Task: Question Answering
Input (English): Context: Given a category C and a morphism f \ colon X \ to Y in C , the image of f is a monomorphism h \ colon I \ to Y satisfying the following universal property : There exists a morphism g \ colon X \ to I such that f = hg . For any object Z with a morphism k \ colon X \ to Z and a monomorphism l \ colon Z \ to Y such that f = lk , there exists a unique morphism m \ colon I \ to Z such that h = lm. Remarks : such a factorization does not necessarily exist g is unique by definition of monic ( = left invertible , abstraction of injectivity ) m is monic . h = lm already implies that m is unique . k = mg The image of f is often denoted by im f or Im ( f ) . One can show that a morphism f is epic if and only if f = im f., 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 formula_1., 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: " (homos)" meaning "same" and " (morphe)" meaning "form" or "shape"., Category theory formalizes mathematical structure and its concepts in terms of a collection of "objects" and of "arrows" (also called morphisms). A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, and groups., Mathematics (from Greek  "máthma", knowledge, study, learning) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics., In various branches of mathematics, a useful construction is often viewed as the most efficient solution to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly., In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. The term "abstract algebra" was coined in the early 20th century to distinguish this area of study from the other parts of algebra., Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular groups as the object of study, in universal algebra one takes "the theory of groups" as an object of study., Subject: image , Relation: part_of, Options: (A) 20th century (B) abstract algebra (C) algebra (D) category theory (E) division (F) learning (G) mathematics (H) theory (I) universal algebra (J) word
Output (English): category theory