $P$ obtains of $x$ and $y$._

Here "property" is interpreted formally as "first-order formula", which makes the axiom (like later the axiom of collection) in fact an axiom scheme.

Axiom 1 basically says that relations are what we expect them to be intuitively.  It should remind the reader of the [[axiom of separation]].  The two important differences are (1) the presence of two sets and two variables, rather than one, and (2) the fact that what is produced is not a *set*, but a *relation* (a different kind of thing). The to-be-introduced Axiom 2 will enable us to make relations into (sub)sets, but first we need some terminology.

+-- {: .num_defn #function}
###### Definition
A relation $\varphi:A\looparrowright B$ is called a **[[function]]** from $A$ to $B$ if for any $x\in A$, there exists exactly one $y\in B$ with $\varphi(x,y)$.
=--

We write $f:A\to B$ for such a function, and for $x\in A$ we write $f(x)$ for the resulting unique $y\in B$.  Note that the uniqueness clause in Axiom 1 implies that functions are extensional: if $f,g:A\to B$ and $f(x)=g(x)$ for all $x\in A$, then $f=g$.

**Axiom 2 (Tabulations):** _For any relation $\varphi:A\looparrowright B$, there exists a set $|\varphi|$ and functions $p:{|\varphi|}\to A$ and $q:{|\varphi|}\to B$ such that: (1) for any $x\in A$ and $y\in B$, we have $\varphi(x,y)$ if and only if there exists $r\in {|\varphi|}$ with $p(r)=x$ and $q(r)=y$, and (2) for any $r\in {|\varphi|}$ and $s\in{|\varphi|}$, if $p(r)=p(s)$ and $q(r)=q(s)$, then $r=s$._

A set $|\varphi|$ equipped with $p$ and $q$ as in Axiom 2 is called a **tabulation** of the relation $\varphi$.  We think of $|\varphi|$ as "the set of pairs $(x,y)$ such that $\varphi(x,y)$ holds," with $p$ and $q$ projecting $(x,y)$ to $x$ and $y$, respectively.  Note that by Axiom 1, any set $R$ equipped with functions $p:R\to A$ and $q:R\to B$ satisfying (2) determines a unique relation $\varphi:A\looparrowright B$ such that $\varphi(x,y)$ holds iff there is an $r\in R$ with $p(r)=x$ and $q(r)=y$; then $R$ is a tabulation of $\varphi$.

### Consequences of the basic axioms

Axioms 0, 1, and 2 alone are very powerful!  Here are some things we can do with them.

+-- {: .num_theorem #emptyset}
###### Theorem
There exists a set $\emptyset$ which has no elements.
=--
+--{: .proof}
###### Proof
By Axiom 0, there exists a set $A$.  By Axiom 1, there exists a relation $\varphi:A\looparrowright A$ such that $\varphi(x,y)$ holds *never*.  Using Axiom 2, let $\emptyset$ be a tabulation of $\varphi$.
=--

From now on we fix a particular set $\emptyset$ having no elements.  By Axiom 1, for any set $A$ there is exactly one relation $\emptyset\looparrowright A$ and exactly one relation $A\looparrowright\emptyset$.

+-- {: .num_theorem #terminalset}
###### Theorem
There exists a set $1$ which has exactly one element.
=--
+--{: .proof}
###### Proof
By Axiom 0, there exists a set $A$ containing an element $x$.  By Axiom 1, there exists a relation $\varphi:A\looparrowright A$ such that $\varphi(y,z)$ holds iff $y=x$ and $z=x$.  Using Axiom 2, let $1$ be a tabulation of $\varphi$.
=--

From now on we fix a particular set $1$ having exactly one element;
we usually denote this element by $\star$.  By Axiom 1, for every set $A$ there exists a unique function $f:A\to 1$, defined by $f(x)=\star$ for every $x\in A$.  Likewise, to every function $g:1\to A$ there corresponds a unique element $x\in A$, by $g(\star)=x$.

We define a **subset** of a set $A$ to be a relation $S:1\looparrowright A$.  By Axiom 1, a subset $S$ of $A$ is determined by precisely those $x\in A$ such that $S(\star,x)$.  If $S(\star,x)$ we write $x\in S$.  Note that this is an *overloading* of the notation $\in$; it can be distinguished from the previous usage because here $S$ is a relation, not a set.  Note also that the previous usage $x\in A$ was a typing declaration (it says what kind of thing $x$ is when we introduce it), whereas $x\in S$ is a statement which can be either true or false.

Now Axiom 2 implies that from any subset $S$ of $A$ we can construct a set $|S|$ with a function $i:{|S|}\to A$ such that (1) $x\in S$ iff there is an $x'\in {|S|}$ with $i(x')=x$, and (2) for $x',x''\in {|S|}$, if $i(x')=i(x'')$ then $x'=x''$.  A function $i$ with property (2) is called **injective**; in this way we set up a correspondence between subsets and injective functions.  (Likewise, a function $f:A\to B$ is **surjective** if for any $y\in B$ there is an $x\in A$ with $f(x)=y$.)  Combined with Axiom 1, this implies the following *separation property*.

+-- {: .num_theorem #sep}
###### Theorem
For any property $P$ of elements of a set $A$, there exists a set $B$ and an injective function $i:B\to A$ such that for $a\in A$, we have $P(a)$ iff $a=i(b)$ for some $b\in B$.
=--

Although the set $|S|$ is not determined uniquely by the subset $S$, it is determined up to [[bijection]].  We define an **bijection** to be a relation $\psi:A\looparrowright B$ such that for every $x\in A$ there is a unique $y\in B$ with $\psi(x,y)$, and for every $y\in B$ there is a unique $x\in A$ with $\psi(x,y)$.  Clearly any bijection is a function, as is its opposite (defined by $\psi^o(y,x) \Leftrightarrow \psi(x,y)$), and these two properties characterize bijections.  It is also easy to see that a function is a bijection iff it is both injective and surjective.

For example, it is easy to see that if $A$ and $B$ both have no elements, there is a unique bijection between them; thus an empty set $\emptyset$ is determined up to unique bijection.  Likewise, if $A$ and $B$ both contain a unique element, then there is a unique bijection between them; thus a set $1$ with one element is also determined up to unique bijection.

+-- {: .num_theorem #tabuniq}
###### Theorem
If $|\varphi|$ and ${|\varphi|}'$ are two tabulations of the same relation $\varphi:A\looparrowright B$, then there is a bijection between $|\varphi|$ and ${|\varphi|}'$.
=--
+--{: .proof}
###### Proof
By Axiom 1, define $B:{|\varphi|} \looparrowright {|\varphi|}'$ such that $B(x,y)$ holds precisely when $p(x)=p'(y)$ and $q(x)=q'(y)$.  The properties of tabulations immediately imply that $B$ is a bijection.
=--

+-- {: .num_cor #sstabuniq}
###### Corollary
If $|S|$ and ${|S|}'$ are two tabulations of the same subset $S\subseteq A$, then there is a bijection between $|S|$ and ${|S|}'$.
=--

The **composite** of two relations $\varphi:A\looparrowright B$ and $\psi:B\looparrowright C$ is defined to be the relation $\psi\varphi:A\looparrowright C$ (also written $\psi\circ\varphi$) such that $\psi\varphi(x,z)$ holds of $x\in A$ and $z\in C$ iff there is a $y\in B$ with $\varphi(x,y)$ and $\psi(y,z)$.  The **identity** relation $id_A:A\looparrowright A$ is defined by $id_A(x,y) \Leftrightarrow x=y$.

+-- {: .num_theorem #category}
###### Theorem
Composition of relations is associative ($\chi(\psi\varphi)=(\chi\psi)\varphi$), and identity relations are identities for composition ($\id_B\circ \varphi = \varphi = \varphi\circ\id_A$).  The composite of functions is a function, and the identity relation is a function.  The composite of bijections is a bijection, and a relation $\varphi:A\looparrowright B$ is a bijection iff there is a relation $\psi:B\looparrowright A$ such that $\psi\varphi = id_A$ and $\varphi\psi=id_B$.
=--
+--{: .proof}
###### Proof
Just as in naive set theory.
=--

Informally speaking, we have two [[categories]] whose objects are the sets: in [[Rel]] the morphisms are the relations, while in [[Set]] the morphisms are the functions.  Formally, all we can say is that these are "meta-categories," in the sense that from any model of the first-order axioms of SEAR we can construct them as models of the first-order axioms of a category.  (This is not that different from the status of large categories such as $Set$ in material set theories such as ZF.  We could also speak about "proper classes" as is done in ZF using first-order formulas.)  See also the section on Universes, below.

In both of these (meta-)categories, the [[isomorphisms]] are the bijections.  We have also already observed that $\emptyset$ and $1$ are an [[initial object]] and a [[terminal object]] in $Set$, respectively, and that $\emptyset$ is both initial and terminal in $Rel$.

For sets $A$ and $B$, let $\top:A\looparrowright B$ denote the relation such that $\top(x,y)$ holds always.  A tabulation of $\top$ is denoted $A\times B$; it is a set equipped with functions $p:A\times B \to A$ and $q:A\times B\to B$ such that for any $x\in A$ and $y\in B$, there exists a unique $z\in A\times B$ with $p(z)=x$ and $q(z)=y$.  It is called the **[[cartesian product]]** of $A$ and $B$.

+-- {: .num_theorem #product}
###### Theorem
For any sets $A$ and $B$, $A\times B$ is a [[product]] of $A$ and $B$ in the category $Set$.
=--
+--{: .proof}
###### Proof
Just as in naive set theory.
=--

In particular, $Set$ has products.

+-- {: .num_theorem #lex}
###### Theorem
The category $Set$ has finite limits.
=--
+--{: .proof}
###### Proof
It remains to construct the [[equalizer]] of two functions $f,g:A\to B$.  Let $S$ be the subset of $A$ so that $x\in S$ just when $f(x)=g(x)$; then ${|S|}\to A$ is easily shown to be an equalizer of $f$ and $g$.
=--

+-- {: .num_theorem #imfact}
###### Theorem
For any function $f:A\to B$ we have $f = m e$, where $m:im(f)\hookrightarrow B$ is an injection and $e:A\twoheadrightarrow im(f)$ is a surjection.  A set $im(f)$ equipped with such $m$ and $e$ is unique up to bijection and is called the **image** of $f$.
=--
+--{: .proof}
###### Proof
Let $im(f)$ be $|S|$ where $S$ is defined by $y\in S$ iff there exists an $x\in A$ with $f(x)=y$.  By definition, we have an injection $m:im(f)\hookrightarrow B$.  And for any $x\in A$, clearly $f(x)\in S$, so there is a unique $y\in im(f)$ with $m(y)=f(x)$; we define $e(x)=y$.  It is easy to verify the rest.
=--

Similarly, we can define the union and intersection of subsets of $A$ in the usual way: we have $x\in S\cup T$ iff $x\in S$ or $x\in T$, and $x\in S\cap T$ iff $x\in S$ and $x\in T$.

+-- {: .num_theorem #heyting}
###### Theorem
$Set$ is a [[well-pointed category|well-pointed]] [[Heyting category]] and $Rel$ is a [[division allegory]].
=--
+--{: .proof}
###### Proof
Left to the reader.
=--

### Power sets and colimits

In structural set theory, sets cannot contain other sets (or relations), so we cannot strictly speaking have "the set of subsets of a set $A$."  What we generally do instead is have a set $P A$ each of whose elements is *associated to* a subset of $A$ in a bijective way.  This association happens via a specified relation between $A$ and $P A$.

**Axiom 3 (power sets):** _For any set $A$, there exists a set $P A$ and a relation $\epsilon:A\looparrowright P A$ such that for any subset $S$ of $A$, there exists a unique $s\in P A$ with the property that for any $x\in A$, we have $x\in S$ iff $\epsilon(x,s)$._

As in both material set theory and ETCS, many useful consequences flow from this axiom.  We first observe that the property of $P A$ stated for subsets actually implies an analogous property for all relations.

+-- {: .num_theorem #topos}
###### Theorem
For any relation $R:B\looparrowright A$, there exists a unique function $f_R:B\to P A$ such that $R(y,x)$ if and only if $\epsilon(x,f_R(y))$.  It follows that $Set$ is a [[topos]] (and $Rel$ is a power allegory).
=--
+--{: .proof}
###### Proof
We simply define $f_R$ elementwise; for each $y$ we define $f_R(y)$ to be the unique element of $P A$ such that $\epsilon(x,f_R(y))$ holds iff $R(y,x)$ holds.  Extensionality of functions implies that it is unique.
=--

As usual, power sets also imply the existence of function sets.

+-- {: .num_theorem #ccc}
###### Theorem
For any two sets $A$ and $B$, there exists a set $B^A$ and a function $ev:B^A \times A\to B$ such that for any function $f:A\to B$ there exists a unique element $s_f\in B^A$ such that $ev(s_f,a)=f(a)$ for all $a\in A$.  It follows that $Set$ is a [[cartesian closed category]].
=--
+--{: .proof}
###### Proof
We take $B^A$ to be a tabulation of the subset of $P(A\times B)$ containing only the functions.  More precisely, define $F\subseteq P(A\times B)$ such that $s\in F$ iff for every $x\in A$, there exists a unique $y\in B$ such that $\epsilon((x,y),s)$, and let $B^A = {|F|}$.
=--

The correspondences in Theorems \ref{topos} and \ref{ccc} are "meta-bijections," but by a standard Yoneda lemma argument they can be converted into actual bijections as defined above in SEAR.  This produces bijections such as $C^{A\times B} \cong (C^A)^B$ and $P(A\times B) \cong (P A)^B$.

We now construct [[quotient set]]s.  Of course, by an **[[equivalence relation]]** on a set $A$ we mean a relation $R:A\looparrowright A$ which is reflexive, transitive, and symmetric.

+-- {: .num_theorem #quotients}
###### Theorem
If $R$ is an equivalence relation on $A$, then there is a surjective function $q:A\twoheadrightarrow B$ such that $R(x,y)$ holds iff $q(x)=q(y)$.
=--
+--{: .proof}
###### Proof
$R$ induces a function $f_R:A\to P A$ as above; let $B$ be $im(f_R)$ and $q$ the surjective part of the image factorization.  If $R(x,y)$ holds, then by symmetry and transitivity, $R(x,z)\Leftrightarrow R(y,z)$ for all $z$; hence $f_R(x)=f_R(y)$ and so $q(x)=q(y)$.  Conversely, if $q(x)=q(y)$, then $f_R(x)=f_R(y)$; but $y\in f_R(y)$ by reflexivity, hence $y\in f_R(x)$ and thus $R(x,y)$ holds.
=--

Such a $B$, which is easily seen to be unique up to bijection, is called a **quotient** of $R$ and often written $A/R$.

Likewise, given sets $A$ and $B$, let $A\sqcup B={|S|}$, where $S$ is the subset of $P A \times P B$ consisting of the pairs $(\{x\},\{\})$ for $x\in A$ and $(\{\},\{y\})$ for $y\in B$.  (Of course, here we are identifying elements of $P A \times P B$ with ordered pairs of subsets of $A$ and $B$ via the projections of the product and the defining relations $\epsilon_A$ and $\epsilon_B$.)  We have evident injections $A\to A\sqcup B$ and $B\to A\sqcup B$ which are jointly surjective and have disjoint image; we call $A\sqcup B$ the **disjoint union** of $A$ and $B$.

In particular, it follows that $Set$ is a [[pretopos]] (as is any topos).

### Infinity

**Axiom 4 (Infinity):** _There exists a set $N$, containing an element $o$, and a function $s:N\to N$ such that $s(n)\neq o$ for any $n\in N$ and $s(n) = s(m)$ only if $n = m$ for any $n, m\in N$._

It is easy to deduce any of the usual consequences of the axiom of infinity.  The set $N$ is not asserted to be minimal, but it is easy to cut it down to be so using separation.  In particular, this axiom implies that $Set$ has a [[natural numbers object]].  We can then proceed to construct the set $\mathbb{Q}$ of rational numbers and the set $\mathbb{R}$ of real numbers in any of the usual ways.

Note also that Axiom 4 implies Axiom 0.

We can however [[natural numbers in SEAR|define sets]] with $n$ elements for each natural number $n$ directly axioms 0, 1, 2 and 3.

### Collection

The final axiom of SEAR is somewhat trickier to motivate.  It corresponds to the [[axiom of replacement]] (or more precisely collection; see [Wikipedia](http://en.wikipedia.org/wiki/Axiom_schema_of_replacement) until we get around to writing our own article) in $\mathbf{ZFC}$, and the reasons that we need it are the same as the reasons that $\mathbf{ZFC}$ needs it.

#### The ghost of a universal set

One way to motivate the collection axiom is as follows.
Suppose for the sake of argument that there existed a *universal set*, meaning a set containing all other sets.  Now in structural set theory this is meaningless, since no set can contain other sets, but there is an easy fix: we consider instead *indexed families* of sets.  There are multiple ways to make this precise, but here is a very simple one: an $A$-indexed [[family of sets]] is simply a relation $M:A\looparrowright X$.  The set indexed by each element $a\in A$ is $M_a = \{x\in X | M(a,x)\}$.  (Often one requires the $M_a$ to be a disjoint decomposition of $X$, so that $M^o$ is a function $X\to A$, but this is unnecessary.  In fact, if $M:A\looparrowright X$ is a general family, then ${|M|} \to A$ is a function that represents a family in this stronger sense which is equivalent to $M$.)

With this definition, a *universal family of sets* would be a set $U$ with a $U$-indexed family of sets $E:U\looparrowright Y$ such that for any set $A$, there exists a unique $u\in U$ and a bijection $A\cong E_u$.  The existence of such a universal set is inconsistent, but the collection axiom can be thought of as the "ghost" of what *would be implied* by Axioms 1 and 2 if a universal set *did* exist.  Let's now unravel that.

If $U$ were a universal set, then any $A$-indexed family $M:A\looparrowright X$ would induce a unique function $f:A\to U$ such that $E_{f(x)} \cong M_x$ for all $x\in A$.  Conversely, any function $f:A\to U$ would induce an $A$-indexed family of sets with this property: just take the composite relation $E \circ f$.  (This family would not be unique, but it would be unique up to a suitable notion of [[equivalence]] between indexed families.)  Thus it is reasonable to consider the "ghost" of a function $f:A\to U$ to be simply an $A$-indexed family of sets.

The ghost of a relation $A\looparrowright U$ is even easier.  By Axiom 1, to give such a relation would be equivalent to describing a property $P$ that can obtain of an element of $A$ and a set.  Thus, the ghost of a relation $A\looparrowright U$ is simply such a property.

If $U$ existed, Axiom 2 would imply that any relation $A\looparrowright U$ had a tabulation consisting of a set $B$ and functions $p:B\to A$ and $q:B\to U$.  We now kill $U$ and inspect the ghostly remnants of this tabulation.

**Axiom 5 (Collection):** _For any set $A$ and any property $P$ which can obtain of an element of $A$ and a set, there exists a set $B$, function $p:B\to A$, and a $B$-indexed family of sets $M:B\looparrowright Y$ such that (1) $P(p(b),M_b)$ holds for any $b\in B$, and (2) for any $a\in A$, if there exists a set $X$ with $P(a,X)$, then $a\in im(p)$._

What this axiom asserts is actually a bit weaker than the precise ghost of a tabulation of $P$; that would require instead of (2) that for any $a\in A$ and any set $X$ with $P(a,X)$, there exists a $b\in B$ with $p(b)=a$ and $M_b\cong X$.  However, this stronger statement is still inconsistent, because if we took $A=1$ and $P=\top$ it would resurrect $U$ in the form of $B$!  The form given above is weak enough to keep the dead in their graves, yet strong enough to be useful.  (The former statement follows from the discussion below about inter-interpretability of SEAR and ZF, at least as long as the reader believes that ZF is consistent.  The latter remains to be gone more into.)

_Exercise: why respect for the [[principle of equivalence]] necessary for Axiom 5 to be reasonable as stated?_

#### Applications of collection

One thing that Axiom 5 is good for is the recursive construction of sets.  For example, using Axioms 1--4 we can construct the sets $N$, $P(N)$, $P(P(N))$, and so on, but we cannot construct anything larger than all the sets in this sequence.  Axiom 5 allows us to do that.

(more detail to be added here...)

## Variants and Comparisons

### The Axiom of Choice

We obtain a theory called **SEARC** (SEAR with Choice) by adding the following axiom:

**Axiom 6 (Choice):** _If $\varphi:A\looparrowright B$ is a relation such that for any $a\in A$, there exists a $b\in B$ with $\varphi(a,b)$, then there exists a function $f:A\to B$ with $\varphi(a,f(a))$ for all $a\in A$._

This is easily seen to be equivalent to asserting that all surjections are [[split epimorphism|split]] in $Set$, which is a more common categorical form of the [[axiom of choice]].

### Constructive SEAR

Axioms 1--5 of SEAR make perfect sense whether the ambient logic is [[classical logic|classical]] or [[intuitionistic logic|constructive]].  By Diaconescu's argument, Choice implies the logic is classical.  We may call SEAR interpreted in intuitionistic logic __ISEAR__.

To obtain a [[predicative mathematics|predicative]] theory, Axiom 3 can be replaced by an appropriate weaker axiom, such as the existence of [[disjoint unions]] and [[quotient sets]].  We may call this variation of SEAR __PSEAR__.

If desired, we can also add [[function sets]] or even a structural version of the axiom of [[subset collection]]; as usual, function sets, subset collection, and power sets are all equivalent using classical logic.  We may call this variation of SEAR __CSEAR__.

Compare [[Erik Palmgren]]\'s [constructive ETCS](http://www.math.uu.se/~palmgren/cetcs.pdf).

### Bounded SEAR and ETCS

By **bounded SEAR** we mean the subtheory consisting of axioms 2--4 of SEAR plus

**Axiom 1B (Bounded relational comprehension):** _The same as Axiom 1, but the property $P$ must be bounded (it can only involve quantifiers over elements, not sets or relations)._

All the theorems cited above remain true with Axiom 1 replaced by Axiom 1B, so bounded SEAR still implies that $Set$ is a well-pointed topos with a NNO.  Therefore, *bounded SEARC* implies [[ETCS]].  Conversely, it is not hard to show that given any [[well-pointed topos]], if we interpret "set" as "object of the topos", "element of $A$" as "[[global element]] $1\to A$", and "relation $A\looparrowright B$" as "[[subobject]] of $A\times B$," then Axioms 0, 1B, 2, and 3 are satisfied.  It follows that ETCS also implies bounded SEARC, so the two are equivalent.

ETCS can also be augmented with additional axioms to make it equivalent to full SEARC, but that is beyond our scope at the moment.

### SEAR and ZF

#### From ZF to SEAR

It is fairly straightforward to construct a model of SEAR from a model of [[ZFC|ZF]].  Given a model of ZF, we define the SEAR-sets to be the ZF-sets, and the SEAR-elements of $A$ to be the ZF-elements of $A$.  If we prefer, we can take the SEAR-elements of $A$ to be pairs $(x,A)$ where $x\in_{ZF} A$, so that the elements of distinct sets will be disjoint---but this is not necessary, since in SEAR the question of whether two distinct sets have elements in common is not even well-posed.  Finally, we of course take the SEAR-relations $A\looparrowright B$ to be the ZF-subsets of $A\times B$, and we let $\varphi(x,y)$ hold in SEAR iff $(x,y)\in_{ZF} \varphi$.  It is then easy to prove Axioms 0, 1, 2, 3, and 4 from the axioms of ZF, and likewise Axiom 6 follows easily from the axiom of choice in ZFC.  (In fact, Z and ZC, where the replacement axiom is omitted, suffice for these conclusions.)  The only axiom which requires some thought is Collection, and it is here that we use replacement.

+-- {: .num_theorem #zf-coll}
###### Theorem
The sets, elements, and relations in a model of ZF satisfy the Collection axiom of SEAR.
=--
+--{: .proof}
###### Proof
Let $A$ be a set and $P$ a property as in Axiom 5.  As given, $P$ is a first-order statement in the language of SEAR, but we can easily translate it into a statement in the language of ZF; from now on we work in ZF.  Define
$$A' = \{a\in A | \exists X. P(a,X)\}.$$
By the [[axiom of foundation]], for each $a\in A'$ there exists an [[ordinal]] $\lambda$ and an $X\in V_{\lambda}$ such that $P(a,X)$; let $\lambda_a$ be the *smallest* such $\lambda$.  Then $a\mapsto V_{\lambda_a}$ is a class function, so by the axiom of replacement, the set $\{V_{\lambda_a} | a\in A'\}$ exists.  Let $Y$ be the union of this set.  Define
$$B = \{ (a,X)\in A\times Y | P(a,X)\},$$
let $p:B\to A$ be the projection onto the first factor, and let $M:B\looparrowright Y$ be defined by
$$M((a,X),x) \Leftrightarrow x\in X.$$
Then $M_{(a,X)} = X$, since each $V_\lambda$ is a transitive set, so $P((a,X), M_{(a,X)})$ holds for any $(a,X)\in B$.  Finally, by construction, if there exists $X$ with $P(a,X)$ then $a\in im(p)$; thus the SEAR axiom of collection is satisfied.
=--

Note the use of the axiom of foundation in addition to the axiom of replacement.  This can be avoided if we use instead the ZF version of the [axiom of collection](http://en.wikipedia.org/wiki/Axiom_of_collection#Axiom_schema_of_collection), which is equivalent to the axiom of replacement over the other ZF axioms (including foundation), by an argument like that above.

#### From SEAR to ZF

Conversely, from any model of SEAR one can *construct* a model of ZF.    The basic idea of this process is described at [[pure set]].  Given a model of SEAR, we define a **ZF-set** to be an equivalence class of well-founded extensional accessible graphs, as described at [[pure set]].  We define the global membership relation $\in$ on ZF-sets to be the "immediate subgraph" relation.

The proofs of most of the axioms of ZF are straightforward, and may eventually be found at [[pure set]].  Here we treat the proof of the replacement/collection axiom of ZF from the Collection axiom of SEAR, since that is specific to SEAR.

+-- {: .num_theorem #coll-zf}
###### Theorem
The ZF-sets constructed from a model of SEAR satisfy the ZF [axiom of collection](http://en.wikipedia.org/wiki/Axiom_of_collection#Axiom_schema_of_collection) (which is equivalent to the axiom of replacement over the other axioms of ZF).
=--
+--{: .proof}
###### Proof
We will prove the version of the ZF axiom of collection which states that for any first-order formula $P$ in the language of ZF, and any ZF-set $A$, there exists a ZF-set $D$ such that for any ZF-element $x\in_{ZF} A$, if there exists a ZF-set $y$ such that $P(x,y)$, then there exists such a $y$ which is a ZF-element of $D$.  Note that any first-order formula in the language of ZF about ZF-sets can easily be translated into a first-order formula in the language of SEAR.

Fix such a $P$ and $A$; thus $A$ is a SEAR-set equipped with a binary relation $\to$ of a certain type.  Let $A'$ be the subset of $A$ consisting of the children of the root of $A$.  For a SEAR-set $X$, let $P'(a,X)$ mean that $X$ admits the structure of a ZF-set such that the ZF-statement $P(down(a),X)$ holds.  By the SEAR axiom of collection, there is a set $B$, a function $p:B\to A'$, and a $B$-indexed family of sets $M:B\looparrowright Y$ such that (1) each $M_b$ admits the structure of a ZF-set with $P(down(p(b)),M_b)$, and (2) if there is a ZF-set $X$ with $P(down(a),X)$, then $a\in im(p)$.

Let $C$ be the subset of $B\times P(Y\times Y)\times Y$ consisting of pairs $(b,r,y)$ such that $y\in M_b$ and $r$ represents a binary relation on $M_b$ making it into a ZF-set such that $P(down(p(b)),M_b)$.  Now $C$ can be equipped with the relation that is the disjoint union of all the $r$s, i.e. we set $(b,r,y)\to (b',r',y')$ iff $b=b'$, $r=r'$, and $y\to_r y'$.  Let $C'=C\sqcup 1$ extend this relation with the additional point being a root, i.e. $(b,r,y)\to \star$ iff $y$ is the root of $r$.  Now $C'$ is a well-founded accessible graph, but it may not be extensional; we take $D$ to be the ZF-set represented by the extensional quotient of $C'$.
Then for any $x\in_{ZF}A$, we have $x\cong down(a)$ for some child $a$ of the root of $A$, i.e. $a\in A'$.  Thus, if there is a ZF-set $y$ with $P(x,y)$, then $a\in im(p)$ and so there is such a ZF-set represented by one of the $r$'s occurring in $C$.  By construction, this ZF-set is then equivalent to $down(d)$ for some child of the root of $D$, hence $down(d) \in_{ZF} D$.  Thus $D$ satisfies the requirement of the ZF axiom of collection.
=--

Thus, SEAR and SEARC are at least equiconsistent with ZF and ZFC, respectively. In fact, SEARC and ZFC are equivalent, in the sense that passage back and forth in either direction yields an equivalent model.  However, passage from SEAR to ZF and back again can lose information, if there are sets in the model of SEAR which do not admit any well-founded extensional relation (this doesn't happen in the presence of choice, since then every set can be well-ordered).

### Type theory and internalization

Every topos has an [[internal logic]], which is a [[type theory]].  However, the line between type theory and structural set theory is fine and sometimes hard to see; the main difference is that structural set theory can involve quantifiers over sets (= types), while type theory only allows ("bounded") quantifiers over elements of types. However, formally, if one has a type judgment $A \; \mathrm{type}$ and a type of sets $Set \; \mathrm{type}$ rather than a set judgment $A \; \mathrm{set}$, one could use quantifers bounded over elements of $Set$ instead of unbounded quantifiers. The type of sets $Set$ in structural set theory behaves as a [[Russell universe]] in [[type theory]], where given a set $A \in Set$ one could derive a type $A \; \mathrm{type}$. Through this correspondence, *Intuitionistic Bounded SEAR* can be treated as a type theory and interpreted internally in any topos with a NNO.  Of course, if the topos is [[boolean topos|boolean]], then the logic can be classical.

One can also write down stronger axioms on a topos such that if they are satisfied, then full Intuitionistic SEAR can be interpreted "internally" in that topos, extending the usual internal logic.

## Making alternate primitive choices

### Three-sorted SEAR

An alternate formulation of the theory uses three [[sorts]], one for [[sets]], [[elements]], and [[relations]], making SEAR into a [[three-sorted set theory]]. The membership $a \in A$ becomes a [[relation]] between the sort of elements and the sort of [[sets]], while the relational domain and codomain notation $\varphi:A \looparrowright B$ also becomes a ternary relation between the sort of relations, and two copies of the sort of sets. There is a quinary relation $\mathrm{holds}(A, B, \varphi, a, b)$ between all three sorts, which says that given sets $A$ and $B$, the relation $\varphi$ holds for element $a$ and $b$ in the [[context]] of $\varphi:A \looparrowright B$, $a \in A$, and $b \in B$.

### SEPS: Using pairs and subsets instead of relations {#SEPS}

An alternate formulation of the theory, suggested by Toby, has four primitive notions: sets, elements, subsets, and a pairing operation.  Sets and elements are as before.  A *subset* is, like an element, attached to a certain set; it is always a subset *of* some set.  Thus we have a typing declaration $S\subseteq A$.  We also have a primitive notion of when an element $x\in A$ *belongs to* a subset $S\subseteq A$; thus now we have $x\in S$ as a possible assertion of the theory (analogous to $R(x,y)$ before).  We allow a typed equality predicate for subsets.  Finally, there is an operation which assigns to every pair of sets $A$ and $B$ a set $A\times B$, and to every pair of elements $x\in A$ and $y\in B$ an element $(x,y)\in A\times B$.

**Axiom 0** is the same as before.

**Axiom $\frac{1}{2}$ (Pairing):** _For every pair of sets $A$ and $B$ and every element $z\in A\times B$, there exists a unique pair of elements $x\in A$ and $y\in B$ such that $z=(x,y)$._

**Axiom $1\frac{1}{2}$ (Subset comprehension):** _For every property $P$ that can obtain of elements $x$ of some set $A$, there exists a unique subset $S\subseteq A$ such that $x\in S$ precisely when $P$ obtains of $x$_

We now define a **relation** $A\looparrowright B$ to be a subset of $A\times B$.  With this definition, Axiom $1\frac{1}{2}$ clearly implies Axiom 1 (relational comprehension).  We define **functions** and their properties as before.  Axiom $\frac{1}{2}$ implies that there are projection functions $A\times B\to A$ and $A\times B\to B$ which make $A\times B$ into a product in $Set$.

**Axiom $2\frac{1}{2}$ (Separation):** _For every set $A$ and every subset $S\subseteq A$, there exists a set $|S|$ and an injective function $i:{|S|}\to A$ such that for any $x\in A$, we have $x\in S$ if and only if there is a $z\in {|S|}$ with $i(z)=x$._

Applying separation to subsets of $A\times B$ and composing $i$ with the product projections, we recover Axiom 2 (tabulations).  We can now go on with the subsequent axioms stated in the same way as before.

### SEPF: Using pairs and functions instead of relations

One could use functions instead of relations. Sets and elements are as before. A function $f:A \to B$ depends on two sets $A$ and $B$, similar to a relation. For every function $f:A \to B$ and element $a \in A$, one could form the element $f(a) \in B$ called the **function evaluation** of $f$ at $a$. Finally, there is an operation which assigns to every pair of sets $A$ and $B$ a set $A\times B$, and to every pair of elements $x\in A$ and $y\in B$ an element $(x,y)\in A\times B$, as well as functions $\pi_A:A \times B \to A$ and $\pi_B:A \times B \to B$.

**Axiom 0** is the same as before.

**Axiom $\frac{1}{4}$ (strong extensionality):** _Given function $i:A \to B$, the following statements are logically equivalent to each other:_

* _there exists a function $i^{-1}:B \to A$ such that for all elements $x \in A$ and $y \in B$, $i^{-1}(i(x)) = x$ and $i(i^{-1}(y)) = y$_
* _for every element $x \in B$ there exists a unique element $y \in A$ such that $i(y) = x$_

A function is a **bijection** if one of the two conditions above hold.

**Axiom $\frac{1}{2}$ (Pairing):** _For every pair of sets $A$ and $B$ and for every element $x \in A$, $y \in B$, and $z\in A\times B$, $\pi_A((x, y)) = x$, $\pi_B((x, y)) = y$, and $z=(\pi_A(z),\pi_B(z))$._

An function $f:A \to B$ is an **injection** if for all elements $x \in A$ and $y \in A$, $x = y$ if and only if $f(x) = f(y)$, and is usually denoted $f:A \hookrightarrow B$. $A$ is a **subset** of $B$ if it comes with an injection $i:A \hookrightarrow B$. We now define a **relation** to be a set $R$ with an injective function $i:R \hookrightarrow A\times B$.

**Axiom $2\frac{1}{2}$ (Separation):** _For every property $P$ that can obtain of elements $x$ of some set $A$, there exists a set $S$ and unique injection $i:S \hookrightarrow A$ such that there exists an element $y \in S$ such that $i(y) = x$ precisely when $P$ obtains of $x$_

A relation $i:R \hookrightarrow A \times B$ is a **total functional relation** if for all elements $a \in A$, there exists a unique element $b \in B$ and a unique element $z \in A \times B$ such that $i(z) = (a, b)$. Since there is a function $\pi_A \circ i:R \to A$, by strong extensionality there is an inverse function $(\pi_A \circ i)^{-1}:A \to R$, and the function $\pi_B \circ (\pi_A \circ i)^{-1}:A \to B$ is the function satisfying $i((\pi_A \circ i)^{-1}(a)) = (a, \pi_B((\pi_A \circ i)^{-1}(a))$. Thus, the [[principle of unique choice]] holds.

Applying separation to relations $i:R \hookrightarrow A \times B$ and using the [[principle of unique choice]], we recover Axioms 1 (relational comprehension) and 2 (tabulations). We can now go on with the subsequent axioms stated in the same way as before.

### Eliminating equality
{#eqfree}

As stated SEAR includes a fundamental "equality" relation on elements of a given set.  However, we can also make equality into *structure*.  This is definitely not along the "more accessible to undergraduates" direction!  But it may sometimes be technically helpful.

Consider a theory of *[[pre-sets]]*, *elements*, and *pre-relations* as in SEAR, with "pre-sets" replacing sets, except that there is no given equality relation on *anything*.  Axiom 0 requires no modification. In Axiom 1, we reinterpret the "uniqueness" clause as a *definition* of what it means for two parallel pre-relations to be equal, i.e. $\varphi=\psi$ if $\varphi(x,y)\Leftrightarrow\psi(x,y)$ for all $x$ and $y$ in the source and target of $\varphi$ and $\psi$.

Before stating the version of Axiom 2 we need some definitions.  We define a **set** to be a preset $A$ equipped with an [[equivalence relation|equivalence pre-relation]] $=_A$ which satisfies the [[principle of substitution]] and the [[identity of indiscernibles]].  A **relation** between sets $(A,{=_A})$ and $(B,{=_B})$ is a pre-relation $\varphi:A\looparrowright B$ which is *extensional* in the sense that if $\varphi(x,y)$, $x'=_A x$, and $y'=_B y$, then $\varphi(x',y')$.  Finally, a **function** $f:(A,{=_A}) \to (B,{=_B})$ is a relation which is (a) *[[entire relation|entire]]:* for any $x\in A$, there is a $y\in B$ with $f(x,y)$, and (b) *[[functional relation|functional]]:* if $f(x,y)$ and $f(x',y')$ and $x=_A x'$, then $y=_B y'$.

Now Axiom 2 reads: for any sets $(A,{=_A})$ and $(B,{=_B})$ and any relation $\varphi:(A,{=_A})\looparrowright (B,{=_B})$, there exists a set $({|\varphi|},{=_{|\varphi|}})$ and functions $p:{|\varphi|}\to A$ and $q:{|\varphi|}\to B$ such that: (1) for any $x\in A$ and $y\in B$, we have $\varphi(x,y)$ if and only if there exists $r\in {|\varphi|}$ with $p(r,x)$ and $q(r,y)$, and (2) for any $r\in {|\varphi|}$ and $s\in{|\varphi|}$, if $p(r)=_A p(s)$ and $q(r)=_B q(s)$, then $r=_{|\varphi|}s$.  Note that it is sufficient to assert merely that $|\varphi|$ is a pre-set and $p$ and $q$ are entire relations satisfying (1), since (2) can then be used to define $=_{|\varphi|}$ in such a way as to make it a set and $p$ and $q$ functions.

The construction of $\emptyset$ is just as in ordinary SEAR.  The construction of $1$ in ordinary SEAR used equality, but we now have a simple replacement for it: let $1$ be any pre-set containing an element, equipped with the equivalence relation such that $x=y$ always for any $x,y\in 1$.  Similarly, in this variant we can construct quotient sets without needing powersets; we simply enlarge the equivalence relation on the underlying pre-set of a set.

Of course, if $(A,{=_A})$ is a set, a **subset** of $A$ is a relation $1\looparrowright (A,{=_A})$.  Axiom 3 can now be translated as-is, or it can be simplified to assert merely the existence of a pre-set $P A$ such that any subset of $A$ is represented by some element of $P A$, with the uniqueness clause turned into a definition of $=_{P A}$.
The same idea applies to all the other axioms.

If to this equality-free version of SEAR we add a primitive notion of (non-extensional) "operation" and a [[choice operator]], we obtain [[SEAR+?]].

## Related pages and spinoffs

The following pages develop various aspects of set theory in SEAR or related theories.

* [[natural numbers in SEAR]]: defining individual natural numbers without the axiom of infinity.
* [[categories in SEAR]]: defining small categories, in response to a query on the blog.
* [[universes in SEAR]]: how to formulate an axiom of [[Grothendieck universes]].
* [[SEAR plus epsilon]]: a variant theory containing a [[choice operator]].
* [[ETCR]]: A categorical version of Bounded SEARC.
* [[ETCS with elements]]: A version of [[ETCS]] that, like SEAR, includes elements as a primitive.

[[!redirects SEPS]]
[[!redirects SEPF]]
[[!redirects three-sorted SEAR]]


SEAR plus epsilon

# SEAR+$\varepsilon$
* table of contents
{: toc}

## Goal

The goal of this page is to prove that a non-extensional [[choice operator]] is conservative over a theory without AC.  I'll take the theory without AC to be [[SEAR]], for definiteness and since that's where [[Mike Shulman|I'm]] most comfortable.

## Defining SEAR+$\varepsilon$

For the theory with a non-extensional choice operator, consider the following theory, which is a variant of the version of SEAR without fundamental equality described at [[SEAR]].  There are four basic sorts: *pre-sets*, *pre-relations*, *elements*, and *operations* (or *pre-functions*).  Each element belongs to a pre-set, and each pre-relation and operation has a source and target which are pre-sets.  For a prerelation $\phi:A\looparrowright B$ and $x\in A$, $y\in B$, we have the assertion $\phi(x,y)$, and for an operation $f:A\rightsquigarrow B$ and $x\in A$ we have an element $f(x)\in B$.  There is no predefined notion of equality of anything.   Axioms 0 and 1 of SEAR are unmodified, except that the uniqueness clause of the latter is interpreted as a *definition* of when two parallel pre-relations are called "equal".  We also impose

+-- {: .un_prop}
###### Axiom $1+\varepsilon$ (Choice operator)
If $\varphi:A\looparrowright B$ is a pre-relation such that for every $x\in A$ there exists a $y\in B$ with $\varphi(x,y)$, then there exists an operation $\varepsilon_\varphi :A\rightsquigarrow B$ such that $\varphi(x,\varepsilon_\varphi(x))$ for all $x\in A$.
=--

A *set* is defined to be a pre-set $A$ equipped with an equivalence pre-relation $=_A$ which satisfies the [[principle of substitution]] and the [[identity of indiscernibles]].  A *relation* between sets is a pre-relation which is *extensional*, i.e. if $\varphi(x,y)$, $x'=_A x$, and $y'=_B y$, then $\varphi(x',y')$.  Likewise, a *function* $f:A\to B$ between sets is an operation $f:A\rightsquigarrow B$ which is extensional, i.e. if $x' =_A x$ then $f(x')=_B f(x)$.  We define two functions $f,g:A\to B$ to be *equal* if $f(x)=_B g(x)$ for all $x\in A$.  Note that we have no notion of equality for arbitrary operations between pre-sets.

For any operation $f:A\rightsquigarrow B$ between *sets*, we have a pre-relation (its *graph*) defined by $\Gamma_f(x,y) \Leftrightarrow (f(x)=_B y)$, which is *[[entire relation|entire]]* in the sense that for any $x\in A$, there is a $y\in B$ with $\Gamma_f(x,y)$.  This pre-relation is extensional in $y$, but not in $x$ unless $f$ is actually a function (in which case $\Gamma_f$ is a *[[functional relation]]* in the usual sense).  Conversely, Axiom $1+\varepsilon$ says that any entire pre-relation induces an operation, and for an entire and functional relation between sets, this induced operation is a function (and is unique, in the sense of equality of functions defined above).

Now Axiom 2 can be taken to read: For any sets $A,B$ and any relation $\varphi:A\looparrowright B$, there exists a pre-set $|\varphi|$ and operations $p:{|\varphi|}\rightsquigarrow A$ and $q:{|\varphi|}\rightsquigarrow B$ such that $\varphi(x,y)$ iff there exists $z\in {|\varphi|}$ with $p(z)=_A x$ and $q(z)=_B y$.  We can then define $z=_{|\varphi|} z'$ iff $p(z)=_A p(z')$ and $q(z)=_B q(z')$ to make $|\varphi|$ into a set and $p,q$ into functions that are jointly injective.  Axioms 3, 4, and 5 are easy to translate.

We call this theory $SEAR+\varepsilon$.  Clearly the sets, elements (with defined equality), and relations in any model of $SEAR+\varepsilon$ satisfy the axioms of SEAR.  Conversely, from any model of SEAR-C we can construct a model of $SEAR+\varepsilon$ by taking the pre-sets, pre-relations, and operations to be the SEAR-C sets, relations, and functions respectively.  With this interpretation axiom $1+\varepsilon$ is precisely the SEAR-C [[axiom of choice]].  The question is whether we can get $SEAR+\varepsilon$ without assuming or implying choice.

## Conservativity over COSHEP

Our goal is to prove that $SEAR+\varepsilon$ is conservative over SEAR.
But since I haven't quite figured out how to do that (or, to be fair, even whether it's true), as a warm-up let's prove that the same thing is true when we add [[COSHEP]], a choice-like axiom notably weaker than full AC, to both sides.

Suppose we have a model of SEAR satisfying COSHEP, call it $V$. Recall that COSHEP means that in $V$, every set admits a surjection from a [[projective object|projective]] one.  We define a model of $SEAR+\varepsilon$ as follows.

* The pre-sets are the projective $V$-sets.
* The elements are the $V$-elements.
* The pre-relations are the $V$-relations.
* The operations are the $V$-functions.

We will continue to qualify the notions in the old model $V$, using unqualified words such as "set" for the new notions defined above.

Axiom 0 of $SEAR+\varepsilon$ is obviously true.  For Axiom 1, we need to translate a first-order formula in the language of $SEAR+\varepsilon$ into a formula in the language of SEAR in order to apply Axiom 1 of $V$, but the above dictionary tells us exactly how to do that.  Axiom $1+\varepsilon$ is true because we have chosen the pre-sets to be the *projective* sets in $V$, so any entire $V$-relation between them contains a $V$-function.

Now a *set* $A$ in our putative model of $SEAR+\varepsilon$ is a $V$-set equipped with an equivalence relation $=_A$, so in particular it has a quotient $A/{=_A}$.  Now any *relation* $A\looparrowright B$, meaning extensional relation, induces a $V$-relation $(A/{=_A}) \looparrowright (B/{=_B})$, and conversely any such $V$-relation induces a relation $A\looparrowright B$, setting up a meta-bijection.  The same is true of functions $A\to B$ and $V$-functions $(A/{=_A}) \to (B/{=_B})$.  It follows that we have meta-functors $Set\to Set_V$ and $Rel\to Rel_V$ given by "take quotients" that are [[full and faithful functor|fully faithful]].

I claim that in fact these functors are [[essentially surjective functor|essentially surjective]] as well.  For given any $V$-set $A$, by COSHEP there is a projective set $P$ and a surjection $e:P\to A$. Since $Set_V$ is a [[regular category]], it follows that $A$ is the [[quotient set]] of the [[kernel pair]] of $e$.  But that means that if we call this kernel pair $=_P$, then $P$ becomes a new-style set which maps onto $A$ under the quotient functor; hence this functor is essentially surjective.

It follows that the meta-categories $Set$ and $Rel$ (in the new sense) are [[equivalence of categories|equivalent]] to the old meta-categories $Set_V$ and $Rel_V$.  Since the remaining axioms of SEAR are essentially just statements about these categories, their truth in the "new world" follows immediately from their truth in $V$.  (There's no need to worry about a meta-axiom-of-choice, since these axioms are just statements about finite numbers of objects and morphisms, so "fully faithful and essentially surjective" is a perfectly sufficient notion of "equivalence.")

Thus, we have constructed a model of $SEAR+\varepsilon$.  Furthermore, its underlying SEAR-model is equivalent to $V$.  (The notion of "equivalence" here means "(weak) equivalence of categories of sets," as above.  But since SEAR intrinsically aheres to the [[principle of equivalence]], this suffices to show that exactly the same first-order statements are true in both.)  It follows that the statements *in the language of SEAR* which are true in this new model of $SEAR+\varepsilon$ are precisely those which are true in $V$.

Therefore, any statement in the language of SEAR which is true in all models of SEAR that underlie models of $SEAR+\varepsilon$ is, in fact, true in *all* models of SEAR+COSHEP.  This is the conservativity result we were looking for.  Its practical upshot is that if you want to work in SEAR+COSHEP, you might as well work in $SEAR+\varepsilon$ if it's more convenient to have a choice operator, since the theorems you can prove in either case will be exactly the same.

Note that the model of $SEAR+\varepsilon$ we've just constructed also satisfies COSHEP (as it must, given its equivalence to $V$).  In fact, if a model of $SEAR+\varepsilon$ admits an "identity" equivalence pre-relation $\equiv_A$ on every pre-set, and relative to which all pre-relations and operations are extensional, then it must satisfy COSHEP---for then every set $(A,{=_A})$ is the surjective image of the set $(A,{\equiv_A})$, which is projective by Axiom $1+\varepsilon$.

## Without COSHEP

Now [[Toby Bartels|I]] step in to say: $\mathbf{SEAR} + \varepsilon \vDash COSHEP$.

The reason is that every preset *does* admit an identity prerelation as in the last paragraph above; let $x \equiv_A y$ if $x \sim_R y$ for every reflexive prerelation $R: A \looparrowright A$ (or even for every equivalence prerelation).  This will work also in $\mathbf{ETCS} - AC + \varepsilon$ by [[quantification|quantifying]] over prefunctions $f: A \to \mathcal{P}1$ and using the kernel of $f$ (relative to the standard equality on truth values) as the equivalence relation.  (Of course, $\mathbf{SEAR}$ is capable of using this method too.)  It will still work in versions with intuitionistic logic, but not (as far as I can see) in $\mathbf{CETCS} - COSHEP + \varepsilon$ (following [Palmgren](http://www.math.uu.se/~palmgren/cetcs.pdf)), where one cannot take power sets or quantify over subsets.

[[Mike Shulman]]: Ah, you're right.  I actually thought of that briefly, but then I didn't immediately see how to prove the following.

+-- {: .num_theorem #equiv-exten}
###### Theorem
Define $x \equiv_A y$ if $R(x,y)$ for every equivalence pre-relation  $R: A \looparrowright A$.  Then every operation and every pre-relation is extensional with respect to these identity relations.
=--
+--{: .proof}
###### Proof
Suppose $f:A\rightsquigarrow B$ is an operation.  Define $K_f:A\looparrowright A$ by $K_f(x,x')$ iff $f(x)\equiv_B f(x')$.  Then $R$ is an equivalence pre-relation, so if $x\equiv_A x'$, then $K_f(x,x')$, and hence $f(x)\equiv_B f(x')$.

Similarly, suppose $\varphi:A\looparrowright B$ is a pre-relation.  Define $R_\varphi:A\looparrowright A$ by $R_\varphi(x,x')$ iff $\varphi(x,y)\Leftrightarrow\varphi(x',y)$ for all $y\in B$.  Then $R_\varphi$ is an equivalence pre-relation, so if $x\equiv_A x'$, then $R_\varphi(x,x')$ and so $\varphi(x,y)\Leftrightarrow\varphi(x',y)$ for all $y\in B$.
=--

Of course, as you point out, this is very impredicative.  Does that mean that your original [suggestion](http://golem.ph.utexas.edu/category/2009/09/towards_a_computeraided_system.html#c026817) would only acceptable be for someone who either (1) accepts COSHEP or (2) doesn't accept (quantification over) powersets?

_Toby_:  That may be so; I hadn\'t thought through all of the implications, so I was hoping otherwise.  However, I\'m not sure that $\varepsilon$ has to work as in Axiom $1 + \varepsilon$; how about this?

+-- {.un_prop}
###### Axiom $\varepsilon$ (Global Choice Operator)

Add a symbol (actually several symbols) to the language for an operation that assigns to each set $A$:

*  a set $A'$,
*  an injective function $i_A: A \to A'$,
*  and an element $\epsilon_A$ of $A'$;

then add the axiom that $\epsilon_A$ belongs to the image of $i_A$ if (hence iff) $A$ is inhabited.
=--

With excluded middle, this is a theorem with $A + 0^A$ for $A'$ as in the discussion at [[choice operator]].  But even this may be too strong; I really want to say that $\epsilon_A$ takes values in $A_\bot$ (the set of subsingleton subsets of $A$) with $\epsilon_A$ an inhabited subsingleton if (hence iff) $A$ is inhabited, but the existence of $A_\bot$ is itself impredicative (in the presence of function sets) since $1_\bot$ is the set of truth values.

## Without COSHEP, predicatively

Okay, second try.  Define **constructive SEAR** to use [[intuitionistic logic]] and consist of Axioms 0, 1B, 2, and 4 of SEAR (no power sets) together with

* The existence of [[quotient sets]] (i.e. $Set$ is an [[exact category]])
* The existence of [[disjoint unions]] (i.e. now $Set$ is a [[pretopos]])
* The existence of [[dependent products]] (i.e. now $Set$ is a $\Pi$-[[Π-pretopos|pretopos]]).

Now modify it as above, removing basic equality and adding a basic notion of "operation," to obtain **CSEAR+$\varepsilon$**.  Note that the axiom of quotient sets becomes redundant, given the definition of "set" as a preset with an equivalence prerelation.

The same argument as above should show that CSEAR+$\varepsilon$ is conservative over CSEAR+COSHEP.  (Just as Bounded SEAR is equivalent to ETCS, CSEAR+COSHEP is equivalent to Palmgren's CETCS.)  However, the converse argument above fails since we cannot define $\equiv_A$ by quantifying over relations (note that avoiding this requires not only throwing out powersets, but restricting axiom 1 to [[bounded quantification]]).

Now: is CSEAR+$\varepsilon$ conservative over CSEAR?

[[!redirects SEAR plus epsilon]]
[[!redirects SEAR plus ?]]
[[!redirects SEAR plus ?]]
[[!redirects SEAR+epsilon]]
[[!redirects SEAR+?]]
[[!redirects SEAR+?]]
[[!redirects SEAR epsilon]]
[[!redirects SEAR ?]]
[[!redirects SEAR ?]]


Searching in the nLab > history

< [[Searching in the nLab]]


Searching the nLab

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

# Contents
* table of contents
{: toc}

## Searching the nLab

There are several methods of searching the nLab:

1. **The built-in search.**  This is via the search box at the top of every page.  The distinguishing characteristics of this search are:
   1. It uses _regular expressions_.
   1. It searches the _source_ of each page.

1. **Formula Search**:  [nLab Formula Search](https://nlabsearch.mathweb.org/) provides an instance of the [MathWebSearch](https://search.mathweb.org) engine for nLab.
To use it enter a formula query (LaTeX with query variables of the
form `?a`, `?b`, ...) into the central input box (or just select one of the
examples), wait until the MathML has been rendered (query variables are
red), and hit search. If you click on one of the hits, you get a URL,
which puts you into nLab and highlights the formula hit (in gray).

1. **An external search engine.**

   Most search engines allow restricting the search to a single site, such as to `ncatlab.org/nlab`.

   The distinguishing characteristics of an external search are:

   1. Usually the search is for alphanumeric characters.

   1. It searches the _rendered version_ of each page, instead of the source code.

## Google Search

+-- {: .standout style="text-align: center"}
 *Search the nLab using Google:*
 <form  name="gsearch" method="get" action="http://www.google.com/search"><input type="text" size="30" name="as_q" style="display: inline"/><input type="hidden" name="as_sitesearch" value="http://ncatlab.org/nlab/"/></form>
=--

## Formula Search

[MathWebSearch](https://search.mathweb.org) (MWS) is a content-based search engine for mathematical formulae.
It indexes MathML formulae, using a technique derived from automated theorem proving: Substitution Tree Indexing.

As the project is still under development, the authors would be happy to hear
your feedback. If you have some comments you can leave it
[here](https://nforum.ncatlab.org/discussion/9792/formula-search/#Item_0). If you
have a bug report or a feature request please open a issue on the github
[repository](https://github.com/MathWebSearch/mws-frontend).

## Regular Expressions

Regular expressions are a powerful way of extending search capabilities to take into account that one often wants to search for more than just a set phrase.  In a regular expression, certain characters are declared to be "special" and have a particular interpretation (somewhat like TeX with its special catcodes).  A special character can always be "escaped" to interpret it as an ordinary character.  Thus `.` means "match any single character" but `\.` means "match a period".

As Instiki is written in ruby, it uses the ruby version of regular expressions (each language has its own version; the differences are usually minor).  The following is based on the list at [ruby-doc](http://www.ruby-doc.org/docs/ProgrammingRuby/html/language.html#UJ).  It has been condensed slightly to those aspects likely to be of use here:

Special characters
: The special characters are: `.`, `|`, `(`, `)`, `[`, `\`, `^`, `{`, `+`, `$`, `*`, and `?`. To match one of these characters, precede it with a backslash.  All other characters ordinarily just match themselves unless they are made somehow special by one of the special characters.

`\b`, `\B`
: Match word boundaries and nonword boundaries respectively.  Thus `cat` matches against `category` and `cat` but `cat\b` only matches `cat` (and `scat`).

`[`...`]`
: This matches against a single character in a list.  The characters `|`, `(`, `)`, `[`, `^`, `$`, `*`, `?` are treated as regular characters in such a list.  You can specify a range using `-`: thus, `a-z`.  To include a `]` or `-` it must come at the start of the list.  A `^` at the start negates the list.

`\d`, `\s`, `\w`
: These match, respectively, digits, spaces, and word characters.

`\D`, `\S`, `\W`
: These are the negations of the lowercase versions.

`.` (period)
: Matches any character except a newline.

`(`...`)`
: Parentheses group pieces of the regular expression.  This is important for the following modifications.  In these, $x$ stands for a sub-expression which can be a single character, a `[`...`]`, or a `(`...`)`.

$x$`*`
: Matches zero or more occurrences of $x$.  Thus `ab*` matches `a`, `ab`, `abb`, and so forth.  Similarly, `(cat)*` matches `cat`, `catcat`, `catcatcat`, and so forth.  This will try to match as much as possible; use $x$`*?` to make it match as little as possible.

$x$`+`
: Matches one or more occurrences of $x$.  Thus `ab+` matches `ab`, `abb`, but not `a`.  This will try to match as much as possible; use $x$`+?` to make it match as little as possible.

$x$`{`$m$`,`$n$`}`
: Matches at least $m$ and at most $n$ occurrences of $re$.  This will try to match as much as possible; use $x$`{`$m$`,`$n$`}?` to make it match as little as possible.

$x$`?`
: Matches zero or one occurrence of $x$.

$x_1$|$x_2$
: Matches either $x_1$ or $x_2$.

[[!redirects Searching nLab]]
[[!redirects searching nLab]]
[[!redirects Searching the nLab]]
[[!redirects searching the nLab]]
[[!redirects Searching in the nLab]]
[[!redirects searching in the nLab]]

category: meta


Sebastian de Haro

* [webpage](http://www.uva.nl/over-de-uva/organisatie/medewerkers/content/h/a/s.deharo/s.de-haro-olle.html)

## related $n$Lab entries

* [[AdS-CFT correspondence]]

* [[Fefferman-Graham ambient construction]]

category: people


Sebastian Dumitru

* [Google Scholar entry](https://scholar.google.co.uk/citations?user=tVcpaWkAAAAJ&hl=en)

## Selected writings

On [[heterotic line bundles]] in the [[hidden sector]] of [[heterotic M-theory]]:

* [[Anthony Ashmore]], [[Sebastian Dumitru]], [[Burt Ovrut]], Section 4.2 of: _Line Bundle Hidden Sectors for Strongly Coupled Heterotic Standard Models_ ([arXiv:2003.05455](https://arxiv.org/abs/2003.05455))

* [[Anthony Ashmore]], [[Sebastian Dumitru]], [[Burt Ovrut]], _Explicit Soft Supersymmetry Breaking in the Heterotic M-Theory B−L MSSM_ ([arXiv:2012.11029](https://arxiv.org/abs/2012.11029))

* [[Anthony Ashmore]], [[Sebastian Dumitru]], [[Burt Ovrut]], *Hidden Sectors from Multiple Line Bundles for the B−L MSSM* ([arXiv:2106.09087](https://arxiv.org/abs/2106.09087))

* [[Sebastian Dumitru]], [[Burt A. Ovrut]], *Heterotic M-Theory Hidden Sectors with an Anomalous $U(1)$ Gauge Symmetry* ([arXiv:2109.13781](https://arxiv.org/abs/2109.13781))

category: people


Sebastian Goette

* [webpage](http://home.mathematik.uni-freiburg.de/goette/)

## Selected writings

On [[quaternionic line bundles]]:

* Diarmuid Crowley, [[Sebastian Goette]], _Kreck-Stolz invariants for quaternionic line bundles_, Transactions of the American Mathematical Society
Vol. 365, No. 6 (2013), pp. 3193-3225 ([arXiv:1012.5237](https://arxiv.org/abs/1012.5237), [jstor:23513538](https://www.jstor.org/stable/23513538))

## Related $n$Lab entries

* [[multiplicative spectral sequence]]

category: people


Sebastian M. Schmidt

* [webpage](https://www.phy.anl.gov/theory/staff/basti.html)

## Selected writings

On [[confinement]] and [[hadron]]-mass generation in [[QCD]] ([[mass gap problem]]):

* {#RobertsSchmidt20} [[Craig Roberts]], [[Sebastian M. Schmidt]], _Reflections upon the Emergence of Hadronic Mass_, The European Physical Journal Special Topics volume 229, pages 3319–3340 (2020) ([arXiv:2006.08782](https://arxiv.org/abs/2006.08782), [doi:10.1140/epjst/e2020-000064-6](https://doi.org/10.1140/epjst/e2020-000064-6))

> More than 98% of visible mass is contained within [[atomic nuclei|nuclei]].  In first approximation, their atomic weights are simply the sum of the masses of all the [[neutrons]] and [[protons]] ([[nucleons]]) they contain.  Each nucleon has a [[mass]] $m_N \sim 1$ [[GeV]], i.e. approximately 2000-times the [[electron]] mass. The [[Higgs  boson]] produces the latter, but  what produces the masses of the neutron and proton?  This is the question posed above, which is pivotal to the development of modern physics: how can science explain the emergence of [[hadron|hadronic]] mass (EHM)? $[\cdots]$

> Modern science is thus encumbered with the fundamental problem of [[gluon]]  and [[quark]] [[confinement]]; and confinement is crucial because it ensures absolute stability of the [[proton]].  $[\cdots]$ Without confinement,our Universe cannot exist.

> As the 21st Century began, the Clay Mathematics Institute established seven Millennium Prize Problems [ [11](mass+gap#ReferencesMassGapProblem) ]. Each represents one of the toughest challenges in mathematics. The set contains the [[mass gap problem|problem of confinement]]; and  presenting a sound solution will win its discoverer 1,000,000 bucks. Even with  such motivation, today, almost fifty years after the discovery of [[quarks]] [12–14], no rigorous solution has been found. Confinement and EHM are inextricably linked. Consequently, as  science plans for the next thirty  years, solving the problem of EHM has become a _grand challenge_. $[\cdots]$

> In trying to match [[QCD]] with Nature, one confronts the many complexities of strong, nonlinear dynamics in relativistic quantum field theory, e.g. the loss of particle number conservation, the frame and scale dependence of the  explanations and interpretations of observable processes, and the evolving character of the relevant degrees-of-freedom. [[electroweak field|Electroweak theory]] and phenomena are essentially [[perturbative quantum field theory|perturbative]]; hence, possess little of this complexity.  Science has never before encountered an [[interaction]] such as that at work in [[QCD]]. Understanding this interaction, explaining everything of which it is capable, can potentially change the way we look at the Universe.

category: people

[[!redirects Sebastian Schmidt]]


Sebastian Mizera

* [personal page](https://www.sns.ias.edu/~smizera/)

* [GoogleScholar page](https://scholar.google.co.uk/citations?user=zoabscUAAAAJ&hl=en)

## Selected writings

On the [[S-matrix]] ([[scattering amplitudes]] in [[quantum field theory]]):

* [[Sebastian Mizera]], *Physics of the Analytic S-Matrix* [[arXiv:2306.05395](https://arxiv.org/abs/2306.05395)]

category: people


Sebastian Pauli

* [webpage](http://www.uncg.edu/mat/faculty/pauli/)

## related $n$Lab entries

* [[congruence subgroup]]

category: people


Sebastian Wolf

* [institute page](https://homepages.uni-regensburg.de/~wos07573/)

## Selected writings

Development of [[(infinity,1)-category theory|$\infty$-category theory]] [[category internal to an (infinity,1)-topos|internal to]] any [[(∞,1)-topos]]:

internal [[(infinity,1)-limits]] and [[(infinity,1)-colimits]]:

* [[Louis Martini]], [[Sebastian Wolf]], *Limits and colimits in internal higher category theory* [[arXiv:2111.14495](https://arxiv.org/abs/2111.14495)]

internal [[presentable (∞,1)-categories]]:

* [[Louis Martini]], [[Sebastian Wolf]], *Presentable categories internal to an ∞-topos* [[arXiv:2209.05103](https://arxiv.org/abs/2209.05103)]

and [[(infinity,1)-toposes|$\infty$-toposes]] internal to [[(infinity,1)-toposes|$\infty$-toposes]]:

* [[Louis Martini]], [[Sebastian Wolf]], *Internal higher topos theory* [[arXiv:2303.06437](https://arxiv.org/abs/2303.06437)]

On [[proper geometric morphisms]] between [[(infinity,1)-toposes|$(\infty,1)$-toposes]]:

* [[Louis Martini]], [[Sebastian Wolf]], _Proper morphisms of ∞-topoi_ [[arXiv:2311.08051](https://arxiv.org/abs/2311.08051)].

category: people


Sebastian Ørsted

* institute page

## Selected writings

On [[homotopy limits]]/[[homotopy colimits]] via [[homotopy ends]]/[[homotopy coends]] and [[homotopy weighted colimit|homotopy weighted (co)limits]]:

* {#ArkhipovOrsted18} [[Sergey Arkhipov]], [[Sebastian Ørsted]], *Homotopy (co)limits via homotopy (co)ends in general combinatorial model categories* ([arXiv:1807.03266](https://arxiv.org/abs/1807.03266))

category: people

[[!redirects Sebastian Orsted]]


Sebastiano Carpi

* [webpage](https://sites.google.com/site/sebastianocarpi/home)

## Selected writings

Relating [[vertex operator algebras]] with [[conformal nets]]:

* {#CarpiKawahigahshiLongoWeiner15} [[Sebastiano Carpi]], [[Yasuyuki Kawahigashi]], [[Roberto Longo]], Mihály Weiner, _From vertex operator algebras to conformal nets and back_, Memoirs of the American Mathematical Society **254** 1213 (2018) [[doi:10.1090/memo/1213](https://doi.org/10.1090/memo/1213), [arXiv:1503.01260](http://arxiv.org/abs/1503.01260)]

* {#Carpi16} [[Sebastiano Carpi]], _Operator algebras and vertex operator algebras_, Contribution to the Proceedings of the 14th Marcel Grossmann Meeting - MG14 (Rome, 2015) [[arXiv:1603.06742](http://arxiv.org/abs/1603.06742), [doi:10.1142/9789813226609_0508](https://doi.org/10.1142/9789813226609_0508)]

and for [[super vertex operator algebras]] and particularly the [[monster vertex operator algebra]]:

* [[Sebastiano Carpi]], [[Tiziano Gaudio]], [[Robin Hillier]], *From vertex operator superalgebras to graded-local conformal nets and back* [[arXiv:2304.14263](https://arxiv.org/abs/2304.14263)]

## Related entries

* [[conformal net]]

* [[vertex operator algebra]]

category: people


second

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

#Contents#
* table of contents
{:toc}

## Idea

The _second_ is a [[physical unit]] of [[time]]. It is defined by taking the fixed numerical value of the caesium [[frequency]] $\Delta \nu_{Cs}$, the unperturbed ground-state hyperfine transition frequency of the caesium-133 [[atom]], to be $9192631770$ when expressed in the [[physical unit|unit]] $Hz$, which is equal to $s^{-1}$

## Related concepts

* [[year]]

## References

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

[[!redirects seconds]]


second Fraenkel model

# The second Fraenkel model
* table of contents
{: toc}

## Idea

The _second Fraenkel model_ is a model of the [[set theory]] [[ZFA]] that doesn't satisfy the [[axiom of choice]]. It was one of the first examples of a [[permutation model]] of set theory.

The [[basic Fraenkel model]] is similar, but uses the automorphism group of a [[countable set]].

## Description

Fraenkel's description used the language of [[material set theory]], and indeed most set theorists would give the description of the Fraenkel model using this language, but it can be described quite simply from a [[structural set theory|structural]] perspective, and then the original version can be recovered by considering [[pure sets]] (allowing [[urelement|atoms]]).

The model is given by the [[topos]] of [[sets]] with an [[action]] of an [[open subset|open]] [[subgroup]] of the [[group]] $(\mathbb{Z}/2)^\mathbb{N}$ for a certain [[topological group|topology]] on this group. Open subgroups are the finite-[[index]] subgroups $\prod_{i\in I} H_i\times (\mathbb{Z}/2)^{(\mathbb{N} - I)}$ for finite $I\subset \mathbb{N}$ and $H_i \le \mathbb{Z}/2$. Arrows in this topos are allowed to be equivariant for an open (possibly proper) subgroup of the groups acting on the domain and codomain.

[[!redirects Fraenkel model of ZFA]]
[[!redirects second Fraenkel model of ZFA]]
[[!redirects Fraenkel model]]
[[!redirects second Fraenkel model]]
[[!redirects Fraenkel model of ZFA]]
[[!redirects Frankel model of ZFA]]


second law of thermodynamics

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* In the context of [[gravity]] there is a [[generalized second law of thermodynamics]].

## References

* Walid K. Abou Salem, [[Jürg Fröhlich]], *Status of the Fundamental Laws of Thermodynamics*, Journal of Statistical Physics volume 126, pages 1045–1068 (2007) ([arXiv:math-ph/0604067](https://arxiv.org/abs/math-ph/0604067), [doi:10.1007/s10955-006-9222-8](https://doi.org/10.1007/s10955-006-9222-8))

See also:

* Wikipedia, *[Second law of thermodynamics](https://en.wikipedia.org/wiki/Second_law_of_thermodynamics)*