HOME

TheInfoList



OR:

In mathematics, a skeleton of a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
is a
subcategory In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, ...
that, roughly speaking, does not contain any extraneous
isomorphism 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 i ...
s. In a certain sense, the skeleton of a category is the "smallest"
equivalent 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 ...
category, which captures all "categorical properties" of the original. In fact, two categories are
equivalent 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 ...
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 bic ...
they have
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 i ...
skeletons. A category is called skeletal if isomorphic objects are necessarily identical.


Definition

A skeleton of a category ''C'' is an equivalent category ''D'' in which no two distinct objects are isomorphic. It is generally considered to be a subcategory. In detail, a skeleton of ''C'' is a category ''D'' such that: * ''D'' is a
subcategory In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, ...
of ''C'': every object of ''D'' is an object of ''C'' :\mathrm(D)\subseteq \mathrm(C) for every pair of objects ''d''1 and ''d''2 of ''D'', the
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
s in ''D'' are morphisms in ''C'', i.e. :\mathrm_D(d_1, d_2) \subseteq \mathrm_C(d_1, d_2) and the identities and compositions in ''D'' are the restrictions of those in ''C''. * The inclusion of ''D'' in ''C'' is full, meaning that for every pair of objects ''d''1 and ''d''2 of ''D'' we strengthen the above subset relation to an equality: :\mathrm_D(d_1, d_2) =\mathrm_C(d_1, d_2) * The inclusion of ''D'' in ''C'' is essentially surjective: Every ''C''-object is isomorphic to some ''D''-object. * ''D'' is skeletal: No two distinct ''D''-objects are isomorphic.


Existence and uniqueness

It is a basic fact that every small category has a skeleton; more generally, every accessible category has a skeleton. (This is equivalent to the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
.) Also, although a category may have many distinct skeletons, any two skeletons are isomorphic as categories, so
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...
isomorphism of categories, the skeleton of a category is unique. The importance of skeletons comes from the fact that they are (up to isomorphism of categories), canonical representatives of the equivalence classes of categories under the
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
of
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 ...
. This follows from the fact that any skeleton of a category ''C'' is equivalent to ''C'', and that two categories are equivalent if and only if they have isomorphic skeletons.


Examples

*The category Set of all sets has the subcategory of all
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The ...
s as a skeleton. *The category ''K''-Vect of all
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
s over a fixed
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
K has the subcategory consisting of all powers K^, where ''α'' is any cardinal number, as a skeleton; for any finite ''m'' and ''n'', the maps K^m \to K^n are exactly the ''n'' × ''m''
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
with entries in ''K''. *
FinSet In the mathematical field of category theory, FinSet is the category whose objects are all finite sets and whose morphisms are all functions between them. FinOrd is the category whose objects are all finite ordinal numbers and whose morphisms a ...
, the category of all
finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. ...
s has
FinOrd In the mathematical field of category theory, FinSet is the category whose objects are all finite sets and whose morphisms are all functions between them. FinOrd is the category whose objects are all finite ordinal numbers and whose morphisms are ...
, the category of all finite
ordinal numbers In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least ...
, as a skeleton. *The category of all well-ordered sets has the subcategory of all
ordinal numbers In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least ...
as a skeleton. *A
preorder In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special ca ...
, i.e. a small category such that for every pair of objects A,B , the set \mbox(A,B) either has one element or is empty, has a
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. A poset consists of a set together with a binary ...
as a skeleton.


See also

* Glossary of category theory * Thin category


References

* Adámek, Jiří, Herrlich, Horst, & Strecker, George E. (1990)
''Abstract and Concrete Categories''
Originally published by John Wiley & Sons. {{isbn, 0-471-60922-6. (now free on-line edition) * Robert Goldblatt (1984). ''Topoi, the Categorial Analysis of Logic'' (Studies in logic and the foundations of mathematics, 98). North-Holland. Reprinted 2006 by Dover Publications. Category theory