strong monomorphism

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##  Definition

A **strong monomorphism** in a [[category]] $C$ is a [[monomorphism]] which is right [[orthogonality|orthogonal]] to any [[epimorphism]].  The dual notion is, of course, [[strong epimorphism]].

## Remarks

* If $C$ has [[coequalizers]], then any morphism which is right orthogonal to epimorphisms must automatically be a monomorphism.

* If $C$ has [[kernel pairs]] and [[coequalizers]] of kernel pairs, then any morphism which is right orthogonal to epimorphisms must automatically be a monomorphism.

* Every [[regular monomorphism]] is strong. The converse is true if $C$ is [[regular category| co-regular]].

* Every strong monomorphism is [[extremal monomorphism|extremal]]; the converse is true if $C$ has [[pushouts]].

## Examples

* A nice example of strong monomorphisms in a category are the subspace inclusions in the category of [[diffeological spaces]].  In this setting, any subset $Y$ of a diffeological space $X$ is again a diffeological space.  If smooth, the inclusion $\iota:Y \rightarrow X$ is always a monomorphism, but it is a strong monomorphism if and only if $Y$ has "enough" plots, that is if $\varphi: U\rightarrow Y$ is a plot if and only if the composite $\iota\varphi: U\rightarrow X$ is a plot.

* Let $C$ be the category whose objects are the integers $\mathbf{Z}$, and whose morphisms are generated from arrows
$$ s_i, t_i : i \to i+1 $$
subject to the relations
$$ s\circ s = s\circ t = t\circ s = t\circ t.$$
Then the only epimorphisms and monomorphisms in $C$ are the identities, thus every map is right orthogonal to all epimorphisms but only the identities are strong monomorphisms.

## Properties

+-- {: .num_prop #PushoutOfStrongMonomorphismInQuasitopos}
###### Proposition
**pushout of [[strong monomorphism]] in [[quasitopos]]**

Suppose that $(\mathrm{T},\mathcal{C})$ is either

* ([[monomorphism]],[[topos]]), or
* ([[strong monomorphism]],[[quasitopos]])

Suppose that
$$\array{
O_{0,1} & \to & O_{1,1} \\
\downarrow m &&\downarrow h  \\
O_{0,0} & \to & O_{1,0}
}$$
is a [[commutative diagram]] in $\mathcal{C}$ such that

* * $m$ is $\mathrm{T}$ in $\mathcal{C}$
* * the diagram is a pushout in $\mathcal{C}$

Then

* * $h$ is $\mathrm{T}$ in $\mathcal{C}$
* * the diagram is a pullback in $\mathcal{C}$

=--

See at _[[quasitopos]]_ [this lemma](quasitopos#PushoutOfStrongMonos). Note that the result for quasitoposes immediately implies the result for toposes, since all monomorphisms $i: A \to B$ in a topos are [[regular monomorphism|regular]] ($i$ being the equalizer of the arrows $\chi_i, t \circ !: B \to \Omega$ in

$$\array{
 & & 1 \\
 & \mathllap{!} \nearrow & \downarrow \mathrlap{t} \\
B & \underset{\chi_i}{\to} & \Omega
}$$

where $\chi_i$ is the classifying map of $i$) and therefore strong.

[[!redirects strong monomorphisms]]


strong natural transformation

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## Idea

The notion of (tensorially) *strong natural transformation* is the compatible notion of [[natural transformation]] on (tensorially) [[strong functors]].

## Properties

### Relation to enriched natural transformations

For [[closed monoidal categories]] $V$ there is a close relation between $V$-strong natural transformations and $V$-[[enriched natural transformations]].

For the moment see at *[enriched monad -- relation to strong monads](enriched+monad#RelationToStrongMonads)* for more

## Selected writings

* [[Kruna S. Ratkovic]], Def. 3.1.4 in: *Tensorial Strength*, Section 3 of: *Morita theory in enriched context* (2012) [[arXiv:1302.2774](https://arxiv.org/abs/1302.2774), [hal:tel-00785301](https://theses.hal.science/tel-00785301/document)]

[[!redirects strong natural transformations]]


strong shape theory

[[!redirects Strong shape theory]]

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## Idea ##

In strong shape theory, you remember all of the [[homotopy theory|homotopy]] coherence information in the approximations to a [[topological space]].

### Background ###
In classical [[shape theory]] or [[Čech homotopy]], the polyhedra or simplicial sets approximating a given space are treated as giving a [[pro-object]] in the homotopy category. In the early strong versions of shape theory the diagrams are thought of as being in the coherent homotopy pro category of simplical sets, or one of the equivalent forms (e.g. that due to Edwards and [[Harold Hastings|Hastings]] (1973)).  This is now probably better treated by the methods discussed in the entry on [[shape theory]].

### History ###

The earliest paper on strong shape would seem to be by D.E. Christie, in 1944. He looked at a 2-truncated version of the later theory. The next appearance of the idea would seem to be in the 1970s when more or less independently, Edwards and Hastings, Porter, Mardesic and Quigley came out with various approaches to the problem of structuring shape theory in a richer manner. The availability of work on  homotopy coherence, homotopy limits and colimits, etc. would seem to have been one of the key steps n this development, whilst problems in handling the naturally occurring  homotopy coherent diagrams of spaces or simplicial sets, led to a clearer understanding of the role of higher homotopy theory in other areas of mathematics.

## Related concepts

* [[shape theory]]

  * [[Borsuk's shape theory]]

## References ##

* D.A. Edwards and [[H. M. Hastings]], (1976), Čech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Maths. 542, Springer-Verlag, [pdf](http://www.math.uga.edu/~davide/Cech_and_Steenrod_Homotopy_Theories_with_Applications_to_Geometric_Topology.pdf)
* J.T. Lisica and [[Sibe Mardesic|S. Mardešić]], Coherent prohomotopy and strong shape theory, Glasnik Mat. 19(39) (1984) 335--399.

* [[Sibe Mardesic|S. Mardešić]], _Strong shape and homology_, Springer monographs in mathematics, Springer-Verlag.

* [[Michael Batanin]], [Categorical strong shape theory](http://www.numdam.org/numdam-bin/fitem?id=CTGDC_1997__38_1_3_0), Cahiers Topologie Géom. Différentielle Catég. 38 (1997), no. 1, 3--66.

* [[T. Porter]], _Stability Results for Topological Spaces_, Math. Zeit. 150, 1974, pp. 1-21.

* [[T. Porter]], _Abstract homotopy theory in procategories_, Cahiers Top. Géom. Diff., 17, 1976, pp. 113-124, [numdam](http://archive.numdam.org/article/CTGDC_1976__17_2_113_0.pdf)

* [[T. Porter]],  _Coherent prohomotopical algebra_, [numdam](http://archive.numdam.org/article/CTGDC_1977__18_2_139_0.pdf), Cahiers Top. Géom. Diff. __18__, (1978) pp. 139-179;

* [[T. Porter]], _Coherent prohomotopy theory_, Cahiers Top. Géom. Diff. __19__, (1978) pp. 3-46, [numdam](http://archive.numdam.org/article/CTGDC_1978__19_1_3_0.pdf)

* F. Cathey, Jack Segal, _Strong shape theory and resolutions_, Topology and its Appl. __15__ (1983) 119--130

* Bernd Günther, _Strong shape of compact Hausdorff spaces_, Topology and its Applications __42__:2, 1991, pp. 165–174, doi; _A Tom Dieck theorem for strong shape theory_, Trans. Amer. Math. Soc. __338__ (1993), 857–870 [doi](https://doi.org/10.1090/S0002-9947-1993-1160155-1); _The use of semisimplicial complexes in strong shape theory_, Glas. Mat. Ser. III 27 (47) 1992, no. 1, 101–144 (contains [[quasicategory]] ideas in strong shape) [gBooks](https://books.google.hr/books?id=FfgyvnqoKBgC&lpg=PA101&ots=zYwj6-RmNo&dq=Gunther%20simplicial%20shape%20Glasnik&pg=PA101) ; _Comparison of the coherent pro-homotopy theories of Edwards-Hastings, Lisica-Mardešić and Günther_, Glas. Mat. Ser. III 26(46) (1991), no. 1-2, 141–176

[[!redirects strong shape theory]]
[[!redirects strong shape]]


strongly compact cardinal

## Definition

A [[cardinal]] $\kappa$ is __strongly compact__ if any $\kappa$-complete [[filter]] can be extended to a $\kappa$-complete [[ultrafilter]].

Here a [[filter]] is $\kappa$-complete if it is closed under intersections
of families with fewer than $\kappa$ elements.

## Properties

Strongly compact cardinals are [[measurable cardinals]].

The existence of a [[proper class]] of strongly compact cardinals
implies that images of [[accessible functors]]
are accessible as long as they are [[complete]] or [[cocomplete]].

## Related notions

* [[accessible functor]]

## References

Strongly compact cardinals were introduced by Keisler and Tarski in 1963.

For a basic theory, see

* [[Thomas Jech]], _Set theory_.

[[!redirects strongly compact cardinals]]


strongly compact topological space

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## Related concepts

* [[compact topological space]], [[sequentially compact topological space]], [[countably compact topological space]], [[paracompact topological space]], [[locally compact topological space]]

## References

* Maximilian Ganster, _Some remarks on strongly compact spaces and semi compact spaces_, Bull. Malaysian Math. Soc. (10) 2 (1987), 67–81. ([pdf](http://www.math.tugraz.at/~ganster/papers/17.pdf))

* S. Jafari and T. Noiri, _More on Strongly Compact Spaces_, Missouri J. Math. Sci. Volume 19, Issue 1 (2007), 52-61. ([Euclid](http://projecteuclid.org/euclid.mjms/1316092238))

[[!redirects stringly compact space]]

[[!redirects stringly compact spaces]]
[[!redirects stringly compact topological spaces]]


strongly connected site

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## Idea

A *strongly connected site* is a [[site]] satisfying sufficient conditions to make its [[topos of sheaves]] into a [[strongly connected topos]].

## Definition

Let $C$ be a [[locally connected site]]; we say it is a **strongly connected site** if it is also a [[cosifted category]]

## Properties

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

If $C$ is strongly connected site, then the [[sheaf topos]] $Sh(C)$ is a [[strongly connected topos]].

=--

Because the left adjoint $\Pi_0$ in the [[sheaf topos]] over a [[locally connected site]] is given by the [[colimit]] functor and colimits preserve finite [[product]]s on the [[sifted category]] $C^{op}$.

## Related concepts

* [[locally connected topos]] / [[locally ∞-connected (∞,1)-topos]]

  * [[connected topos]] / [[∞-connected (∞,1)-topos]]

  * [[strongly connected topos]] / [[strongly ∞-connected (∞,1)-topos]]

  * [[totally connected topos]] / [[totally ∞-connected (∞,1)-topos]]

* [[local topos]] / [[local (∞,1)-topos]].

* [[cohesive topos]] / [[cohesive (∞,1)-topos]]

and

* [[locally connected site]] / [[locally ∞-connected site]]

  * [[connected site]] / [[∞-connected site]]

  * **strongly connected site** / [[strongly ∞-connected site]]

  * [[totally connected site]] / [[totally ∞-connected site]]

* [[local site]] / [[∞-local site]]

* [[cohesive site]], [[∞-cohesive site]]

[[!redirects locally connected sites]]


strongly correlated system

under construction

Many-body systems, that is physical systems comprising many interacting particles are called strongly correlated if the interactions are strong in the sense that they are causing configurations which are not just perturbations of the non-interacting or weakly interacting system; and in which the correlations between different particles are also strong. Examples include physical [[plasma]], strongly correlated electron systems, many models of magnetism, frustrated magnetism, High-Tc superconductivity. Because of involved nonlinearity and or

* [pdf](http://www.isis.stfc.ac.uk/groups/theory/downloads/the-strong-correlations-puzzle8120.pdf)

* [html](http://cmt.harvard.edu/demler/layman.html)

category: physics


strongly extensional function

# Strongly extensional functions
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{: toc}

## Idea

In [[constructive mathematics]], one sometimes calls a [[function]] 'extensional' to stress that it preserves [[equality]] (as opposed to a [[prefunction]], which need not do so).  A function is _strongly_ extensional if it also reflects inequality in a relevant sense.  One usually employs this in a context where the principle of [[excluded middle]] is enough to prove that every function is strongly extensional.

## Definitions

This is probably not the only context where the term 'strongly extensional' is used, but it\'s the one that I know.

Let $X$ and $Y$ be [[sets]], each equipped with a [[tight apartness]] $\ne$.  Let $f$ be a [[function]] from $X$ to $Y$.  Then $f$ is __strongly extensional__ if $a \ne b$ whenever $f(a) \ne f(b)$.

Note that if $f$ is only a [[prefunction]] (in a [[foundation]] where this makes sense) and we impose the condition of strong extensionality, it follows (upon taking the [[contrapositive]], since [[equality]] is the [[negation]] of any tight apartness) that $f$ is extensional (and so a function).  Conversely, if $\ne$ is also the [[negation]] of [[equality]] (which is always the unique tight apartness in [[classical mathematics]]), then any extensional $f$ is also strongly extensional.  In this way, strong extensionality is a constructively stronger but classically equivalent variation of extensionality.

We have a [[category]] $Set_{\ne}$ whose [[objects]] are sets equipped with tight apartnesses and whose [[morphisms]] are strongly extensional functions.

A strongly extensional function is __strongly [[injective function|injective]]__ if it also preserves $\ne$.  (These are the [[regular monomorphisms]] of $Set_{\ne}$, the [[monomorphisms]] being merely injective in the usual sense of reflecting equality.)

## Examples

If $X$ has [[decidable equality]], then the [[negation]] of [[equality]] is a (in fact the unique) tight apartness on $X$, and any function from $X$ to any set $Y$ (with any tight apartness on $Y$) must be strongly extensional.

Any pointwise-[[continuous function]] between [[metric spaces]] is strongly extensional.  Using [[countable choice]] and [[Markov's principle]] (or actually weak versions thereof), it follows that any function to a metric space from a [[complete metric space]] (regardless of continuity!?) is strongly extensional.

A weak counterexample:  [[internal logic|Internal to]] the [[topos of sheaves]] on the [[unit interval]] $\mathbb{I}$, where an internal [[real number]] is (externally) a [[continuous map]] from $\mathbb{I}$ to the [[real line]] $\mathbb{R}$, consider the internal number $c$ given by the function $(t \mapsto t - t^2)$.  Then the [[subspace]] $\{0,c\}$ of the internal real line is an internal complete metric space (as proved by [Richman](#Richman)), we may define an internal continuous map $f\colon \{0,c\} \to \mathbb{R}$ by $f(0) = 0$ and $f(c) = 1$, and of course we have $f(0) \ne f(c)$ internally.  But the internal [[truth value]] $\{0 \ne c\}$ is only the open interval $]0,1[$, not all of $[0,1]$, so it\'s not completely true that $f$ is strongly extensional.  (However, this $f$ is [[double negation|not not]] strongly extensional, so a stronger counterexample would be nice.)

## References

* [[Fred Richman]]; Weak Markov\'s principle, strong extensionality, and countable choice; available from [Fred Richman\'s documents](http://math.fau.edu/richman/html/docs.htm).
{#Richman}

[[!redirects strongly extensional function]]
[[!redirects strongly extensional functions]]
[[!redirects strongly extensional map]]
[[!redirects strongly extensional maps]]
[[!redirects strongly extensional mapping]]
[[!redirects strongly extensional mappings]]


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## Definition

A [[locally ∞-connected (∞,1)-topos]]

$$
  (\Pi \dashv \Delta \dashv \Gamma) : \mathbf{H} \to \infty Grpd
$$

is called **strongly connected** if $\Pi$ preserves [[finite limit|finite]] [[(∞,1)-product]]s (hence in particular the [[terminal object]], which makes it also an [[∞-connected (∞,1)-topos]]).

Similarly for an $n$-connected $(n,1)$-topos.

For $n = 1$ this yields the notion of [[strongly connected topos]].

If in addition $\mathbf{H}$ is a [[local (∞,1)-topos]] then it is a [[cohesive (∞,1)-topos]].

## Related concepts

* [[locally connected topos]] / [[locally ∞-connected (∞,1)-topos]]

  * [[connected topos]] / [[∞-connected (∞,1)-topos]]

  * [[strongly connected topos]] / **strongly $\infty$-connected $(\infty,1)$-topos**

  * [[totally connected topos]] / [[totally ∞-connected (∞,1)-topos]]

* [[local topos]] / [[local (∞,1)-topos]].

* [[cohesive topos]] / [[cohesive (∞,1)-topos]]

and

* [[locally connected site]], [[locally ∞-connected site]]

* [[connected site]]

* [[local site]]

* [[cohesive site]]

[[!redirects strongly ∞-connected (∞,1)-topos]]

[[!redirects strongly ∞-connected (∞,1)-toposes]]
[[!redirects strongly ∞-connected (∞,1)-topoi]]

[[!redirects strongly infinity-connected (infinity,1)-toposes]]
[[!redirects strongly infinity-connected (infinity,1)-topoi]]

[[!redirects strongly connected (∞,1)-topos]]


strongly infinity-connected site

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## Idea

A *strongly $\infty$-connected site* is a [[site]] satisfying sufficient conditions to make the [[(∞,1)-sheaf (∞,1)-topos]] over it a [[strongly ∞-connected (∞,1)-topos]].

## Definition

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

Let $C$ be a [[∞-connected site|∞-connected locally ∞-connected site]]; we say it is a **strongly $\infty$-connected site** if it is also a [[cosifted (∞,1)-category]].

=--

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

If $C$ is in addition an [[∞-local site]] then it is an [[∞-cohesive site]].

=--

## Properties

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

If $C$ is a strongly $\infty$-connected site, then the [[(∞,1)-sheaf (∞,1)-topos]] $Sh_{(\infty,1)}(C)$ over it is a [[strongly ∞-connected (∞,1)-topos]].

=--

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

We need to check that the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]]-functor $\Pi : Sh_{(\infty,1)}(C) \to \infty Grpd$ preserves finite [[(∞,1)-product]]s.

By the discussion at [[∞-connected site]] we have that $\Pi$ is given by the [[(∞,1)-colimit]] [[(∞,1)-functor]] $\lim_\to : Func(C^{op}, \infty Grpd) \to \infty Grpd$. On the [[opposite (∞,1)-category|opposite]] and therefore [[sifted (∞,1)-category]] $C^{op}$ these preserve finite [[(∞,1)-product]]s.

=--

## Examples

* [[CartSp]], [[ThCartSp]].

## Related concepts

* [[locally connected topos]] / [[locally ∞-connected (∞,1)-topos]]

  * [[connected topos]] / [[∞-connected (∞,1)-topos]]

  * [[strongly connected topos]] / [[strongly ∞-connected (∞,1)-topos]]

  * [[totally connected topos]] / [[totally ∞-connected (∞,1)-topos]]

* [[local topos]] / [[local (∞,1)-topos]].

* [[cohesive topos]] / [[cohesive (∞,1)-topos]]

and

* [[locally connected site]] / [[locally ∞-connected site]]

  * [[connected site]] / [[∞-connected site]]

  * [[strongly connected site]] / **strongly ∞-connected site**

  * [[totally connected site]] / [[totally ∞-connected site]]

* [[local site]], [[∞-local site]]

* [[cohesive site]], [[∞-cohesive site]]

[[!redirects strongly ∞-connected site]]

[[!redirects strongly ∞-connected sites]]
[[!redirects strongly infinity-connected sites]]


Structural Complexity of Proofs

[[!redirects Structural Complexity Of Proofs]]
_Structural Complexity of Proofs_ (1974) is the title of a Ph.D. thesis by [[Richard Statman]], who was a student of [[Georg Kreisel]] at Stanford University.  Among other contributions, the thesis is notable for defining a notion of "genus of a proof", as a global measure of complexity more interesting than (but analogous to) number of inferences, depth of the proof tree, etc.

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## Genus of a proof

Any proof $\pi$ in [[natural deduction]] induces a [[rooted tree]], by representing the judgments $B$ occurring in the proof as nodes $x_B$ (note that different occurrences of the same judgment are represented as distinct nodes), and drawing a directed edge

$$x_B \to x_C$$

just in case $B$ occurs as a premise of the inference rule with conclusion $C$.  This rooted tree can then be turned into a directed [[graph]] $G(\pi)$ by adding more edges

$$x_A \to x_C$$

just in case the assumption $A$ is _discharged_ by the inference rule with conclusion $C$ (see [[natural deduction#hypothetical_reasoning|natural deduction: hypothetical reasoning]]).

The **genus** of the proof $\pi$ is then defined to be the graph genus of $G(\pi)$, i.e., the minimal [[genus of a surface]] in which $G(\pi)$ embeds without crossing edges.

## Chapters

### 1: The Effect of the Elimination of Explicit Definitions on the Complexity of Proofs

### 2: The Effect of the Choice of Recursion Equations and Rules on the Lengths of Equational Proofs

### 3: Strong Normalization of Classical Analysis with Some Applications to the Length of Proofs

## References

* Richard Statman. _Structural Complexity of Proofs_, Ph.D. Thesis (1974), Stanford University.

* Alessandra Carbone. _Logical structures and the genus of proofs_, APAL 161:2, 2009. ([doi](http://dx.doi.org/10.1016/j.apal.2009.05.007))

## Related pages

* [[proof net]]
* [[natural deduction]]
* [[lambda calculus]]
* [[proof theory]]

[[!redirects Structural Complexity Of Proofs]]

category:reference


structural realism

The position in the philosophy of science which maintains either that all we can know of the world is its structure ( _epistemological_ structural realism) or that the world just is structure ( _ontological_ structural realism).

category: philosophy


structural rule

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# Structural rules
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## Idea

\begin{imagefromfile}
    "file_name": "Gentzen-StrukturSchlussfiguren.jpg",
    "float": "right",
    "width": 400,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "(from [Gentzen 1935](#Gentzen35))"
\end{imagefromfile}

### General

In [[formal logic]] and [[type theory]], by the _structural inference rules_ [[Gentzen 1935 §1.2.1](#Gentzen35), [1969, §1.21](#Gentzen69)] of a [[deductive system]]  one means those [[inference rules]] which make no reference to [logical operations](formal+logic#LogicalOperations) but only to the unstructured [[premises]] of a [[deduction]].

In standard [[intuitionistic logic]] ([[intuitionistic type theory]]) the structural rules notably include the [[weakening rule]] and the [[contraction rule]] in the [[antecedent]], which exhibit [[context extensions]] $\Gamma \,\mapsto\, \Gamma, P$ as admitting [[natural transformation|natural]] [[diagonal map|diagonal]] and [[projection]] [[maps]], respectively, hence as admitting [[categorical semantics|interpretation]] as *[[cartesian products]]* $\Gamma \times P$ (cf. [Jacobs 1994](structural+rule#Jacobs94)):

\begin{imagefromfile}
    "file_name": "WeakeningContractionRules-230330d.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Discarding these rules leads to [[linear logic]] ([[linear type theory]]).

Generally, logical systems discarding some structural rules are called *[[substructural logics]]*.

### In dependent type theory
 {#InDependentTypeTheory}

In ([[intuitionistic type theory|intuitionistic]]) [[dependent type theory]] the structural rules include (e.g. [Jacobs (1998), p. 122](#Jacobs98), [UFP13, A.2.2](#UFP13), [Rijke (2018), Sec. 1.4](#Rijke18)):

* the [[variable rule]],

* the [[weakening rule]],

* the [[substitution rule]]

shown in the following table together with their [[categorical semantics]] [[categorical semantics of dependent types|of dependent types]]:

## Examples

* [[structural rules]]

  * [[identity rule]]

  * [[weakening rule]]

  * [[contraction rule]]

  * [[exchange rule]]

  * [[substitution rule]]

  * [[cut rule]]

## Sub-structural logic

If some of the structural rules are *not* imposed in a [[formal logic]] one speaks of *[[substructural logic]]*.

For instance, if the [[weakening rule]] and [[contraction rule]] are omitted, one speaks of [[linear logic]]/[[linear type theory]] (see [there](linear+logic#AbsenceOfWeakeningAndContraction)), since then the [[logical conjunction]] is no longer consrained to behave like a [[Cartesian product]] but may behave like a non-cartesian [[tensor product]].

## References

The notion of the structural inference rules originates (under the German name *Struktur-Schlußfiguren*) with

* {#Gentzen35} [[Gerhard Gentzen]], §1.2.1 in: _Untersuchungen über das logische Schließen. I_, _Mathematische Zeitschrift_ **39** 1 (1935) [[doi:10.1007/BF01201353](http://dx.doi.org/10.1007/BF01201353)]

* {#Gentzen69} [[Gerhard Gentzen]], §1.21 in: *Investigations into Logical Deduction*, in M. E. Szabo (ed.), *The Collected Papers of Gerhard Gentzen*, Studies in Logic and the Foundations of Mathematics **55**, Springer (1969) 68-131 [[ISBN:978-0-444-53419-4](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/55), [pdf](https://logic-teaching.github.io/prop/texts/Gentzen%201969%20-%20Investigations%20into%20Logical%20Deduction.pdf)]

Review:

* [[Anne Sjerp Troelstra]], §1.5 in: *Lectures on Linear Logic*, CSLI Lectures notes **29** (1992) [[ISBN:0937073776](https://web.stanford.edu/group/cslipublications/cslipublications/site/0937073776.shtml)]

Their [[categorical semantics]] is made explicit in:

* {#Jacobs94} [[Bart Jacobs]], *Semantics of weakening and contraction*, Annals of Pure and Applied Logic **69** 1 (1994) 73-106 [doi:10.1016/0168-0072(94)90020-5]

Discussion in [[dependent type theory]]:

* {#Jacobs98} [[Bart Jacobs]], p. 122, 585 in: *Categorical Logic and Type Theory*, Studies in Logic and the Foundations of Mathematics **141**, Elsevier (1998)  [[ISBN:978-0-444-50170-7](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/141), [pdf](https://people.mpi-sws.org/~dreyer/courses/catlogic/jacobs.pdf)]

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

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

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

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

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

[[!redirects structural rule]]
[[!redirects structural rules]]

[[!redirects structural inference rule]]
[[!redirects structural inference rules]]


structural set theory

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## Idea

A **structural set theory** is a [[set theory]] which describes [[structuralism|structural]] mathematics, and *only* [[structuralism|structural]] mathematics: Sets are conceived as [[objects]] that have [[elements]], and are related to each other by [[functions]] or [[relations]].  In the most common structural set theories such as [[ETCS]], sets are characterized by the functions between them, i.e. by the [[category]] ("[[Set]]") which they form ([Lawvere 65](#Lawvere65)).  This is what essentially all the application of [[set theory]] in the practice of [[mathematics]] actually uses -- a point amplified by the approach of the introductory textbook [Lawvere-Rosebrugh 03](#LawvereRosebrugh03).

This is in contrast to traditional *[[material set theory]]* (cf *[[material versus structural]]*) such as [[ZFC]] or [[ZFA]], where sets are characterized by the membership relation "$\in$" and [[propositional equality]] of sets "$=$" alone, and where sets can be elements of other sets, hence where there are sequences of sets which are elements of the next set in the sequence. This latter feature, however, is almost never actually used or even considered in [[mathematics]], away from the study of [[material set theory]] itself.

As conceived by the [[structuralism|structuralist]], [[mathematics]] is the study of structures whose form is independent of the particular attributes of the things that make them up.  For instance, in [[ZF]] [[set theory]], "the set of natural numbers" can be defined as the von Neumann naturals $\omega_N = \{ \emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}, \dots \}$ or alternately as the Zermelo naturals $\omega_Z = \{ \emptyset, \{\emptyset\}, \{\{\emptyset\}\}, \dots \}$, but either definition has all the necessary properties (namely, the properties making it a [[natural numbers object]] in [[Set|the category of sets]], when equipped with an appropriate successor operation).  See, for instance, [Benacerraf's paper](#WhatNumbersCouldNotBe).

The structuralist says, essentially, that the number "$3$" should denote "the third place in a [[natural numbers object]]" rather than some particular set such as $\{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}$ as it does in any definition of "the set of natural numbers" in ZF.

*Structural set theory* provides a [[foundation of mathematics]] which is free of this "superfluous baggage" attendant on theories such as ZF, in which there is lots of information such as whether or not $3\in 17$ (yes, says von Neumann; no, says Zermelo) which is never used in mathematics.  In a structural set theory, the elements (such as $3$) of a set (such as $\mathbb{N}$) *have* no identity apart from their existence as elements of that set, and whatever structure is given to that set by the functions and relations placed upon it.  That is, sets (together with other attendant concepts such as [[elements]], [[functions]], and [[relations]]) are the "raw material" from which mathematical structures are built.  By contrast, theories such as ZF may be called [[material set theories]] or *membership-based set theories*.

Thus, somewhat paradoxically, it turns out that one of the primary attributes of a structural set theory is that the *elements* of a set have *no* "internal" structure; they are only given structure by means of functions and relations.  In particular, they are not themselves sets, and by default cannot be elements of any other set (not in the sense that it is false that they are, but in the sense that it is meaningless to ask whether they are), so that elements of different sets cannot be compared (unless and until extra structure is imposed).  Structural set theory thus looks very much like [[type theory]].  We contrast it with [[material set theory|material set theories]] such as ZF, in which the elements of sets can have internal structure, and are often (perhaps always) themselves sets.

Structural set theories can be distinguished by whether they include a formal notion of [[family]] of [[sets]]. Those that do are known as [[first-order set theory|first-order set theories]], while those that do not are known as [[zeroth-order set theory|zeroth-order set theories]]. [[higher-order set theory|Higher-order set theories]] have a notion of families of families in addition to families of sets. This parallels [[logic]], where the role of [[proposition]]s is played by sets and the role of [[predicate]]s is played by families. Many structural set theories are only zeroth-order set theories, but the hSets and functions from hSets to the type hSet in [[homotopy type theory]] form a higher-order set theory.

It is hard to say precisely what makes a set theory "structural", but one attempt is the notion of a [[structurally presented set theory]].

## Examples
 {#Examples}

### Elementary theory of the category of sets (ETCS)
 {#ETCS}

The original, and most commonly cited, categorially presented structural set theory is the _[[elementary theory of the category of sets]]_, or _[[ETCS]]_ for short ([Lawvere 65](#Lawvere65)).

Therefore structural set theory is also called **categorial set theory**.

[[ETCS]] axiomatizes the [[category]] [[Set]] of [[sets]] as a [[well-pointed topos]] and thus lends itself to [[foundations of mathematics]] in [[topos theory]]. As such this is similar to the [[h-set]]-theory found in [[homotopy type theory]] ([below](#InHomotopyTypeTheory)), which forms a [[ΠW-pretopos]].

[[ETCS]] is weaker than [[ZFC]]. To handle some esoteric parts of modern mathematics such as Borel determinacy, it must be supplemented with an [[axiom of collection]], although it suffices for most everyday uses.

[McLarty 93](#McLarty93) argues that ETCS resolves the issues originally raised by [Benacerraf 65](#Benacerraf65).

### Sets, Elements and Relations (SEAR)

Another structural set theory, which is stronger than ETCS (since it includes the axiom of collection by default) and also less closely tied to category theory, is [[SEAR]].

### Structural ZFC

One could reformulate [[ZFC]] as a three-sorted or dependently sorted [[structural set theory]] consisting of [[sets]], [[elements]], [[functions]], and structural versions of the 10 standard [[ZFC]] axioms. This set theory is called [[structural ZFC]].

It should also be possible to create an [[allegory|allegorical]] version of ZFC, but nobody has done so yet.

### Local set theory

Local set theory avoids the use of any global universe but instead is formulated in a many-sorted language that has various forms of sorts including, for each sort a power-sort; see [Bell](#Bell) and [Aczel](#Aczel).

### Setoids

Set theory set up in [[extensional type theory|extensional]] [[intuitionistic type theory]] via [[setoids]] is structural.

### Homotopy-sets in homotopy type theory
  {#InHomotopyTypeTheory}

The collection of  [[h-sets]] in [[homotopy type theory]] constitute a [[ΠW-pretopos]], hence a structural set theory ([Rijke-Spitters 13](#RijkeSpitter13)). If the type theory also has a [[type of all propositions]], then the collections of [[h-sets]] constitute an [[elementary topos]].

## Related Pages

* [[material set theory]]

* [[material versus structural set theory]]

* [[structurally presented set theory]]

* [[material-structural adjunction]]

* [[categorical set theory]], [[allegorical set theory]]

## References ##

A textbook that introduces the [[foundations of mathematics]] informally via structural set theory is

* {#LawvereRosebrugh03} [[William Lawvere]], [[Robert Rosebrugh]], _[[Sets for Mathematics]]_, Cambridge UP 2003.

The formal account of [[ETCS]] originates with

* {#Lawvere65} [[William Lawvere]], _An elementary theory of the category of sets_, Proceedings of the National Academy of Science of the U.S.A **52** pp.1506-1511 (1965). ([pdf](http://www.ncbi.nlm.nih.gov/pmc/articles/PMC300477/pdf/pnas00186-0196.pdf))

Structural set theory found in [[homotopy type theory]] is discussed in

* {#RijkeSpitters13} [[Egbert Rijke]], [[Bas Spitters]], _Sets in homotopy type theory_, Mathematical Structures in Computer Science, Volume 25, Issue 5 (From type theory and homotopy theory to Univalent Foundations of Mathematics)  ([arXiv:1305.3835](http://arxiv.org/abs/1305.3835))

which became one chapter in

* [[Univalent Foundations Project]], _[[Homotopy Type Theory – Univalent Foundations of Mathematics]]_

Relation to [[material set theory]] is discussed in

* {#Shulman18} [[Michael Shulman]], _Comparing material and structural set theories_ ([arXiv:1808.05204](https://arxiv.org/abs/1808.05204))

See also

* {#Benacerraf65} P. Benacerraf, _What numbers could not be_, The Philosophical Review Vol. 74, No. 1, Jan., 1965  {#WhatNumbersCouldNotBe}

* {#McLarty93} [[Colin McLarty]], _Numbers can be just what they have to_, Noûs, Vol 27, No. 4, 1993.

* John Bell, _Notes on toposes and local set theories_ [PDF](http://publish.uwo.ca/~jbell/topnotes.pdf)

* Peter Aczel, _Local Constructive Set Theory and Inductive Definitions_, [PDF](http://www.cs.man.ac.uk/~petera/sommaruga-book-paper.pdf)

* [[Mike Shulman]], *Syntax, Semantics, and Structuralism II* [blog](https://golem.ph.utexas.edu/category/2009/12/syntax_semantics_and_structura_1.html)

* [[Tom Leinster]], *Rethinking set theory* [blog](https://golem.ph.utexas.edu/category/2012/12/rethinking_set_theory.html), [arXiv](http://arxiv.org/abs/1212.6543)

[[!redirects structural set theory]]
[[!redirects structural set theories]]


structural ZFC

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## Idea

In this article we introduce a [[structural set theory]] consisting of four basic primitives, [[sets]], [[elements]], [[functions]], and [[relations]], as well as function evaluation, and relation holding, and structural versions of 9 of the 10 axioms of [[ZFC]] presented in the nLab article of [[ZFC]], making the set theory equivalent in strength to ZFC. We propose that [[structural ZFC]] is a suitable theory for [[practical foundations]], as the primitives and axioms are in alignment with the usage of such objects in mathematical practice, rather than coming with its own additional baggage.

## Presentation

We work in a [[first-order logic]] with [[equality]]. In structural ZFC, there are four basic primitives:

* **[[sets]]** $A$

* for every set $A$, **[[elements]]** $a \in A$ (the membership $\in$ here is a typing [[judgment]], rather than a relation)

* for every set $A$ and $B$, **[[functions]]** $f:A \to B$ with **[[domain]]** $A$ and **[[codomain]]** $B$ (the function notation $f:A \to B$ here is a [[judgment]], rather than a relation)

* for every set $A$ and $B$, **[[relations]]** $\varphi:A \looparrowright B$ with **[[domain]]** $A$ and **[[codomain]]** $B$ (the relation notation $\varphi:A \looparrowright B$ here is a [[judgment]], rather than a relation)

In addition,

* for every function $f:A \to B$ and element $a \in A$, one could form the element $f(a) \in B$; this operation is called **function evaluation**,
* for every relation $\varphi:A \looparrowright B$ and element $a \in A$ and $b \in B$, one could form that $\varphi(a, b)$ **holds** of $a$ and $b$.

The axioms of structural ZFC are as follows:

**Axiom 0A (Axiom of equality preservation):** _For all sets $A$ and $B$, functions $f:A \to B$, and elements $a \in A$ and $b \in A$, if $a = b$, then $f(a) = f(b)$.

**Axiom 0B ([[axiom of unique choice|Axiom of unique choice]]):** _For all functions $f:A \to B$, there is a unique relation $\varphi:A \looparrowright B$ such that for all $a \in A$ and $b \in B$, $\varphi(a, b)$ iff $b = f(a)$, and for all relations $\varphi:A \looparrowright B$ such that for all $a \in A$, there is a unique $b \in B$ such that $\varphi(a, b)$ holds of $a$ and $b$, there is a function $f:A \to B$ such that $f(a) = b$._

A function $f:A \to B$ is an **[[injection]]** if for all elements $a \in A$ and $b \in A$, $a = b$ if and only if $f(a) = f(b)$. Injections are written as $f:A \hookrightarrow B$.

**Axiom 1 ([[axiom of strong extensionality|Axiom of strong extensionality]]):** _Given sets $A$ and $B$ and an [[injection]] $i:A \hookrightarrow B$, the following statements are logically equivalent to each other:_

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

One uses the axiom of strong extensionality as the definition of a [[bijection]] between sets.

**Axiom 2 (Axiom of [[empty set]]):** _There exists a set $\emptyset$ such that for every other set $A$, there is a unique relation $u_A^\emptyset:\emptyset \looparrowright A$._

**Axiom 3 (Axiom of [[Cartesian products]]):** _For every set $A$ and $B$, there exists a set $A \times B$ with a function $\pi_A:A \times B \to A$ and a function $\pi_B:A \times B \to B$ such that for every element $a \in A$ and $b \in B$ there is a unique element $(a, b) \in A \times B$, such that $\pi_A((a, b)) = a$ and $\pi_B((a, b)) = b$._

Given a set $I$, an $I$-indexed [[family of subsets]] is a relation $M:I \looparrowright A$ to a set $A$ called the **[[union]]** of $M$.

**Axiom 4 (Axiom of [[indexed subsets]]):** _For every set $I$ and $I$-indexed family of subsets $M:I \looparrowright A$ with union $A$, and for every element $i \in I$, there exists a set $M_i$ and an injection $m:M_i \hookrightarrow A$, such that for every element $a \in M_i$, $M(i, m(a))$ holds for $i$ and $m(a)$, and for every other set $B$ with injection $f:B \hookrightarrow A$ such that for every element $b \in B$, $M(i, f(b))$ holds for $i$ and $f(b)$, there is a unique function $u_B^{M_i}:B \to M_i$ such that for every element $b \in B$, $f(b) = m(u_B^{M_i}(b))$._

**Axiom 5 ([[axiom schema of separation|Axiom schema of separation]]):** _For any set $B$ and any formula $\phi(x)$ with free variable $x \in B$, there exists a set $\{B \vert \phi\}$ with an injection $m:\{B \vert \phi\} \hookrightarrow B$ such that for every element $x \in B$, $\phi(x)$ holds if and only if there exists an element $y \in \{B \vert \phi\}$ such that $m(y) = x$._

**Axiom 6 ([[axiom schema of collection|Axiom schema of collection]]):** _For any set $A$ and formula $\phi(x, X)$ with free variables $x \in A$ and $X$, there exists a set $B$, function $p:B \to A$, and a $B$-indexed family of sets $M:B \looparrowright C$ with union $C$, such that for every $b \in B$, $\phi(p(b), M_b)$, and for every $a \in A$, if there exists a set $X$ with $\phi(a, X)$, then $a \in \mathrm{im}(p)$._

**Axiom 7 (Axiom of [[power sets]]):** _For any set $A$, there exists a set $\mathcal{P}(A)$ and a relation $(-)\in_A(-):A\looparrowright \mathcal{P}(A)$ such that for any set $B$ with injection $i:B \hookrightarrow A$, there exists a unique $b \in \mathcal{P}(A)$ such that for any $x \in A$, $x \in_A b$ holds of $x$ and $b$ if and only if there exists an element $y \in B$ such that $i(y) = x$._

**Axiom 8 (Axiom of [[natural numbers]]):** _There exists a set $\mathbb{N}$ with an element $0 \in \mathbb{N}$ and a function $s:\mathbb{N} \to \mathbb{N}$, such that for all sets $A$ with an element $0_A:A$ and function $s_A:A \to A$, there is a unique function $u_A^\mathbb{N}:\mathbb{N} \to A$ such that $u_A^\mathbb{N}(0) = 0_A$ and for all elements $n \in \mathbb{N}$, $u_A^\mathbb{N}(s(n)) = s_A(u_A^\mathbb{N}(n))$._

**Axiom 9 ([[axiom of choice|Axiom of choice]]):** _If $f:A \to B$ is a function such that for every element $x \in B$ there exists an element $y \in A$ such that $f(y) = x$, then there is a function $g:B \to A$ such that for all elements $x \in B$, $f(g(x)) = x$._

The structural version of the [[axiom of foundation]], the [[axiom of well-founded materialization]], states that every set can be embedded in some [[well-founded relation|well-founded]] [[extensional relation|extensional]] [[graph]]. This axiom follows from the [[axiom of choice]], and well-founded extensional graphs are not a fundamental concept in structural set theory, so it is not listed here as a separate axiom.

## Making alternate primitive choices

One could choose to make either the relations or the functions a derived object of the other, leading to **categorical ZFC** and **allegorical ZFC** respectively.

### Dependently-sorted categorical ZFC

Categorical ZFC follows [[ETCS with elements]] in taking [[functions]] to be fundamental and [[relations]] as defined.

We work in a [[first-order logic]] with [[equality]]. In structural ZFC, there are three basic primitives:

* **[[sets]]** $A$

* for every set $A$, **[[elements]]** $a \in A$ (the membership $\in$ here is a typing [[judgment]], rather than a relation)

* for every set $A$ and $B$, **[[functions]]** $f:A \to B$ with **[[domain]]** $A$ and **[[codomain]]** $B$ (the function notation $f:A \to B$ here is a [[judgment]], rather than a relation)

In addition, for every function $f:A \to B$ and element $a \in A$, one could form the element $f(a) \in B$; this operation is called **function evaluation**. A relation between sets $A$ and $B$ is a set $R$ with an injection $(s, t):R \hookrightarrow A \times B$.

### Three-sorted categorical ZFC

An alternate formulation of categorical ZFC three [[sorts]], one for [[sets]], [[elements]], and [[functions]], making structural ZFC 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 functional domain and codomain notation $f:A \to B$ also becomes a ternary relation between the sort of functions, and two copies of the sort of sets. There is a quinary relation $\mathrm{eval}(A, B, f, a, b)$ between all three sorts, which says that given sets $A$ and $B$, the function $f$ can be evaluated at element $a$ and the result of the evaluation is the element $b$, in the [[context]] of $\varphi:A \looparrowright B$, $a \in A$, and $b \in B$.

### Dependently sorted allegorical ZFC

Allegorical ZFC follows [[SEAR]] in taking [[relations]] to be fundamental and [[functions]] as defined.

We work in a [[first-order logic]] with [[equality]]. In structural ZFC, there are three basic primitives:

* **[[sets]]** $A$

* for every set $A$, **[[elements]]** $a \in A$ (the membership $\in$ here is a typing [[judgment]], rather than a relation)

* for every set $A$ and $B$, **[[relations]]** $\varphi:A \looparrowright B$ with **[[domain]]** $A$ and **[[codomain]]** $B$ (the relation notation $\varphi:A \looparrowright B$ here is a [[judgment]], rather than a relation)

In addition, for every relation $\varphi:A \looparrowright B$ and element $a \in A$, one could form the judgment that $\varphi(a, b)$ **holds**. A function is a relation $\varphi:A \looparrowright B$ such that for all $a \in A$, there is a unique element $b \in B$ such that $\varphi(a, b)$ holds, and $b$ is said to be the evaluation of $a$ at $\varphi$.

### Three-sorted allegorical ZFC

An alternate formulation of allegorical ZFC 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 $b \in B$.

## See also

* [[ZFC]]

* [[SEAR]]

* [[ETCS with elements]]

## References

The axiom of well-founded materialization, the structural axiom of extensionality (the strong terminal generator condition of well-pointedness), and the structural axiom schemata of separation and collection could be found in:

* [[Michael Shulman]] (2018). Comparing material and structural set theories. [arXiv:1808.05204](https://arxiv.org/abs/1808.05204).

The rest of the axioms are standard axioms from categorical set theory (or the type theoretic equivalent).

[[!redirects structural ZFC]]

[[!redirects dependently sorted structural ZFC]]
[[!redirects dependently-sorted structural ZFC]]
[[!redirects four sorted structural ZFC]]
[[!redirects four-sorted structural ZFC]]
[[!redirects three sorted structural ZFC]]
[[!redirects three-sorted structural ZFC]]

[[!redirects categorical ZFC]]
[[!redirects dependently sorted categorical ZFC]]
[[!redirects dependently-sorted categorical ZFC]]
[[!redirects three sorted categorical ZFC]]
[[!redirects three-sorted categorical ZFC]]

[[!redirects allegorical ZFC]]
[[!redirects dependently sorted allegorical ZFC]]
[[!redirects dependently-sorted allegorical ZFC]]
[[!redirects three sorted allegorical ZFC]]
[[!redirects three-sorted allegorical ZFC]]


structuralism

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## Idea

> These many different things stand in essential reciprocal action via their properties; the property is this reciprocal relation itself and apart from it the thing is nothing; ([WdL §1065a](Science+of+Logic#1065a))

In [[philosophy]], _structuralism_ is the point of view which emphasises that the entities considered are parts of a system and that their very meaning and identity is defined according to its relation to the rest of the system.  For instance, if one part of the system changes in time, all other parts change as well. The system has features which are not just the composition of features of its constituent parts.

### In the humanities

In the 20th century, structuralism in the humanities is associated with Emile Durkheim and Georg Simmel in sociology, Ferdinand de Saussure (and later Roman Jakobson) in linguistics, and [[Claude Lévi-Strauss]] in anthropology.

### In philosophy of mathematics
 {#InThePhilosophyOfMathematics}

Structuralism is also the term given to a position in the [[philosophy of mathematics]] which holds that the entities treated in [[mathematics]] are simply structures. It is associated also with [[structural realism]] in the [[philosophy of science]], which maintains either that all we can know of the world is its structure (epistemological structural realism) or that indeed the world just is structure (ontological structural realism).

One may argue that [[category theory]] and [[type theory]] serve as formalization of these intuitive notions, see [below](#FormalizationInCategoryTheoryAndTypeTheory).

## Formalization of structuralism in category theory and type theory
 {#FormalizationInCategoryTheoryAndTypeTheory}

Since in a [[category]], the nature of any [[object]] is determined only by the system of _[[morphisms]]_ which relate this object to itself and to the other objects  of the category (by the [[Yoneda lemma]]), and not by how the object itself is [[generators and relations|presented]] (if it is at all), it may be argued that [[category theory]] provides a formalization of philosophical structuralism, at least in the [[philosophy of mathematics]] ([Awodey 96](#Awodey96), [Awodey 03](#Awodey03)). [[Yuri Manin]] said (citation?) that [[category theory]] regards [[objects]] as part of a "society". (See also at _[[objective logic]]_ for more on [[categorical logic]] and [[philosophy|philosophical]] notions.)

Notice that the terminology _[[morphism]]_ of course originates in _[[homomorphism]]_, since the morphisms in a category of [[mathematical structures]] (such as [[groups]], [[modules]], etc.) are precisely the maps of the underlying [[sets]] that preserve this structure.

This is essentially [[Bourbaki]]'s emphasis on [[mathematics]] as the science of abstract [[structures]], where structures are important only up to isomorphism and the emphasis is on relations (including) functions which are part of the "structure" (e.g. [Kantor 11](#Kantor11)).

However, a category need not be such a [[concrete category]] of sets with structure. Even when there is no such explicit structure on the objects, the same category-theoretic reasoning still applies and one may detect structure on objects from the morphisms between them.

For instance, while in general an object $X$ in a category does not have an underlying set, if the category has a [[terminal object]] $\ast$ one may still find a set of [[global elements]] of $X$ as the set of morphisms of the form $\ast \to X$.

Accordingly, in a category one only talks about properties invariant under [[isomorphisms]]; from such a point of view an [[object]] is determined by the morphisms to or from other objects.

This perspective of the nature of objects of a category being determined only by the morphisms between them is fully embodied by the concept of [[equivalence of categories]]: two categories are to be regarded as equivalent if there is a [[functor]] between them that is [[essentially surjective]] in that it does not omit any [[isomorphism class]] of objects and which otherwise is a [[bijection]] on morphisms (precisely: on the [[hom-sets]] between any ordered pair of objects, a [[fully faithful functor]]).

Given that [[type theories]] of sorts are the [[internal language]] of suitable kinds of [[categories]], with morphisms now being [[terms]] of [[function types]] (see at _[[relation between category theory and type theory]]_), these comments also give that type theory offers some kind of formalization of structuralism ([Awodey 15](#Awodey15), [Corfield 15](#Corfield15)).

Notably in [[homotopy type theory]] with [[univalence]], which is the [[internal language]] of categories called [[(infinity,1)-toposes]], the [[type universe]] reflects all the [[isomorphisms]] ([[equivalences]]) in the category in its [[identity type]], which therefore may be thought of as providing a structuralist concept of [[identity]].

(Notice however that the categories of [[mathematical structures]] such as [[groups]], [[modules]], etc., even when regarded as [[(infinity,1)-categories]], i.e. of [[infinity-groups]], [[(infinity,n)-modules]] etc.) are rarely [[(infinity,1)-toposes]]. The "structure" carried by an object in an $(\infty,1)$-topos is generically more a kind of geometric structure.

## Related entries

* [[structure]]

* [[structure identity principle]]

* [[idealism]]

  * [[objective idealism]], [[objective logic]]

  * [[subjective idealism]]

  * [[absolute idealism]]

* [[Salomon Maimon]]

## References

See also

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

Discussion in view of [[category theory]]:

* {#Awodey96} [[Steve Awodey]], _Structure in mathematics and logic: a categorical perspective_, Philosophia Mathematica (3), vol. 4,  p. 209-237, 1996 ([doi:10.1093/philmat/4.3.209](https://doi.org/10.1093/philmat/4.3.209))

* {#Awodey03} [[Steve Awodey]], _An Answer to Hellman's Question: "Does Category Theory Provide a Foundation for Mathematical Structuralism?"_, 2003 ([pdf](http://www.andrew.cmu.edu/user/awodey/preprints/awodeyVhellman.pdf), [doi:10.1093/philmat/12.1.54](https://doi.org/10.1093/philmat/12.1.54))

* {#MacLarty04} [[Colin McLarty]], 2004, _Exploring Categorical Structuralism_, Philosophia Mathematica, 12, 37–53 ([pdf](http://www.cwru.edu/artsci/phil/PMExploring.pdf))

* {#Kantor11} [[Jean-Michel Kantor]], _Bourbaki's Structures and Structuralism_, The Mathematical Intelligencer __33__:1 (2011), 1, [doi](http://dx.doi.org/10.1007/s00283-010-9173-4)

Discussion in view of [[univalent foundations of mathematics]] ([[homotopy type theory]] with the [[univalence axiom]]):

* {#Awodey15} [[Steve Awodey]], _Structuralism, Invariance and Univalence_, 2014 ([pdf](https://www.andrew.cmu.edu/user/awodey/preprints/siu.pdf))

* {#Corfield15} [[David Corfield]], _Expressing 'The Structure of' in Homotopy Type Theory_, 2015 ([pdf](https://dl.dropboxusercontent.com/u/16936016/The_Structure.pdf))

* [[Dimitris Tsementzis]], *Univalent foundations as structuralist foundations*, Synthese **194** 9 (2017) 3583–3617 [[jstor:26748765](https://www.jstor.org/stable/26748765), [doi:10.1007/s11229-016-1109-x](https://doi.org/10.1007/s11229-016-1109-x), [pdf](https://core.ac.uk/download/pdf/157866891.pdf)]

category: philosophy


structurally presented set theory

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

# Contents
* table of contents
{: toc}

## Idea

A [[structural set theory]] is a set theory which describes *only* structural mathematics.  A **structural presentation** is an attempt to describe formally when a particular (presentation of a) set theory is *structural* in this sense.  Paradoxically, this is achieved by requiring that the elements of sets have *no* internal structure.

## Semiformal Definition

The following is an attempted formal definition of when a set theory is structural.  It should be regarded as extremely tentative.

### Notions of set

First of all, in order to give a definition pertaining to all set theories, we need to know: what is a set theory?  Probably no theory can *intrinsically* be called a set theory; it is only given that distinction by our intent to regard some of its objects of study as "sets" and others as their "elements."  Thus we make the following definition.

+-- {: .un_defn}
###### Definition
A **notion of set** in a typed first-order theory consists of:

* A type $\mathbb{S}$ called the *type of potential sets*.
* A unary predicate $set(-)$ on $\mathbb{S}$ called *being a set*.
* A type $\mathbb{E}_A$, possibly depending on $A\in\mathbb{S}$, called the *type of potential elements of $A$*.
* A unary predicate $in_A(-)$ on $\mathbb{E}_A$ called *being an element of $A$*.
=--

We have split the notions of sethood and elementhood into both a type and a predicate to allow for variation in theories which model them by one or the other method or a combination.  Here are some examples.

* [[ZF]] is a one-typed theory; we call the objects in the one type  "ZF-sets" for clarity.  The type of ZF-sets is the type of potential sets, and we take the predicate $set(-)$ to be $\top$ (every potential set is a set).  For such a set $A$, the type of potential elements of $A$ must once again be the type of ZF-sets, and the elementhood predicate is simply $in_A(x) \equiv (x\in A)$.

* ZF can be augmented by the presence of [[urelement|urelements]] or atoms.  One way to represent this is with two types, the type of ZF-sets and the type of ZF-atoms.  Again we take the type of ZF-sets to be the type of potential sets, and the sethood predicate $set(-)$ is $\top$.  The type of potential elements of $A$ is now the sum type (ZF-sets + ZF-atoms), with the membership predicate again being the ordinary membership predicate of $A$.

* Atoms can be added to ZF in another way: we maintain only a single type, say the type of "ZF-objects," but we add a "sethood" predicate which distinguishes the sets from the atoms.  Now the type of potential sets is of course the type of ZF-objects, and the predicate $set(-)$ is the ZF-sethood predicate.

* In [[NBG]] or [[MK]] set-class theory, there are two possible "notions of set": the sets and the classes.  The way to formalize this depends on how the set-class theory is stated.  For instance, one way to state NBG is with a single type of classes, defining a class $A$ to be an NBG-set when $A\in B$ for some class $B$.  In this case the sethood predicate (for the "NBG-set" notion of set) will be $set(A) \equiv (\exists B. A\in B)$.

* In [[SEAR]], which is a dependently typed theory, we take the type of potential sets to be the type of SEAR-sets, with the sethood predicate $set(-)$ being again $\top$.  For each set $A$, of course $\mathbb{E}_A$ is the type of SEAR-elements of $A$.  (In contrast to ZF, these types are now nontrivially dependent on $A$.)  The elementhood predicate is $\top$, since every object in the type of SEAR-elements of $A$ is an element of $A$ by definition (that is, in SEAR elementhood is a typing assertion).

* In the variant of SEAR without primitive equality (and thus also in [[SEAR+ε]]), the type of potential sets is the type of "SEAR-sets equipped with an endo-relation" (a [[dependent sum]] type).  The sethood predicate $set(A,R)$ then asserts that $R$ is an equivalence pre-relation, with the other structure as in ordinary SEAR.

* Since [[ETCS]] is an extension of the first-order theory of a [[category]], it can be stated in several ways.

   * If ETCS is stated with two types (objects and arrows), then the type of potential sets is the type of objects (i.e. ETCS-sets) and the sethood predicate is $\top$.  The type of potential elements of each set $A$ is the type of arrows (i.e. ETCS-functions), and $in_A(x)$ holds when the target of $x$ is $A$ and its source is a [[terminal object]].

   * If ETCS is stated with [[single-sorted definition of a category|one type]], then the type of potential sets is that type, the sethood predicate is $set(A) \equiv (A=s(A))$, the type of potential elements is of course again the single type, and the elementhood predicate is as in the two-typed version.

   * If ETCS is stated with dependent types (a type $hom(A,B)$ for every pair of objects $A$ and $B$), then the type of potential sets is the type of objects, the sethood predicate is $\top$, the type of potential elements of $A$ is $hom(1,A)$, and the elementhood predicate is $\top$.

### Structural Presentation

Now we make the following definition.

+-- {: .un_defn}
###### Definition
A notion of set in a typed first-order theory is called **structurally presented** if for any first-order formula $\varphi$ containing free variables $A:\mathbb{S}$ and $x:\mathbb{E}_A$ (and possibly others), if the only free occurrence of $A$ in $\varphi$ is in the (possibly) dependent type $\mathbb{E}_A$ of the variable $x$, then
\[ (\forall \vec{z})(\forall A:\mathbb{S})(\forall x,y:\mathbb{E}_A) \Big(set(A) \Rightarrow in_A(x) \Rightarrow in_A(y) \Rightarrow \big(\varphi \Leftrightarrow \varphi[y/x]\big)\Big)
\label{star} \]
holds (where $\vec{z}$ denotes all the other free variables of $\varphi$). In words, for every potential set $A$ and every potential elements $x$ and $y$ of $A$, if $A$ is a set and $x$ and $y$ are elements of $A$, then $\varphi$ holds of $x$ if and only if it holds of $y$.
=--

Equation (eq:star) says that all elements of any set $A$ are completely indistinguishable by the formula $\varphi$.  This is intended to approximate the idea of a structural set theory: the elements of $A$ have no independent structure (and thus cannot be distinguished from each other) apart from their identity as elements of $A$.  Of course, we can't do mathematics without some way to distinguish the elements of a set, but the idea of structural set theory is that this happens only through external structure imposed *on* the set, such as functions and relations, rather than through intrinsic properties of the elements themselves.  This is the idea behind the restrictions placed on $\varphi$.

Before we go on, let's first see that ZF is definitely *not* structurally presented.  Let $A=\{\emptyset, \{\emptyset\}\}$ be the von Nemuann numeral $2$, and let $\varphi \equiv (x = \emptyset)$.  Clearly there are some $x\in A$ such that $\varphi$ holds and others such that it fails, so (eq:star) is false.  Moreover, $\varphi$ satisfies the condition: $A$ does not occur in it at all.  For the same reasons, NBG and MK are not structurally presented.

Now, why can't we do the same thing in a structural set theory?  Consider SEAR, and let $A$ be a set containing two distinct elements $a$ and $a'$.  By analogy to what we did in ZF, we'd like to take $\varphi \equiv (x = a)$, but in SEAR this violates the required condition: $a$ must also be an element of $\mathbb{E}_A$, so $A$ appears elsewhere in $\varphi$ than in the type of $x$ (namely, in the type of $a$).  In fact, we can show:

+-- {: .un_theorem}
###### Theorem
SEAR is structurally presented.
=--
+-- {: .proof}
###### Proof
Let $\varphi$ be a formula as in the above definition.  Since it does not contain $A$ free (except in the type of $x$), it cannot contain any variables (free *or* bound) denoting elements of $A$ (other than $x$) or relations whose source or target is $A$.  But the only atomic formulas in the language of SEAR are equalities of set-elements, equalities of relations, and assertions that a relation holds of a pair of set-elements, and SEAR has no term constructors that could produce an element of some other set from an element of $A$.  Thus, the only atomic formula involving $x$ that can appear in $\varphi$ is $x=x$, whose truth is clearly independent of the value of $x$.  It follows that the truth value of $\varphi$ must also be so.
=--

For similar reasons, we have:

+-- {: .un_theorem}
###### Theorem
When ETCS is written with dependent hom-types, it is structurally presented.
=--
+-- {: .proof}
###### Proof
Let $\varphi$ be a formula as above.  Similarly to the case of SEAR, $\varphi$ cannot contain any variables other than $x$ that denote functions whose source or target is $A$.  Since the only term-constructor in ETCS is function composition, the only terms containing $x$ that can occur in $\varphi$ are composites ending in $x:1\to A$ followed by zero or more applications of $id_A$, such as $x$ itself, $x\circ f$, $\id_A\circ x$, $(x\circ g)\circ h$, or $(id_A\circ x)\circ (i\circ j)$.  Note that all of these terms denote functions with target $A$.  Moreover, the only possible terms denoting functions of target $A$ are terms of this sort, or else composites of some number of copies of $id_A$ (and nothing else).

Now the only atomic formulas of ETCS are equality between parallel function terms.  Since the source of a composite involving $x$ cannot be $A$ (otherwise $A$ would have to occur as the source of the first function-variable in that composite), the only possible such formulas involving $x$ will be equality between two composites containing $x$.  But since $1$ is a terminal object, any two functions that factor through it will always be equal.  Thus, the truth value of such an atomic formula is independent of the value of $x$, and so the truth value of $\varphi$ must also be so.
=--

However, if ETCS is written using a two-sorted or one-sorted definition of a category, then it is not structurally presented.  For instance, we can take the formula
$$\varphi \equiv \big((s(f)=t(x)) \wedge (f\circ x = z)\big)$$
for variables $f$ and $z$ of type "arrow".  This formula does not contain $A$ at all, yet its truth value (for a fixed $f$ and $z$) is not independent of $x$.  This is why we have defined "structurally presented" rather than "structural"---the presentation matters.

It is, however, easy to modify the two- or one-sorted version of ETCS to make it structurally presented: we simply require the equality relation on functions to be decorated by the source and target sets, and the composition operation on functions to be decorated with the set along which composition occurs.  That is, instead of $f=g$ we write $equal(f,g,A,B)$ (which can only hold when $s(f)=s(g)=A$ and $t(f)=t(g)=B$), and instead of $g\circ f$ we write $g\circ_B f$ where $t(f)=s(g)=B$.  Now the same argument as for the dependently-typed version applies.

It would be nice to have a definition according to which any "presentation" of ETCS would be structural; intuitively, a theory should be structural if it is "equivalent" in some sense to a structurally presented one.  However, I haven't yet been able to come up with a notion of "equivalence" which includes the above modifications of ETCS, yet excludes the equivalence between ETCS and BZC (or ZFC and ETCS+R or SEARC).

### Consequences of structural presentation

From our definition of structural presentation we can deduce some of the other standard properties of structural set theories.  The first is that "elements of different sets cannot be compared for equality."

+-- {: .un_theorem}
###### Theorem
If a structurally presented notion of set includes a notion of equality between the elements of distinct sets, which is symmetric and transitive, then all of its sets are [[subsingletons]].
=--
+-- {: .proof}
###### Proof
Suppose that we have an equality relation that can relate elements of two distinct sets $A$ and $B$.  Thus, for variables $x$ of type $A$ and $z$ of type $B$, we would have an formula $x=z$.  By symmetry and transitivity, if $x=z$ and $x'=z$, then $x=x'$.  But then (eq:star) applied to $\varphi\equiv (x=z)$ implies that for any $x,y\colon A$ we have $x=y$, so $A$ must be a subsingleton.  Since $A$ was arbitrary, all sets must be subsingletons.
=--

The second is "elements of sets are not themselves sets."

+-- {: .un_theorem}
###### Theorem
Suppose that a structurally presented notion of set includes relations $is_A:\mathbb{E}_A\looparrowright\mathbb{S}$, which we regard as giving a way to interpret some elements as sets.  Suppose moreover that each $\mathbb{E}_A$ has an equality relation $=_A$, and that $is_A(x,B)$ and $is_A(y,B)$ imply $x =_A y$.  Then for any set $A$, if there exist $a:\mathbb{E}_A$ and $B:\mathbb{S}$ such that $is_A(a,B)$, then $A$ must be a subsingleton.
=--
+-- {: .proof}
###### Proof
Suppose that $is_A(a,B)$ and choose $\varphi\equiv is_A(x,B)$.  By (eq:star) it follows that $is_A(x,B)$ holds for all $x:\mathbb{E}_A$, hence $x =_A y$ for all $x,y:\mathbb{E}_A$, i.e. $A$ is a subsingleton.
=--

Note that the assumption about equality is quite reasonable if we want to interpret "$is_A$" as asserting that a given element *is* a given set (rather than merely being related to one in some way).

[[!redirects structurally presented set theory]]
[[!redirects structurally presented set theories]]
[[!redirects structurally-presented set theory]]
[[!redirects structurally-presented set theories]]


structure

> This entry is about a general concepts of "mathematical structure" such as formalized by [[category theory]] and/or [[dependent type theory]]. This subsumes but is more general than the concept of [[structure in model theory]].

+-- {: .rightHandSide}
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### Context
#### Mathematics
+-- {: .hide}
[[!include mathematicscontents]]
=--
#### Mathematics
+-- {: .hide}
[[!include mathematicscontents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

It is common in informal language to speak of mathematical objects "equipped with [[extra structure]]" of some sort. The archetypical examples are [[algebras over a Lawvere theory]] in [[Set]]: these are [[sets]] equipped with the structure of certain algebraic operations. For instance a [[group]] $(G, e, {\cdot})$ is a [[set]] $G$ equipped with a binary operation ${\cdot} : G \times G \to G$, etc.

One may formalize the notion of structure using the language of [[category theory]]. This is discussed at _[[stuff, structure, property]]_. In that formalization [[objects]] in some [[category]] $D$ are objects in some category $C$ _equipped with extra structure_ if there is a [[functor]] $p \colon D \to C$ such that

* $p$ is a [[faithful functor]].

{#InCategoryTheory} Depending on author and situation, more properties are required of this functor ([Ehresmann 57](#Ehresmann57), [Ehresmann 65](#Ehresmann65), [Adamek-Rosicky-Vitale 09, remark 13.18](#AdamekRosickyVitale)):

* $p$ is an [[amnestic functor]] ($p$-vertical [[isomorphisms]] are [[identities]]),

* $p$ is an [[isofibration]] (isomorphisms can be lifted along $p$).

{#StrictStructure} However, notice that these two conditions violate the [[principle of equivalence]] for [[categories]]. In the terminology of _[[strict categories]]_ one might hence refer to these conditions as expressing "strict extra structure".

## Notions of structure

A special class of examples of this is the notion of [[structure in model theory]]. In this case one defines a "language" $L$ that describes the constants, functions (say operations) and relations with which we want to equip sets, and then sets equipped with those operations and relations are called $L$-[[structure in model theory|structures]] for that language. (Equivalently one might say "sets with $L$-structure". Or one might generally say "$X$-structure" for "set with $X$-structure".) In this case there is a [[faithful functor]] from $L$-structures to their underlying sets, and so this is a special case of the general definition.

We instead say [[model]] of a [[theory]] when we restrict to those structures which satisfy the axioms of a theory (in other words, satisfy _properties_ specified by the axioms). In this case there is a full and faithful functor from the category of models of a theory $T$ to the category of structures of the underlying language $L(T)$, while the composition of forgetful functors

$$Mod_T \to Struct_{L(T)} \to Set$$

is again faithful.

+-- {: .num_remark}
###### Remarks
Thus, the English word "structure" is used in several slightly differing mathematical senses.

1. Within category theory itself, "structure" can function as a kind of mass noun, as in a phrase like "forgetting structure". Here it refers to data comprising operations, relations, constants, and _also properties_ borne by models of a theory or relative theory, considered abstractly (for example, the functor $Grp \to Set$ which forgets group structure, or the functor $Ring \to Ab$ which forgets multiplicative structure). On the other hand, it can also operate in the singular, where one says for example "a topological group is a topological space equipped with a group structure, such that..."

1. In model theory, however, the term _structure_ is not a mass noun; it refers to a _particular_ set (or "structures" for a family of sets) together with functions, relations, and elements that interpret the symbols of operations, predicates, and constants of a _language_. When one adds axioms to a language to make a _theory_, then a structure of the language where those axioms get interpreted as properties _satisfied_ by the structure is called a _model_ of the theory. Thus, in summary, the category theorist might refer to "the structure of a group" as consisting of a multiplication, a unit, etc., satisfying group axioms, while the model theorist would say that each particular group (like $\mathbb{Z}$) is a model of a theory of groups.  For a model theorist, being a model does entail being a structure for the language of groups, but she would also say that a structure for the language of groups need not satisfy any of the axioms of a group (like associativity or unitality).
=--

## Examples

There are gazillions of examples of objects equipped with extra structure. The most familiar is maybe

* [[algebraic structure]].

Generally the [[forgetful functor]] from a category of algebras over an [[algebraic theory]] down to the base category exhibits the equipment with the corresponding algebraic structure.

## Structures in dependent type theory
 {#InDependentTypeTheory}

In [[dependent type theory]] the notion of "mathematical structure" and/or *data structure* on a base $B \,\colon\, Type$ is given by iterated [[dependent pairs|dependent pairings]] (hence forming "[[type telescopes]]" also called  "records" in [[Coq]]) with $B$-dependent types encoding operations on/with this type together with their behavioural specification.

The following shows some examples, using the notation for [[dependent pairs]] from [[dependent functions and dependent pairs -- table|here]].

{#StructureIdentityPrinciple} Via the [[extension (semantics)|extensionality]] principles for [[dependent pairs]] ([here](dependent+sum+type#ExtensionalityPrinciple)) and for [[dependent functions]] ([here](function+extensionality#StatementForDependentFunctions)) such type theoretic structure automatically obey the *[[structure identity principle]]*:

In that any equivalence/identification between pairs of data of the following types are isomorphisms in the sense of bijective [[homomorphisms]].

### Lensed data structure

To say that a given data (base) type $B \,\colon\, Set$ is

1. equipped with

   1. a [[function type|function]] $read_D \,\colon\, B \to D $ reading out $D$-data;

   1. a [[function type|function]] $write_D \,\colon\, D \times B \to B$ (over-)writing $D$-data;

1. such that this does behave as expected, namely as a [well-behaved](lens+in+computer+science#LensesAreCostateCoalgebras) $D$-[[lens (in computer science)|lens]]-structure on $B$

means to declare it to be of the following iterated [[dependent pair]]-[[telescope]] [[type]]:

### Group data structure
 {#GroupDataStructure}

The [[dependent pair]]-[[telescope]] [[type]] declaration of [[group objects|group structure]]:

Further restriction to [[abelian groups]]:

Via the [[structure identity principle]] ([above](#StructureIdentityPrinciple)), the [[identification type|identitifications]]/[[type equivalence|equivalences]] between such group data types are indeed bijective [[group homomorphisms]], hence group-[[isomorphisms]]:

The structure of [[subgroups]] of a given group structure:

and their [[underlying]] abstract group structure:

### Group action structure
 {#GroupActionStructure}

The [[dependent pair]]-[[telescope]] [[type]] declaration of [[group actions]] on [[sets]] ([[G-sets]]):

Specialization to [[torsors]]:

### Ring data structure
 {#RingDataStructure}

The [[dependent pair]]-[[telescope]] [[type]] declaration of [[unital ring|unital]] [[ring objects|ring structure]]:

For examples see at *[[numbers]]* -- *[In dependent type theory](https://ncatlab.org/nlab/show/number#InDependentTypeTheory)*.

### Module data structure
 {#ModuleDataStructure}

Given a [[unital ring|unital]] [[ring]] type $R \colon Ring$ (as [above](#RingDataStructure)), the type of $R$-[[module objects]] is the following [[dependent pair]]-[[telescope]]:

## Higher structures in homotopy type theory
 {#HigherStructuresInHoTT}

The [above](#InDependentTypeTheory) examples of structures formulated in dependent type theory all had data [[h-set|sets]] as their base, which is the classical situation. But in [[dependent type theory]] with untruncated [[identification types]], hence in [[homotopy type theory]], we may simply drop this constraint and consider structures whose base is any higher [[homotopy type]]: This yields the notion of *[[higher structures]]*.

### Delooping

Given a set-based [[group object|group structure]] (as [above](#GroupDataStructure)), [[delooping]] structure is

(...)

### Higher delooping

More generally, one may consider delooping of [[n-groups|$n$-groups]], but this is a lot of (higher) structure if spelled out in detail. Yet more generally and again more readily axiomatized there is the notion of deloopings of any [[pointed object|
pointed]] types (cf. [Wärn (2023, §2)](delooping#Wärn23), [BCFR23](delooping#BCFR23)), which we may write as follows:

Similarly there is [[iterated loop space|iterated delooping structure]]:

## Related entries

* Evident as the notion of _mathematical structure_ may seem these days, it was at least not made explicit until the middle of the 20th century. Then it was the influence of the _[[Bourbaki]]_-project (see there for more) and then later the development of [[category theory]] which made the notion explicit and finally led to the above formalization.

* [[functions]] that preserves extra structure are called _[[homomorphisms]]_; [[relations]] that preserve extra structure are called [[logical relations]]_

* [[Birkhoff's HSP theorem]]

* [[structuralism]], [[structure identity principle]]

* [[exceptional structure]]

## References

### General

On the history of the notion of "mathematical structure" with some focus on what [[Bourbaki]] did and did not contribute to the subject:

* [[Leo Corry]], *Mathematical Structures from Hilbert to Bourbaki: The Evolution of an Image of Mathematics*, in: *Changing Images of Mathematics in History. From the French Revolution to the new Millenium* Harwood Academic Publishers (2001) 167-186 [[ISBN:9780415868273](https://www.routledge.com/Changing-Images-in-Mathematics-From-the-French-Revolution-to-the-New-Millennium/Bottazini-Dalmedico/p/book/9780415868273), [pdf](https://www.leocorry.com/_files/ugd/e6fef7_15c7077b08ec4f249fb40eb961256ced.pdf)]

* [[Leo Corry]], *Modern Algebra and the Rise of Mathematical Structures*, Springer (2004) [[doi:10.1007/978-3-0348-7917-0](https://doi.org/10.1007/978-3-0348-7917-0)]

See also

* Wikipedia, *[Mathematical structure](https://en.wikipedia.org/wiki/Mathematical_structure)*

* Wikipedia, *Structure_(mathematical_logic)*

Early proposal to grasp the notion of mathematical structures via [[category theory]] (specifically via [[forgetful functors]] between [[groupoids]]):

* {#Ehresmann57} [[Charles Ehresmann]], *Gattungen in Lokalen Strukturen*, Jahresbericht der Deutschen Mathematiker-Vereinigung **60** (1958) 49-77 [[dml:146434](https://eudml.org/doc/146434)]

* {#Ehresmann65} [[Charles Ehresmann]], _Catégories et Structures_, Séminaire Ehresmann. Topologie et géométrie différentielle **6** (1964) 1-31 [[numdam:SE_1964__6__A8_0](http://www.numdam.org/item?id=SE_1964__6__A8_0), [dml:112200](https://eudml.org/doc/112200)]

For modern accounts on mathematical structures via [[categorical algebra]] see also at *[[algebraic theory]]*, *[[monad]]*, *[[sketch]]*.

### In dependent type theory
 {#ReferencesInDependentTypeTheory}

Discussion of mathematical structures via [[dependent type theory]]:

the general idea of representing mathematical structures as [[type telescopes]]:

* [[Jeffery Zucker]], *Formalization of Classical Mathematics in Automath*, Colloques Internationaux du Centre National de la Recherche Scientifique **249** (1975) 135-145 [[web](https://www.win.tue.nl/automath/archive/webversion/aut042/aut042.html), [pdf](https://www.win.tue.nl/automath/archive/pdf/aut042.pdf)]

  also in: Studies in Logic and the Foundations of Mathematics **133** (1994) 127-139 [doi:10.1016/S0049-237X(08)70202-7]

* [[Nicolaas de Bruijn]], *Telescopic mappings in typed lambda calculus*, Information and Computation **91** 2 (1991) 189-204 [doi:10.1016/0890-5401(91)90066-B]

and with emphasis of the notion of [[isomorphism]] between such structures:

* [[David McAllester]], *Dependent Type Theory as Related to the Bourbaki Notions of Structure and Isomorphism* [[arXiv:2104.08958](https://arxiv.org/abs/2104.08958)]

in [[Coq]]:

* [[Herman Geuvers]], [[Randy Pollack]], [[Freek Wiedijk]], [[Jan Zwanenburg]], *A Constructive Algebraic Hierarchy in Coq*, Journal of Symbolic Computation **34** 4 (2002) 271-286 [[doi:10.1006/jsco.2002.0552](https://doi.org/10.1006/jsco.2002.0552), [pdf](http://www.cs.ru.nl/~herman/PUBS/JSC2002-GeuversPollackWiedijkZwanenburg-alghier1.pdf)]

* Claudio Sacerdoti Coen, Enrico Tassi, *Working with Mathematical Structures in Type Theory*, in: *Types for Proofs and Programs. TYPES 2007*, Lecture Notes in Computer Science **4941** Springer (2008) [[doi:10.1007/978-3-540-68103-8_11](https://doi.org/10.1007/978-3-540-68103-8_11), [pdf](https://www.cs.unibo.it/~sacerdot/PAPERS/types07.pdf)]

* François Garillot, Georges Gonthier, Assia Mahboubi & Laurence Rideau, *Packaging Mathematical Structures*, in: *Theorem Proving in Higher Order Logics. TPHOLs 2009*, Lecture Notes in Computer Science **5674**, Springer (2009) [[doi:10.1007/978-3-642-03359-9_23](https://doi.org/10.1007/978-3-642-03359-9_23)]

* [[Bas Spitters]], Eelis van der Wegen *Type classes for mathematics in type theory*, Mathematical Structures in Computer Science **21** 4 "Interactive Theorem Proving and the Formalisation of Mathematics" (2011) 795-825 [[doi:10.1017/S0960129511000119](https://doi.org/10.1017/S0960129511000119), [arXiv:1102.1323](https://arxiv.org/abs/1102.1323)]

  > (via [[ufias2012:Type classes]])

* Kazuhiko Sakaguchi, *Validating Mathematical Structures*, in *Automated Reasoning. IJCAR 2020*, Lecture Notes in Computer Science **12167**, Springer (2020) [[doi:10.1007/978-3-030-51054-1_8](https://doi.org/10.1007/978-3-030-51054-1_8)]

in [[Lean]]:

* [The mathlib Community](https://leanprover-community.github.io/mathlib-overview.html), §4 in: *The Lean mathematical library*, in *CPP 2020: Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs* (2020) 367–381 [[arXiv:1910.09336](https://arxiv.org/abs/1910.09336), [doi:10.1145/3372885.3373824](https://doi.org/10.1145/3372885.3373824)]

Discussion of classes of mathematical structures in [[univalence|univalent]] [[homotopy type theory]] ([[univalent foundations of mathematics]]) which satisfy a [[structure identity principle]]:

* {#CoquandDanielsson13} [[Thierry Coquand]], [[Nils Anders Danielsson]], *Isomorphism is equality*, Indagationes Mathematicae **24** 4 (2013) 1105-1120 [[doi:10.1016/j.indag.2013.09.002](https://doi.org/10.1016/j.indag.2013.09.002)]

* {#UFP13} [[Univalent Foundations Project]], §9.8 but also §2 of: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

* {#Escardó19} [[Martín Escardó]], *[A structure identity principle for a standard notion of structure](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html#sns)*, §3.33.1 in: *Introduction to Univalent Foundations of Mathematics with Agda* [[arXiv:1911.00580](https://arxiv.org/abs/1911.00580),  [webpage](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html)]

  > (formalized in the [[Agda]] [[proof assistant]])

* [[Benedikt Ahrens]], [[Paige Randall North]], [[Michael Shulman]], [[Dimitris Tsementzis]], *A Higher Structure Identity Principle*, LICS '20 (2020) 53–66 [[arXiv:2004.06572](https://arxiv.org/abs/2004.06572), [doi:10.1145/3373718.3394755](https://doi.org/10.1145/3373718.3394755)]

Discussion in the context of the [[philosophy]] of [[structuralism]]:

* [[Dimitris Tsementzis]], §5.1 in: *Univalent foundations as structuralist foundations*, Synthese **194** 9 (2017) 3583–3617 [[jstor:26748765](https://www.jstor.org/stable/26748765), [doi:10.1007/s11229-016-1109-x](https://doi.org/10.1007/s11229-016-1109-x), [pdf](https://core.ac.uk/download/pdf/157866891.pdf)]

Expressions towards the notion that, thereby the notions of *mathematical structures* are *data structures* may be understood to coincide (just along the lines of *[[types]]* being *[[data types]]*):

* [[Thomas Kehrenberg]], *Basic building blocks of dependent type theory* (Dec. 2022) [[post on LessWrong](https://www.lesswrong.com/posts/ccbsYSpTcTqXwukH8/basic-building-blocks-of-dependent-type-theory), [post on personal blog](https://tm.kehrenberg.net/a/type-theory1/)]

> "An example of a use case for non-binary sum types is the definition of mathematical “data structures” like a ring or a group. These data structures are usually defined as..."

[[!redirects structure]]
[[!redirects structures]]

[[!redirects mathematical structure]]
[[!redirects mathematical structures]]


structure (model theory)

A **structure** on a [[set]] $X$ consists of a collection

$$\mathcal{S}_n \subseteq P(X^n)$$

for each [[natural number]] $n$ such that

* $\mathcal{S}_n$ is a [[Boolean algebra|Boolean subalgebra]] of $P(X^n)$,

* If $A \in \mathcal{S}_n$, then $A \times X$ and $X \times A$ belong to $\mathcal{S}_{n+1}$,

* The set $\Delta = \{(x_1, \ldots, x_n): \forall_{1 \leq i \leq n} x_i = x_1\}$ belongs to $\mathcal{S}_n$,

* If $\pi: X^{n+1} \to X^n$ denotes the projection onto the first $n$ coordinates, then the image $\pi(A)$ belongs to $\mathcal{S}_n$ whenever $A \in \mathcal{S}_{n+1}$.

Morally, a structure on a set is the collection of subsets of a finite power of $X$ which are definable with respect to a [[first-order language]] that has been interpreted in $X$ (such an interpretation being called a structure of the language). Indeed, the definable sets of such a language do form a structure in the present sense. Conversely, to any structure in this sense, we may introduce a relational language whose $n$-ary relation symbols are named by the elements of $\mathcal{S}_n$, and then the tautological structure of this language on $X$, where each relation is interpreted as the set that names it, is a structure of this language.

Each structure $\mathcal{S}$ on $X$ induces a [[bicategory of relations]]: the objects are natural numbers, and 1-cells $m \to n$ are triples $(m, n, R)$ where $R \in \mathcal{S}_{m+n}$, ordered by inclusion. (It is indeed not difficult to show that 1-cells are closed under set-theoretic relational composition.)

[[!redirects structure (model theory)]]
[[!redirects structures (model theory)]]
[[!redirects model-theoretic structure]]
[[!redirects model-theoretic structures]]


structure formation

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

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[cosmology]], _structure formation_ refers to the process by which cosmological structures such as [[galaxy clusters]] and [[galaxies]] themselves form under [[gravity|gravitational]] collapse of a primordial gas.

Various computer simulations ([Millennium I](#MillenniumI), [Millennium II](#MillenniumII), [Millennium XXL](#MillenniumXXL), [Illustris](#Illustris)) test theory against observations, such as those by the [[Planck Collaboration]], see [Springel 13](#Springel13) for survey.

> graphics from _[The Millennium Simulation Project](https://wwwmpa.mpa-garching.mpg.de/galform/virgo/millennium/)_

## Feedback mechanisms and realistic dark matter models
 {#Feedback}

A revolution in realism of computer simulations of galaxy structure formation occurred after ~2010 when hardware and software advances allowed to resolve various feedback mechanisms, such as notably the effect of star-formation inside a gas cloud collapsing to a galaxy. This advance is reflected in the naming of the [FIRE](#ReferencesFIRE)-project: _F_eedback _I_n _R_ealistic _E_nvironments.

The resolution of such feedback mechanisms in simulation of  [[galaxy]] formation seems to resolve long-standing apparent discrepancies of [[dark matter]] models with observed galaxy-scale phenomena: For example [[galactic rotation curves]] are qualitatively reproduced  ([Hopkins 17, Figure 4, Figure 5](#Hopkins17)) as is the baryonic [[Tully-Fisher relation]] ([El-Badry 18, Figure 4](#ElBadry18)).

For more on this see also at _[[dark matter]]_ the section _[On galactic scales](dark+matter#Properties)_.

From [Vogelsberger et al. 14](#VogelsbergerEtAl14):

> {#TheSimulationReproduces} The simulation reproduces reasonably well the cosmic star formation rate density, the galaxy luminosity function, and baryon conversion efficiency at $z=0$. It also qualitatively captures the impact of galaxy environment on the red fractions of galaxies. The internal velocity structure of selected well-resolved disk galaxies obeys the stellar and baryonic [[Tully-Fisher relation]] together with flat circular velocity curves. In the well-resolved regime the simulation reproduces the observed mix of early-type and late-type galaxies.

See also [Illustris Project Website](#IllustrisProjectWebsite): _[What are we learning?](http://www.illustris-project.org/about/#public-three)_ and _[Preliminary Results](http://www.illustris-project.org/about/#astronomers-four)_.

## Related concepts

* [[inhomogeneous cosmology]]

  [[KBC void]]

* [[observable universe]]

* [[QCD cosmology]]

* [[dark matter]]

## References

### General

Review includes

* [[Joel Primack]], _Dark Matter and Structure Formation in the Universe_, Chapter 1 of _Formation of Structure in the Universe_, Proceedings of the Jerusalem Winter School 1996, edited by A. Dekel and J.P. Ostriker ([arXiv:astro-ph/9707285](https://arxiv.org/abs/astro-ph/9707285))

* [[Daniel Baumann]], _The big bang and the origin of structure_, 2019 ([web](https://www.dropbox.com/s/as3vb4jpv65822k/BigHistory-2019.pdf?dl=0))

Galaxy formation:

* Julio Navarro, _Simulations of Galaxy Formation:  the State of the Art_ ([pdf](http://www.physics.ox.ac.uk/iau311/navarro_iau311.pdf))

* {#ReadEtAl19} [[Justin Read]] et al. _EDGE I: the mass-metallicity relation as a critical test of galaxy formation physics_ ([arXiv:1904.02723](https://arxiv.org/abs/1904.02723))

See also

* Wikipedia, _[Structure formation](https://en.wikipedia.org/wiki/Structure_formation)_

Application of [[persistent homology]] to cosmological structure formation:

* Matteo Biagetti, [[Alex Cole]], [[Gary Shiu]], _The Persistence of Large Scale Structures I: Primordial non-Gaussianity_ ([arXiv:2009.04819](https://arxiv.org/abs/2009.04819))

On structure formation during [[cosmic inflation]] via [[inhomogeneous cosmology|inhomogeneous]] [[quantum cosmology]]:

* [[Martin Bojowald]], Ding Ding, *Canonical description of cosmological backreaction*, JCAP **03** (2021) 083
([arXiv:2011.03018](https://arxiv.org/abs/2011.03018))

### Computer simulation

There are various computer simulations for cosmic structure formation, listed below. Review includes the following:

* {#Springel13} Volker Springel, _Simulations of Cosmic Structure Formation_, USP Cosmology Conference Sao Paulo, February 2013 ([pdf](http://astro.dur.ac.uk/~cmb/LACEGAL/Talks/talk_springel.pdf), [[Springel13.pdf:file]])

* [Kavli Institute for Cosmological Physics](http://cfcp.uchicago.edu/), _[Formation of the large-scale structure in the Universe: simulations](cosmicweb.uchicago.edu/sims.html)_

* {#Springel19} Volker Springel, _Hydrodynamical galaxy formation simulations_, talk, December 2019 ([CCA announcement page](https://www.simonsfoundation.org/event/cca-colloquium-volker-springel),
[video recording](https://youtu.be/inbfOSzsGgA))

* Mark Vogelsberger, Federico Marinacci, Paul Torrey, Ewald Puchwein, _Cosmological Simulations of Galaxy Formation_, Nature Reviews Physics volume 2, pages 42–66 (2020)  ([arXiv:1909.07976](https://arxiv.org/abs/1909.07976), [doi:10.1038/s42254-019-0127-2](https://doi.org/10.1038/s42254-019-0127-2))

#### Millennium Project

* Wikipedia, _[Millennium Run](https://en.wikipedia.org/wiki/Millennium_Run)_

##### Millennium I
 {#MillenniumI}

* _[The Millennium Simulation Project](https://wwwmpa.mpa-garching.mpg.de/galform/virgo/millennium/)_

##### Millennium II
 {#MillenniumII}

*  Michael Boylan-Kolchin, Volker Springel, Simon D. M. White, Adrian Jenkins, Gerard Lemson, _Resolving Cosmic Structure Formation with the Millennium-II Simulation_, Mon.Not.Roy.Astron.Soc.398:1150, 2009 ([arXiv:0903.3041](https://arxiv.org/abs/0903.3041))

##### Millennium XXL
  {#MillenniumXXL}

* _The Millennium-XXL Project:  Simulating the Galaxy Population  of dark Energy Universes_ ([pdf](https://www.h-its.org/wp-content/uploads/2014/10/inSiDE_autumn2010_p20.pdf))

* R. E. Angulo, V. Springel, S. D. M. White, A. Jenkins, C. M. Baugh, C. S. Frenk, _Scaling relations for galaxy clusters in the Millennium-XXL simulation_, Monthly Notices of the Royal Astronomical Society, Volume 426, Issue 3, 1 November 2012, Pages 2046–2062, (doi:10.1111/j.1365-2966.2012.21830.x, [arXiv:1203.3216](https://arxiv.org/abs/1203.3216))

#### Illustris Project
 {#Illustris}

* {#IllustrisProjectWebsite} _[Illustris Project Website](http://www.illustris-project.org/about/)_

* Wikipedia, _[Illustris project](https://en.wikipedia.org/wiki/Illustris_project)_

* {#VogelsbergerEtAl14} Mark Vogelsberger, Shy Genel, Volker Springel, Paul Torrey, Debora Sijacki, Dandan Xu, Greg Snyder, Dylan Nelson, Lars Hernquist, _Introducing the Illustris Project: simulating the coevolution of dark and visible matter in the Universe_ ([arXiv:1405.2921](https://arxiv.org/abs/1405.2921))

#### FIRE
 {#ReferencesFIRE}

[[galaxy]] formation

* [FIRE: Feedback In Realistic Environments
Increasing the predictive power of galaxy formation simulations
](https://fire.northwestern.edu/)

  * [visualization](https://fire.northwestern.edu/visualizations/)

* {#GarrisonKimmel18} Shea Garrison-Kimmel, _Next-generation Galaxy Formation Simulations with FIRE_, 2018 ([video recording](https://youtu.be/sSkrm66uDvw))

* {#Fitts16} Fitts et al. _FIRE in the Field: Simulating the Threshold of Galaxy Formation_, Mon.Not.Roy.Astron.Soc.471:3547-3562, 2017 ([arXiv:1611.02281](https://arxiv.org/abs/1611.02281))

* {#Hopkins17} Hopkins et al. _FIRE-2 Simulations: Physics versus Numerics in Galaxy Formation_. Monthly Notices of the Royal Astronomical Society, Volume 480, Issue 1, 11 October 2018, Pages 800–863 ([arXiv:1702.06148](https://arxiv.org/abs/1702.06148), [doi:10.1093/mnras/sty1690](https://doi.org/10.1093/mnras/sty1690))

* {#ElBadry18} El-Badry et al. _Gas Kinematics in FIRE Simulated Galaxies Compared to Spatially Unresolved HI Observations_, Monthly Notices of the Royal Astronomical Society, Volume 477, Issue 2, 21 June 2018, Pages 1536–1548 ([arXiv:1801.03933](https://arxiv.org/abs/1801.03933), [doi:10.1093/mnras/sty730](https://doi.org/10.1093/mnras/sty730))

[[galactic rotation curves]] qualitatively reproduced in [Hopkins 17, Figure 4, Figure 5](#Hopkins17)

baryonic [[Tully-Fisher relation]] qualitatively reproduced in [El-Badry 18, Figure 4](#ElBadry18)

* Daniel Angles-Alcazar, _Multi-scale physical modeling in galaxy formation and evolution_, 2019 ([web](http://www.simonsfoundation.org/event/cca-seminar-daniel-angles-alcazar/))

#### VINTERGATAN
 {#ReferencesVintergatan}

Simulation of Milky Way-like [[galaxies]]:

* [VINTERGATAN simulation](http://www.astro.lu.se/~florent/vintergatan.php)

#### Cosmic Dawn
 {#CosmicDawn}

Simulation of [[galaxy formation]] in feedback with [[cosmic reionization]]:

* [[Pierre Ocvirk]] et al. , _Cosmic Dawn (CoDa): the First Radiation-Hydrodynamics Simulation of Reionization and Galaxy Formation in the Local Universe_ ([arXiv:1511.00011](https://arxiv.org/abs/1511.00011))

* [[Pierre Ocvirk]], et al., _Cosmic Dawn II (CoDa II): a new radiation-hydrodynamics simulation of the self-consistent coupling of galaxy formation and reionization_ ([arXiv:1811.11192](https://arxiv.org/abs/1811.11192))

#### Other

* _The Quijote simulations_ ([arXiv:1909.05273](https://arxiv.org/abs/1909.05273))

* Tomoaki Ishiyama  et al., *The Uchuu Simulations: Data Release 1 and Dark Matter Halo Concentrations* ([arXiv:2007.14720](https://arxiv.org/abs/2007.14720))

* [[gevolution]] (for [[inhomogeneous cosmology]])

[[!redirects galaxy formation]]

[[!redirects cosmic structure formation]]
[[!redirects cosmological structure formation]]

[[!redirects FIRE]]
[[!redirects FIRE-2]]


structure group

#Contents#
* table of contents
{:toc}

## Idea

For $G$ a geometric [[group]] (e.g. a [[topological group]] or [[Lie group]]) and $P \to X$ a $G$-[[principal bundle]], one says that $G$ is the **structure group** of $P$.

More generally, and less tautologically, if $V \to X$ is a $G$-[[associated bundle]] then one says that $V$ has **structure group** $G$.

In [[physics]] this is called the _[[gauge group]]_.

## Related concepts

* [[reduction of structure group]]

* [[gauge group]]

* [[equivariance group]]

[[!redirects structure groups]]