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,^{ [1] } 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.

The dual concept to a subobject is a **quotient object**. This generalizes concepts such as quotient sets, quotient groups, quotient spaces, quotient graphs, etc.

In detail, let * be an object of some category. Given two monomorphisms *

with codomain *, we write if factors through **—that is, if there exists such that . The binary relation defined by*

is an equivalence relation on the monomorphisms with codomain *, and the corresponding equivalence classes of these monomorphisms are the ***subobjects** of *. (Equivalently, one can define the equivalence relation by if and only if there exists an isomorphism with .)*

The relation ≤ induces a partial order on the collection of subobjects of .

The collection of subobjects of an object may in fact be a proper class; this means that the discussion given is somewhat loose. If the subobject-collection of every object is a set, the category is called *well-powered* or sometimes *locally small*.

To get the dual concept of **quotient object**, replace "monomorphism" by "epimorphism" above and reverse arrows. A quotient object of *A* is then an equivalence class of epimorphisms with domain *A.*

- In
**Set**, the category of sets, a subobject of*A*corresponds to a subset*B*of*A*, or rather the collection of all maps from sets equipotent to*B*with image exactly*B*. The subobject partial order of a set in**Set**is just its subset lattice. - In
**Grp**, the category of groups, the subobjects of*A*correspond to the subgroups of*A*. - Given a partially ordered class
**P**= (*P*, ≤), we can form a category with the elements of*P*as objects, and a single arrow from*p*to*q*iff*p*≤*q*. If**P**has a greatest element, the subobject partial order of this greatest element will be**P**itself. This is in part because all arrows in such a category will be monomorphisms. - A subobject of a terminal object is called a subterminal object.

- ↑ Mac Lane, p. 126

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 (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).

In mathematics, an **isomorphism** is a 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 isomorphism is derived from the Ancient Greek: ἴσος *isos* "equal", and μορφή *morphe* "form" or "shape".

In mathematics, specifically abstract algebra, the **isomorphism theorems** are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.

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

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 *X* ↪ *Y*.

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 *g*_{1}, *g*_{2}: *Y* → *Z*,

An **exact sequence** is a concept in mathematics, especially in group theory, ring and module theory, homological algebra, as well as in differential geometry. An exact sequence is a sequence, either finite or infinite, of objects and morphisms between them such that the image of one morphism equals the kernel of the next.

In mathematics, a **direct limit** is a way to construct a object from many 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 between those smaller objects. The direct limit of the objects , where ranges over some directed set , is denoted by .

The **ultraproduct** is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature. The **ultrapower** is the special case of this construction in which all factors are equal.

In mathematics, 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 **subobject classifier** is a special object Ω of a category such that, intuitively, the subobjects of any object *X* in the category correspond to the morphisms from *X* to Ω. In typical examples, that morphism assigns "true" to the elements of the subobject and "false" to the other elements of *X.* Therefore, a subobject classifier is also known as a "truth value object" and the concept is widely used in the categorical description of logic. Note however that subobject classifiers are often much more complicated than the simple binary logic truth values {true, false}.

In category theory, a **coequalizer** is a generalization of a quotient by an equivalence relation to objects in an arbitrary category. It is the categorical construction dual to the equalizer.

In mathematics, the **category of topological spaces**, often denoted **Top**, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. The study of **Top** and of properties of topological spaces using the techniques of category theory is known as **categorical topology**.

In set theory, a **prewellordering** is a binary relation that is transitive, connex, and wellfounded. In other words, if is a prewellordering on a set , and if we define by

This is a glossary of properties and concepts in category theory in mathematics.

In mathematics, particularly in homotopy theory, a **model category** is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category of topological spaces or of chain complexes, via the acyclic model theorem. The concept was introduced by Daniel G. Quillen (1967).

**Constructive set theory** is an approach to mathematical constructivism following the program of axiomatic set theory. The same first-order language with and of classical set theory is usually used, so this is not to be confused with a constructive types approach. On the other hand, some constructive theories are indeed motivated by their interpretability in type theories.

In category theory, a **regular category** is a category with finite limits and coequalizers of a pair of morphisms called **kernel pairs**, satisfying certain *exactness* conditions. In that way, regular categories recapture many properties of abelian categories, like the existence of *images*, without requiring additivity. At the same time, regular categories provide a foundation for the study of a fragment of first-order logic, known as regular logic.

In computer science, a **trace** is a set of strings, wherein certain letters in the string are allowed to commute, but others are not. It generalizes the concept of a string, by not forcing the letters to always be in a fixed order, but allowing certain reshufflings to take place. Traces were introduced by Pierre Cartier and Dominique Foata in 1969 to give a combinatorial proof of MacMahon's Master theorem. Traces are used in theories of concurrent computation, where commuting letters stand for portions of a job that can execute independently of one another, while non-commuting letters stand for locks, synchronization points or thread joins.

In Category theory and related fields of mathematics, an **envelope** is a construction that generalizes the operations of "exterior completion", like completion of a locally convex space, or Stone–Čech compactification of a topological space. A dual construction is called refinement.

- Mac Lane, Saunders (1998),
*Categories for the Working Mathematician*, Graduate Texts in Mathematics,**5**(2nd ed.), New York, NY: Springer-Verlag, ISBN 0-387-98403-8, Zbl 0906.18001 - Pedicchio, Maria Cristina; Tholen, Walter, eds. (2004).
*Categorical foundations. Special topics in order, topology, algebra, and sheaf theory*. Encyclopedia of Mathematics and Its Applications.**97**. Cambridge: Cambridge University Press. ISBN 0-521-83414-7. Zbl 1034.18001.

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