Skeleton (category theory)
Encyclopedia
In mathematics
, a skeleton of a category is a subcategory
which, roughly speaking, does not contain any extraneous isomorphism
s. In a certain sense, the skeleton of a category is the "smallest" equivalent
category which captures all "categorical properties". In fact, two categories are equivalent
if and only if
they have isomorphic
skeletons. A category is called skeletal if isomorphic objects are necessarily identical.
, isomorphism-dense subcategory
D in which no two distinct objects are isomorphic. In detail, a skeleton of C is a category D such that:
has a skeleton. (This is equivalent to the axiom of choice.) Also, although a category may have many distinct skeletons, any two skeletons are isomorphic as categories
, so up to
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
of equivalence of categories
. 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a skeleton of a category is a subcategory
Subcategory
In mathematics, 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, a subcategory of C is a category obtained from C by "removing" some of its objects and...
which, roughly speaking, does not contain any extraneous isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
s. In a certain sense, the skeleton of a category is the "smallest" equivalent
Equivalence of categories
In category theory, an abstract branch of 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 from many areas of mathematics...
category which captures all "categorical properties". In fact, two categories are equivalent
Equivalence of categories
In category theory, an abstract branch of 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 from many areas of mathematics...
if and only if
IFF
IFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...
they have isomorphic
Isomorphism of categories
In category theory, two categories C and D are isomorphic if there exist functors F : C → D and G : D → C which are mutually inverse to each other, i.e. FG = 1D and GF = 1C. This means that both the objects and the morphisms of C and D stand in a one to one correspondence to each other...
skeletons. A category is called skeletal if isomorphic objects are necessarily identical.
Definition
A skeleton of a category C is a fullSubcategory
In mathematics, 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, a subcategory of C is a category obtained from C by "removing" some of its objects and...
, isomorphism-dense subcategory
Subcategory
In mathematics, 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, a subcategory of C is a category obtained from C by "removing" some of its objects and...
D in which no two distinct objects are isomorphic. In detail, a skeleton of C is a category D such that:
- Every object of D is an object of C.
- (Fullness) For every pair of objects d1 and d2 of D, the morphismMorphismIn mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
s in D are precisely the morphisms in C, i.e.
- For every object d of D, the D-identity on d is the C-identity on d.
- The composition law in D is the restriction of the composition law in C to the morphisms in D.
- (Isomorphism-dense) Every C-object is isomorphic to some D-object.
- 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 categoryAccessible category
The theory of accessible categories was introduced in 1989 by mathematicians Michael Makkai and Robert Paré in the setting of category theory. While the original motivation came from model theory, a branch of mathematical logic, it turned out that accessible categories have applications in homotopy...
has a skeleton. (This is equivalent to the axiom of choice.) Also, although a category may have many distinct skeletons, any two skeletons are isomorphic as categories
Isomorphism of categories
In category theory, two categories C and D are isomorphic if there exist functors F : C → D and G : D → C which are mutually inverse to each other, i.e. FG = 1D and GF = 1C. This means that both the objects and the morphisms of C and D stand in a one to one correspondence to each other...
, so up to
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
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
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
of equivalence of categories
Equivalence of categories
In category theory, an abstract branch of 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 from many areas of mathematics...
. This follows from the fact that any skeleton of a category C is equivalent
Equivalence of categories
In category theory, an abstract branch of 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 from many areas of mathematics...
to C, and that two categories are equivalent if and only if they have isomorphic skeletons.
Examples
- The category SetCategory of setsIn the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...
of all sets has the subcategory of all cardinal numberCardinal numberIn mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
s as a skeleton. - The category K-VectCategory of vector spacesIn mathematics, especially category theory, the category K-Vect has all vector spaces over a fixed field K as objects and K-linear transformations as morphisms...
of all vector spaceVector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s over a fixed fieldField (mathematics)In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
has the subcategory consisting of all powers 
, where n is any cardinal number, as a skeleton. - FinSetFinSetIn the mathematical field of category theory, FinSet is the category whose objects are all finite sets and whose morphisms are all functions between them...
, the category of all finite sets has FinOrd, the category of all finite ordinal numbers, as a skeleton. - The category of all well-ordered setsWell-orderIn mathematics, a well-order relation on a set S is a strict total order on S with the property that every non-empty subset of S has a least element in this ordering. Equivalently, a well-ordering is a well-founded strict total order...
has the subcategory of all ordinal numbers as a skeleton. - A preorderPreorderIn mathematics, especially in order theory, preorders are binary relations that are reflexive and transitive.For example, all partial orders and equivalence relations are preorders...
, i.e. a small category such that for every pair of objects
, the set 
either has one element or is empty, has a partially ordered setPartially ordered setIn mathematics, especially order theory, a partially ordered set 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 relation that indicates that, for certain pairs of elements in the...
as a skeleton.