Equivalent definitions of mathematical structures
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In mathematics, equivalent definitions are used in two somewhat different ways. First, within a particular mathematical theory (for example, Euclidean geometry), a notion (for example,
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
or minimal surface) may have more than one definition. These definitions are equivalent in the context of a given
mathematical structure In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additional ...
(
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, in this case). Second, a mathematical structure may have more than one definition (for example,
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
has at least seven definitions; ordered field has at least two definitions). In the former case, equivalence of two definitions means that a mathematical object (for example, geometric body) satisfies one definition
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
it satisfies the other definition. In the latter case, the meaning of equivalence (between two definitions of a structure) is more complicated, since a structure is more abstract than an object. Many different objects may implement the same structure.


Isomorphic implementations

Natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...
may be implemented as 0 = , 1 = = , 2 = = , 3 = = and so on; or alternatively as 0 = , 1 = =, 2 = = and so on. These are two different but
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
implementations of natural numbers in set theory. They are isomorphic as models of Peano axioms, that is, triples (''N'',0,''S'') where ''N'' is a set, 0 an element of ''N'', and ''S'' (called the
successor function In mathematics, the successor function or successor operation sends a natural number to the next one. The successor function is denoted by ''S'', so ''S''(''n'') = ''n'' +1. For example, ''S''(1) = 2 and ''S''(2) = 3. The successor functio ...
) a map of ''N'' to itself (satisfying appropriate conditions). In the first implementation ''S''(''n'') = ''n'' ∪ ; in the second implementation ''S''(''n'') = . As emphasized in
Benacerraf's identification problem In the philosophy of mathematics, Benacerraf's identification problem is a philosophical argument developed by Paul Benacerraf against set-theoretic Platonism and published in 1965 in an article entitled "What Numbers Could Not Be".Paul Benacerraf ...
, the two implementations differ in their answer to the question whether 0 ∈ 2; however, this is not a legitimate question about natural numbers (since the relation ∈ is not stipulated by the relevant signature(s), see the next section).Technically, "0 ∈ 2" is an example of a non-transportable relation, see , . Similarly, different but isomorphic implementations are used for complex numbers.


Deduced structures and cryptomorphisms

The successor function ''S'' on natural numbers leads to arithmetic operations, addition and multiplication, and the total order, thus endowing ''N'' with an ordered semiring structure. This is an example of a deduced structure. The ordered semiring structure (''N'', +, ·, ≤) is deduced from the Peano structure (''N'', 0, ''S'') by the following procedure: ''n'' + 0 = ''n'',   ''m'' + S (''n'') = S (''m'' + ''n''),   ''m'' · 0 = 0,   ''m'' · S (''n'') = ''m'' + (''m'' · ''n''), and ''m'' ≤ ''n'' if and only if there exists ''k'' ∈ ''N'' such that ''m'' + ''k'' = ''n''. And conversely, the Peano structure is deduced from the ordered semiring structure as follows: ''S'' (''n'') = ''n'' + 1, and 0 is defined by 0 + 0 = 0. It means that the two structures on ''N'' are equivalent by means of the two procedures. The two isomorphic implementations of natural numbers mentioned in the previous section are isomorphic as triples (''N'',0,''S''), that is, structures of the same
signature A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
(0,''S'') consisting of a constant symbol 0 and a unary function ''S''. An ordered semiring structure (''N'', +, ·, ≤) has another signature (+, ·, ≤) consisting of two binary functions and one binary relation. The notion of isomorphism does not apply to structures of different signatures. In particular, a Peano structure cannot be isomorphic to an ordered semiring. However, an ordered semiring deduced from a Peano structure may be isomorphic to another ordered semiring. Such relation between structures of different signatures is sometimes called a
cryptomorphism In mathematics, two objects, especially systems of axioms or semantics for them, are called cryptomorphic if they are equivalent but not obviously equivalent. In particular, two definitions or axiomatizations of the ''same'' object are "cryptomorp ...
.


Ambient frameworks

A structure may be implemented within a set theory ZFC, or another set theory such as NBG, NFU, ETCS. Alternatively, a structure may be treated in the framework of first-order logic, second-order logic, higher-order logic, a
type theory In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a foundat ...
, homotopy type theory etc.A reasonable choice of an ambient framework should not alter basic properties of a structure, but can alter provability of finer properties. For example, some theorems about the natural numbers are provable in set theory (and some other strong systems) but not provable in first-order logic; see
Paris–Harrington theorem In mathematical logic, the Paris–Harrington theorem states that a certain combinatorial principle in Ramsey theory, namely the strengthened finite Ramsey theorem, is true, but not provable in Peano arithmetic. This has been described by some (suc ...
and Goodstein's theorem. The same applies to definability; see for example
Tarski's undefinability theorem Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1933, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics. Informally, the theorem states that ''arithmetical truth ...
.


Structures according to Bourbaki

:"Mathematics ..cannot be explained completely by a single concept such as the mathematical structure. Nevertheless, Bourbaki's structuralist approach is the best that we have." () :"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 and then later the development of category theory which made the notion explicit"
nLab
. According to Bourbaki, the scale of sets on a given set ''X'' consists of all sets arising from ''X'' by taking
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...
s and power sets, in any combination, a finite number of times. Examples: ''X''; ''X'' × ''X''; ''P''(''X''); ''P''(''P''(''X'' × ''X'') × ''X'' × ''P''(''P''(''X''))) × ''X''. (Here ''A'' × ''B'' is the product of ''A'' and ''B'', and ''P''(''A'') is the powerset of ''A''.) In particular, a pair (0,''S'') consisting of an element 0 ∈ ''N'' and a unary function ''S'' : ''N'' → ''N'' belongs to ''N'' × ''P''(''N'' × ''N'') (since a function is a subset of the Cartesian product). A triple (+, ·, ≤) consisting of two binary functions ''N'' × ''N'' → ''N'' and one binary relation on ''N'' belongs to ''P''(''N'' × ''N'' × ''N'') × ''P''(''N'' × ''N'' × ''N'') × ''P''(''N'' × ''N''). Similarly, every algebraic structure on a set belongs to the corresponding set in the scale of sets on ''X''. Non-algebraic structures on a set ''X'' often involve sets of subsets of ''X'' (that is, subsets of ''P''(''X''), in other words, elements of ''P''(''P''(''X''))). For example, the structure of a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
, called a topology on ''X'', treated as the set of "open" sets; or the structure of a measurable space, treated as the σ-algebra of "measurable" sets; both are elements of ''P''(''P''(''X'')). These are second-order structures. More complicated non-algebraic structures combine an algebraic component and a non-algebraic component. For example, the structure of a
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...
consists of a topology and the structure of a group. Thus it belongs to the product of ''P''(''P''(''X'')) and another ("algebraic") set in the scale; this product is again a set in the scale.


Transport of structures; isomorphism

Given two sets ''X'', ''Y'' and a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
''f'' : ''X'' → ''Y'', one constructs the corresponding bijections between scale sets. Namely, the bijection ''X'' × ''X'' → ''Y'' × ''Y'' sends (''x''1,''x''2) to (''f''(''x''1),''f''(''x''2)); the bijection ''P''(''X'') → ''P''(''Y'') sends a subset ''A'' of ''X'' into its
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
''f''(''A'') in ''Y''; and so on, recursively: a scale set being either product of scale sets or power set of a scale set, one of the two constructions applies. Let (''X'',''U'') and (''Y'',''V'') be two structures of the same signature. Then ''U'' belongs to a scale set ''S''''X'', and ''V'' belongs to the corresponding scale set ''S''''Y''. Using the bijection ''F'' : ''S''''X'' → ''S''''Y'' constructed from a bijection ''f'' : ''X'' → ''Y'', one defines: : ''f'' is an ''isomorphism'' between (''X'',''U'') and (''Y'',''V'') if ''F''(''U'') = ''V''. This general notion of isomorphism generalizes many less general notions listed below. * For algebraic structures: isomorphism is a bijective homomorphism. * In particular, for
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s: linear bijection. * For
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
s: order isomorphism. * For
graphs Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
: graph isomorphism. * More generally, for sets endowed with a binary relation: relation-preserving isomorphism. * For topological spaces: homeomorphism or topological isomorphism or bi continuous function. * For uniform spaces:
uniform isomorphism In the mathematical field of topology a uniform isomorphism or is a special isomorphism between uniform spaces that respects uniform properties. Uniform spaces with uniform maps form a category. An isomorphism between uniform spaces is called a un ...
. * For
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s: bijective isometry. * For topological groups: group isomorphism which is also a homeomorphism of the underlying topological spaces. * For topological vector spaces: isomorphism of vector spaces which is also a homeomorphism of the underlying topological spaces. * For
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s: bijective linear isometry. * For
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s:
unitary transformation In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation. Formal definition More precisely, ...
. * For Lie groups: a bijective smooth group homomorphism whose inverse is also a smooth group homomorphism. * For smooth manifolds: diffeomorphism. * For
symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sympl ...
s: symplectomorphism. * For
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
s: isometric diffeomorphism. * For conformal spaces: conformal diffeomorphism. * For probability spaces: a bijective measurable and measure preserving map whose inverse is also measurable and measure preserving. * For affine spaces: bijective
affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, ...
. * For
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
s:
homography In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, ...
. In fact, Bourbaki stipulates two additional features. First, several sets ''X''1, ..., ''X''''n'' (so-called principal base sets) may be used, rather than a single set ''X''. However, this feature is of little use. All the items listed above use a single principal base set. Second, so-called auxiliary base sets ''E''1, ..., ''E''''m'' may be used. This feature is widely used. Indeed, the structure of a vector space stipulates not only addition ''X'' × ''X'' → ''X'' but also scalar multiplication R × ''X'' → ''X'' (if R is the field of scalars). Thus, R is an auxiliary base set (called also "external"). The scale of sets consists of all sets arising from all base sets (both principal and auxiliary) by taking Cartesian products and power sets. Still, the map ''f'' (possibly an isomorphism) acts on ''X'' only; auxiliary sets are endowed by identity maps. (However, the case of ''n'' principal sets leads to ''n'' maps.)


Functoriality

Several statements formulated by Bourbaki without mentioning categories can be reformulated readily in the language of
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
. First, some terminology. * The scale of sets is indexed by "echelon construction schemes", called also "types". One may think of, say, the set ''P''(''P''(''X'' × ''X'') × ''X'' × ''P''(''P''(''X''))) × ''X'' as a set ''X'' substituted into the formula "''P''(''P''(''a'' × ''a'') × ''a'' × ''P''(''P''(''a''))) × ''a''" for the variable ''a''; this formula is the corresponding echelon construction scheme.In order to be more formal, Bourbaki encodes such formulas with sequences of ordered pairs of natural numbers. (This notion, defined for all structures, may be thought of as a generalization of the signature defined only for algebraic structures.)On one hand, it is possible to exclude the Cartesian products, treating a pair (''x'',''y'') as just the set . On the other hand, it is possible to include the set operation ''X'',''Y''->''Y''''X'' (all functions from ''X'' to ''Y''). "It is possible to simplify the matter by considering operations and functions as a special kind of relations (for example, a binary operation is a ternary relation). However, quite often, it is an advantage to have operations as a primitive concept." * Let Set* denote the groupoid of sets and bijections. That is, the category whose objects are (all) sets, and morphisms are (all) bijections. ''Proposition.'' Each echelon construction scheme leads to a functor from Set* to itself. In particular, the
permutation group In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to it ...
of a set ''X''
acts The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
on every scale set ''S''''X''. In order to formulate one more proposition, the notion "species of structures" is needed, since echelon construction scheme gives only preliminary information on a structure. For example, commutative groups and (arbitrary) groups are two different species of the same echelon construction scheme. Another example: topological spaces and measurable spaces. They differ in the so-called axiom of the species. This axiom is the conjunction of all required properties, such as "multiplication is associative" for groups, or "the union of open sets is an open set" for topological spaces. * A species of structures consists of an echelon construction scheme and an axiom of the species. ''Proposition.'' Each species of structures leads to a functor from Set* to itself. Example. For the species of groups, the functor ''F'' maps a set ''X'' to the set ''F''(''X'') of all group structures on ''X''. For the species of topological spaces, the functor ''F'' maps a set ''X'' to the set ''F''(''X'') of all topologies on ''X''. The morphism ''F''(''f'') : ''F''(''X'') → ''F''(''Y'') corresponding to a bijection ''f'' : ''X'' → ''Y'' is the transport of structures. Topologies on ''Y'' correspond bijectively to topologies on ''X''. The same holds for group structures, etc. In particular, the set of all structures of a given species on a given set is invariant under the action of the permutation group on the corresponding scale set ''S''''X'', and is a fixed point of the action of the group on another scale set ''P''(''S''''X''). However, not all fixed points of this action correspond to species of structures.The set of all possible axioms of species is countable, while the set of all fixed points of the considered action may be uncountable. Tarski's " logical notions of higher order" are closer to the fixed points than to species of structures, see and references therefrom. Given two species, Bourbaki defines the notion "procedure of deduction" (of a structure of the second species from a structure of the first species). A pair of mutually inverse procedures of deduction leads to the notion "equivalent species". Example. The structure of a topological space may be defined as an open set topology or alternatively, a closed set topology. The two corresponding procedures of deduction coincide; each one replaces all given subsets of ''X'' with their complements. In this sense, these are two equivalent species. In the general definition of Bourbaki, deduction procedure may include a change of the principal base set(s), but this case is not treated here. In the language of category theory one have the following result. ''Proposition.'' Equivalence between two species of structures leads to a natural isomorphism between the corresponding functors. However, in general, not all natural isomorphisms between these functors correspond to
equivalences Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry *Equivalence class (music) *''Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *'' Equival ...
between the species.The set of all possible deduction procedures is countable, while the set of all natural isomorphisms between the considered functors may be uncountable (see an example in Section #Canonical, not just natural).


Mathematical practice

:"We often do not distinguish structures that are isomorphic and often say that ''two structures are the same, up to isomorphism''." :"When studying structures we are interested only in their form, but when we prove their existence we need to construct them." :'Mathematicians are of course used to identifying isomorphic structures in practice, but they generally do so by "abuse of notation", or some other informal device, knowing that the objects involved are not "really" identical.' (A radically better approach is expected; but for now, Summer 2014, the definitive book quoted above does not elaborate on structures.) In practice, one makes no distinction between equivalent species of structures. Usually, a text based on natural numbers (for example, the article "
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
") does not specify the used definition of natural numbers. Likewise, a text based on topological spaces (for example, the article "
homotopy In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
", or "
inductive dimension In the mathematical field of topology, the inductive dimension of a topological space ''X'' is either of two values, the small inductive dimension ind(''X'') or the large inductive dimension Ind(''X''). These are based on the observation that, in ...
") does not specify the used definition of a topological space. Thus, it is possible (and rather probable) that the reader and the author interpret the text differently, according to different definitions. Nevertheless, the communication is successful, which means that such different definitions may be thought of as equivalent. A person acquainted with topological spaces knows basic relations between neighborhoods, convergence, continuity, boundary, closure, interior, open sets, closed sets, and does not need to know that some of these notions are "primary", stipulated in the definition of a topological space, while others are "secondary", characterized in terms of "primary" notions. Moreover, knowing that subsets of a topological space are themselves topological spaces, as well as products of topological spaces, the person is able to construct some new topological spaces irrespective of the definition. Thus, in practice a topology on a set is treated like an abstract data type that provides all needed notions (and constructors) but hides the distinction between "primary" and "secondary" notions. The same applies to other kinds of mathematical structures. "Interestingly, the formalization of structures in set theory is a similar task as the formalization of structures for computers."


Canonical, not just natural

As was mentioned, equivalence between two species of structures leads to a natural isomorphism between the corresponding functors. However, "
natural Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are p ...
" does not mean " canonical". A natural transformation is generally non-unique. Example. Consider again the two equivalent structures for natural numbers. One is the "Peano structure" (0,''S''), the other is the structure (+, ·, ≤) of ordered semiring. If a set ''X'' is endowed by both structures then, on one hand, ''X'' = where ''S''(''a''''n'') = ''a''''n''+1 for all ''n'' and 0 = ''a''0; and on the other hand, ''X'' = where ''b''''m''+''n'' = ''b''''m'' + ''b''''n'', ''b''''m''·''n'' = ''b''''m'' · ''b''''n'', and ''b''''m'' ≤''b''''n'' if and only if ''m'' ≤ ''n''. Requiring that ''a''''n'' = ''b''''n'' for all ''n'' one gets the canonical equivalence between the two structures. However, one may also require ''a''0 = ''b''1, ''a''1 = ''b''0, and ''a''''n'' = ''b''''n'' for all ''n'' > 1, thus getting another, non-canonical, natural isomorphism. Moreover, every
permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
of the index set leads to a natural isomorphism; they are uncountably many! Another example. A structure of a (simple) graph on a set ''V'' = of vertices may be described by means of its
adjacency matrix In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite simp ...
, a (0,1)-matrix of size ''n''×''n'' (with zeros on the diagonal). More generally, for arbitrary ''V'' an adjacency function on ''V'' × ''V'' may be used. The canonical equivalence is given by the rule: "1" means "connected" (with an edge), "0" means "not connected". However, another rule, "0" means "connected", "1" means "not", may be used, and leads to another, natural but not canonical, equivalence. In this example, canonicity is rather a matter of convention. But here is a worse case. Instead of "0" and "1" one may use, say, the two possible orientations of the plane R2 ("clockwise" and "counterclockwise"). It is difficult to choose a canonical rule in this case! "Natural" is a well-defined mathematical notion, but it does not ensure uniqueness. "Canonical" does, but generally is more or less conventional. A consistent choice of canonical equivalences is an inevitable component of equivalent definitions of mathematical structures.


See also

* Concrete category * Abuse of terminology#Equality vs. isomorphism *
Equivalence of categories In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences fro ...


Notes


Footnotes


References

*. *.


Further reading

*. *. *.


External links


nLab:structured set
"Almost everything in contemporary mathematics is an example of a structured set." (quoted from Section "Examples")
nLab: structure in model theory

nLab: stuff, structure, property

MathOverflow: What is the definition of “canonical”?
"a rule of thumb: there is a canonical isomorphism between X and Y if and only if you would feel comfortable writing X = Y" (Reid Barton)

"When you think of a structure it is best to think of it as containing all that information, not just the stuff in the definition" (Charles Wells)
MathStackExchange: A pedantic question about defining new structures in a path-independent way
`We would continue making statements like, "A topological space is determined by its open sets," but would never make a statement like, "A topological space is an ordered pair (X,\mathcal O) such that..."'
MathStackExchange: Does there exist another way of obtaining a topological space from a metric space equally deserving of the term “canonical”?
{{Mathematical logic Equivalence (mathematics) Mathematical structures