Morphism in the context of Product (category theory)

In category theory, a branch of mathematics, given a morphism f: X → Y and a morphism g: Z → Y, a lift or lifting of f to Z is a morphism h: X → Z such that f = g ∘ h. We say that f factors through h.

Lifts are ubiquitous; for example, the definition of fibrations (see Homotopy lifting property) and the valuative criteria of separated and proper maps of schemes are formulated in terms of existence and (in the last case) uniqueness of certain lifts.

In mathematics, an isomorphism is a structure-preserving mapping or morphism 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, and this is often denoted as ⁠${\displaystyle A\cong B}$⁠. The word is derived from Ancient Greek ἴσος (isos) 'equal' and μορφή (morphe) 'form, shape'.

The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may often be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism. A common example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension one of a vector space are isomorphic and cannot be identified.

In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

When some object ${\displaystyle X}$ is said to be embedded in another object ${\displaystyle Y}$, the embedding is given by some injective and structure-preserving map ${\displaystyle f:X\rightarrow Y}$. The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which ${\displaystyle X}$ and ${\displaystyle Y}$ are instances. In the terminology of category theory, a structure-preserving map is called a morphism.

In mathematics, a map or mapping is a function in its general sense. These terms may have originated as from the process of making a geographical map: mapping the Earth surface to a sheet of paper.

The term map may be used to distinguish some special types of functions, such as homomorphisms. For example, a linear map is a homomorphism of vector spaces, while the term linear function may have this meaning or it may mean a linear polynomial. In category theory, a map may refer to a morphism. The term transformation can be used interchangeably, but transformation often refers to a function from a set to itself. There are also a few less common uses in logic and graph theory.

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word homomorphism comes from the Ancient Greek language: ὁμός (homos) meaning "same" and μορφή (morphe) 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).

Homomorphisms of vector spaces are also called linear maps, and their study is the subject of linear algebra.

In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any category. The way they are put together is specified by a system of homomorphisms (group homomorphism, ring homomorphism, or in general morphisms in the category) between those smaller objects. The direct limit of the objects ${\displaystyle A_{i}}$, where ${\displaystyle i}$ ranges over some directed set ${\displaystyle I}$, is denoted by ${\displaystyle \varinjlim A_{i}}$. This notation suppresses the system of homomorphisms; however, the limit depends on the system of homomorphisms.

Direct limits are a special case of the concept of colimit in category theory. Direct limits are dual to inverse limits, which are a special case of limits in category theory.

In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the preimage of any open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable.

In mathematics, the automorphism group of an object X is the group consisting of automorphisms of X under composition of morphisms. For example, if X is a finite-dimensional vector space, then the automorphism group of X is the group of invertible linear transformations from X to itself (the general linear group of X). If instead X is a group, then its automorphism group ${\displaystyle \operatorname {Aut} (X)}$ is the group consisting of all group automorphisms of X.

Especially in geometric contexts, an automorphism group is also called a symmetry group. A subgroup of an automorphism group is sometimes called a transformation group.

In algebraic geometry, a semistable abelian variety is an abelian variety defined over a global or local field, which is characterized by how it reduces at the primes of the field.

For an abelian variety ${\displaystyle A}$ defined over a field ${\displaystyle F}$ with ring of integers ${\displaystyle R}$, consider the Néron model of ${\displaystyle A}$, which is a 'best possible' model of ${\displaystyle A}$ defined over ${\displaystyle R}$. This model may be represented as a scheme over ${\displaystyle \mathrm {Spec} (R)}$ (cf. spectrum of a ring) for which the generic fibre constructed by means of the morphism${\displaystyle \mathrm {Spec} (F)\to \mathrm {Spec} (R)}$gives back ${\displaystyle A}$. The Néron model is a smooth group scheme, so we can consider ${\displaystyle A^{0}}$, the connected component of the Néron model which contains the identity for the group law. This is an open subgroup scheme of the Néron model. For a residue field ${\displaystyle k}$, ${\displaystyle A_{k}^{0}}$ is a group variety over ${\displaystyle k}$, hence an extension of an abelian variety by a linear group. If this linear group is an algebraic torus, so that ${\displaystyle A_{k}^{0}}$ is a semiabelian variety, then ${\displaystyle A}$ has semistable reduction at the prime corresponding to ${\displaystyle k}$. If ${\displaystyle F}$ is a global field, then ${\displaystyle A}$ is semistable if it has good or semistable reduction at all primes.

In the mathematical field of category theory, the category of sets, denoted by Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are the functions from A to B, and the composition of morphisms is the composition of functions.

Many other categories (such as the category of groups, with group homomorphisms as arrows) add structure to the objects of the category of sets or restrict the arrows to functions of a particular kind (or both).

In category theory, a branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X.

The dual notion is that of a terminal object (also called terminal element): T is terminal if for every object X in C there exists exactly one morphism X → T. Initial objects are also called coterminal or universal, and terminal objects are also called final.

In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) 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

Usually the morphisms f and g are omitted from the notation, and then the pullback is written

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 C. 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 C. (C is composed by reversing every morphism of C.) Duality, as such, is the assertion that truth is invariant under this operation on statements. In other words, if a statement S is true about C, then its dual statement is true about C. Also, if a statement is false about C, then its dual has to be false about C. (Compactly saying, S for C is true if and only if its dual for C is true.)

Given a concrete category C, it is often the case that the opposite category C per se is abstract. C need not be a category that arises from mathematical practice. In this case, another category D is also termed to be in duality with C if D and C are equivalent as categories.

In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is the category-theoretic dual notion to the categorical product, which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products within a given category.

In category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain. The pushout consists of an object P along with two morphisms X → P and Y → P that complete a commutative square with the two given morphisms f and g. In fact, the defining universal property of the pushout (given below) essentially says that the pushout is the "most general" way to complete this commutative square. Common notations for the pushout are ${\displaystyle P=X\sqcup _{Z}Y}$ and ${\displaystyle P=X+_{Z}Y}$.

The pushout is the categorical dual of the pullback.