In category theory, the concept of an element, or a point, generalizes the more usual set theoretic concept of an element of a set to an object of any category. This idea often allows restating of definitions or properties of morphisms (such as monomorphism or product) given by a universal property in more familiar terms, by stating their relation to elements. Some very general theorems, such as Yoneda's lemma and the Mitchell embedding theorem, are of great utility for this, by allowing one to work in a context where these translations are valid. This approach to category theory – in particular the use of the Yoneda lemma in this way – is due to Grothendieck, and is often called the method of the functor of points.
Suppose C is any category and A, T are two objects of C. A T-valued point of A is simply a morphism . The set of all T-valued points of A varies functorially with T, giving rise to the "functor of points" of A; according to the Yoneda lemma, this completely determines A as an object of C.
Many properties of morphisms can be restated in terms of points. For example, a map is said to be a monomorphism if
Suppose and in C. Then g and h are A-valued points of B, and therefore monomorphism is equivalent to the more familiar statement
Some care is necessary. f is an epimorphism if the dual condition holds:
In set theory, the term "epimorphism" is synonymous with "surjection", i.e.
This is clearly not the translation of the first statement into the language of points, and in fact these statements are not equivalent in general. However, in some contexts, such as abelian categories, "monomorphism" and "epimorphism" are backed by sufficiently strong conditions that in fact they do allow such a reinterpretation on points.
Similarly, categorical constructions such as the product have pointed analogues. Recall that if A, B are two objects of C, their product A × B is an object such that
In this definition, f and g are T-valued points of A and B, respectively, while h is a T-valued point of A × B. An alternative definition of the product is therefore:
This is the more familiar definition of the product of two sets.
The terminology is geometric in origin; in algebraic geometry, Grothendieck introduced the notion of a scheme in order to unify the subject with arithmetic geometry, which dealt with the same idea of studying solutions to polynomial equations (i.e. algebraic varieties) but where the solutions are not complex numbers but rational numbers, integers, or even elements of some finite field. A scheme is then just that: a scheme for collecting together all the manifestations of a variety defined by the same equations but with solutions taken in different number sets. One scheme gives a complex variety, whose points are its -valued points, as well as the set of -valued points (rational solutions to the equations), and even -valued points (solutions modulo p).
One feature of the language of points is evident from this example: it is, in general, not enough to consider just points with values in a single object. For example, the equation (which defines a scheme) has no real solutions, but it has complex solutions, namely . It also has one solution modulo 2 and two modulo 5, 13, 29, etc. (all primes that are 1 modulo 4). Just taking the real solutions would give no information whatsoever.
The situation is analogous to the case where C is the category Set, of sets of actual elements. In this case, we have the "one-pointed" set {1}, and the elements of any set S are the same as the {1}-valued points of S. In addition, though, there are the {1,2}-valued points, which are pairs of elements of S, or elements of S × S. In the context of sets, these higher points are extraneous: S is determined completely by its {1}-points. However, as shown above, this is special (in this case, it is because all sets are iterated coproducts of {1}).
In mathematics, given two groups, (G,∗) and (H, ·), a group homomorphism from (G,∗) to (H, ·) is a function h : G → H such that for all u and v in G it holds that
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type. The word homomorphism comes from the Ancient Greek language: ὁμός meaning "same" and μορφή meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning "similar" to ὁμός meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).
In mathematics, the Yoneda lemma is a fundamental result in category theory. It is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory. It allows the embedding of any locally small category into a category of functors defined on that category. It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category. It is an important tool that underlies several modern developments in algebraic geometry and representation theory. It is named after Nobuo Yoneda.
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties.
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation .
In category theory, an epimorphism is a morphism f : X → Y that is right-cancellative in the sense that, for all objects Z and all morphisms g1, g2: Y → Z,
In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams. The five lemma is not only valid for abelian categories but also works in the category of groups, for example.
An exact sequence is a sequence of morphisms between objects such that the image of one morphism equals the kernel of the next.
In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result. It is said that commutative diagrams play the role in category theory that equations play in algebra.
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert.
The cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y / im(f) of the codomain of f by the image of f. The dimension of the cokernel is called the corank of f.
In category theory, a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the opposite category Cop. Given a statement regarding the category C, by interchanging the source and target of each morphism as well as interchanging the order of composing two morphisms, a corresponding dual statement is obtained regarding the opposite category Cop. Duality, as such, is the assertion that truth is invariant under this operation on statements. In other words, if a statement is true about C, then its dual statement is true about Cop. Also, if a statement is false about C, then its dual has to be false about Cop.
In category theory, a branch of mathematics, the image of a morphism is a generalization of the image of a function.
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 from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation.
In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of model categories. The dual notion is that of a projective object.
In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion is a generalization of concepts such as subsets from set theory, subgroups from group theory, and subspaces from topology. Since the detailed structure of objects is immaterial in category theory, the definition of subobject relies on a morphism that describes how one object sits inside another, rather than relying on the use of elements.
In category theory, a branch of mathematics, a pullback is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. The pullback is written
This is a glossary of properties and concepts in category theory in mathematics.
In category theory, a branch of mathematics, a section is a right inverse of some morphism. Dually, a retraction is a left inverse of some morphism. In other words, if and are morphisms whose composition is the identity morphism on , then is a section of , and is a retraction of .
In category theory, an abstract mathematical discipline, a nodal decomposition of a morphism is a representation of as a product , where is a strong epimorphism, a bimorphism, and a strong monomorphism.
