In mathematics, specifically category theory, 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 arrows.

Let *C* be a category. A **subcategory***S* of *C* is given by

- a subcollection of objects of
*C*, denoted ob(*S*), - a subcollection of morphisms of
*C*, denoted hom(*S*).

such that

- for every
*X*in ob(*S*), the identity morphism id_{X}is in hom(*S*), - for every morphism
*f*:*X*→*Y*in hom(*S*), both the source*X*and the target*Y*are in ob(*S*), - for every pair of morphisms
*f*and*g*in hom(*S*) the composite*f*o*g*is in hom(*S*) whenever it is defined.

These conditions ensure that *S* is a category in its own right: its collection of objects is ob(*S*), its collection of morphisms is hom(*S*), and its identities and composition are as in *C*. There is an obvious faithful functor *I* : *S* → *C*, called the **inclusion functor** which takes objects and morphisms to themselves.

Let *S* be a subcategory of a category *C*. We say that *S* is a **full subcategory of***C* if for each pair of objects *X* and *Y* of *S*,

A full subcategory is one that includes *all* morphisms in *C* between objects of *S*. For any collection of objects *A* in *C*, there is a unique full subcategory of *C* whose objects are those in *A*.

- The category of finite sets forms a full subcategory of the category of sets.
- The category whose objects are sets and whose morphisms are bijections forms a non-full subcategory of the category of sets.
- The category of abelian groups forms a full subcategory of the category of groups.
- The category of rings (whose morphisms are unit-preserving ring homomorphisms) forms a non-full subcategory of the category of rngs.
- For a field
*K*, the category of*K*-vector spaces forms a full subcategory of the category of (left or right)*K*-modules.

Given a subcategory *S* of *C*, the inclusion functor *I* : *S* → *C* is both a faithful functor and injective on objects. It is full if and only if *S* is a full subcategory.

Some authors define an **embedding** to be a full and faithful functor. Such a functor is necessarily injective on objects up to isomorphism. For instance, the Yoneda embedding is an embedding in this sense.

Some authors define an **embedding** to be a full and faithful functor that is injective on objects.^{ [1] }

Other authors define a functor to be an **embedding** if it is faithful and injective on objects. Equivalently, *F* is an embedding if it is injective on morphisms. A functor *F* is then called a **full embedding** if it is a full functor and an embedding.

With the definitions of the previous paragraph, for any (full) embedding *F* : *B* → *C* the image of *F* is a (full) subcategory *S* of *C*, and *F* induces an isomorphism of categories between *B* and *S*. If *F* is not injective on objects then the image of *F* is equivalent to *B*.

In some categories, one can also speak of morphisms of the category being embeddings.

A subcategory *S* of *C* is said to be ** isomorphism-closed ** or **replete** if every isomorphism *k* : *X* → *Y* in *C* such that *Y* is in *S* also belongs to *S*. An isomorphism-closed full subcategory is said to be **strictly full**.

A subcategory of *C* is **wide** or **lluf** (a term first posed by Peter Freyd ^{ [2] }) if it contains all the objects of *C*.^{ [3] } A wide subcategory is typically not full: the only wide full subcategory of a category is that category itself.

A **Serre subcategory** is a non-empty full subcategory *S* of an abelian category *C* such that for all short exact sequences

in *C*, *M* belongs to *S* if and only if both and do. This notion arises from Serre's C-theory.

Look up in Wiktionary, the free dictionary. subcategory |

- Reflective subcategory
- Exact category, a full subcategory closed under extensions.

In mathematics, especially in category theory and homotopy theory, a **groupoid** generalises the notion of group in several equivalent ways. A groupoid can be seen as a:

In category theory, a branch of mathematics, a **Grothendieck topology** is a structure on a category *C* that makes the objects of *C* act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a **site**.

In mathematics, the **Yoneda lemma** is arguably the most important 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. The motivating prototypical example of an abelian category is the category of abelian groups, **Ab**. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. Abelian categories are very *stable* categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory. Abelian categories are named after Niels Henrik Abel.

In mathematics, specifically category theory, **adjunction** is a relationship that two functors may have. Two functors that stand in this relationship are known as **adjoint functors**, one being the **left adjoint** and the other the **right adjoint**. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems, such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology.

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*,

In mathematics, a **concrete category** is a category that is equipped with a faithful functor to the category of sets. This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions. Many important categories have obvious interpretations as concrete categories, for example the category of topological spaces and the category of groups, and trivially also the category of sets itself. On the other hand, the homotopy category of topological spaces is not **concretizable**, i.e. it does not admit a faithful functor to the category of sets.

**Mitchell's embedding theorem**, also known as the **Freyd–Mitchell theorem** or the **full embedding theorem**, is a result about abelian categories; it essentially states that these categories, while rather abstractly defined, are in fact concrete categories of modules. This allows one to use element-wise diagram chasing proofs in these categories. The theorem is named after Barry Mitchell and Peter Freyd.

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 mathematics, in the area of category theory, a **forgetful functor** 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given signature, this may be expressed by curtailing the signature: the new signature is an edited form of the old one. If the signature is left as an empty list, the functor is simply to take the **underlying set** of a structure. Because many structures in mathematics consist of a set with an additional added structure, a forgetful functor that maps to the underlying set is the most common case.

In category theory, a **faithful functor** is a functor that is injective when restricted to each set of morphisms that have a given source and target.

In mathematics, the **derived category***D*(*A*) of an abelian category *A* is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on *A*. The construction proceeds on the basis that the objects of *D*(*A*) should be chain complexes in *A*, with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described by complicated spectral sequences.

In mathematics, a **triangulated category** is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy category. The exact triangles generalize the short exact sequences in an abelian category, as well as fiber sequences and cofiber sequences in topology.

In the branch of mathematics called homological algebra, a ** t-structure** is a way to axiomatize the properties of an abelian subcategory of a derived category. A

**Fibred categories** are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which *inverse images* of objects such as vector bundles can be defined. As an example, for each topological space there is the category of vector bundles on the space, and for every continuous map from a topological space *X* to another topological space *Y* is associated the pullback functor taking bundles on *Y* to bundles on *X*. Fibred categories formalise the system consisting of these categories and inverse image functors. Similar setups appear in various guises in mathematics, in particular in algebraic geometry, which is the context in which fibred categories originally appeared. Fibered categories are used to define stacks, which are fibered categories with "descent". Fibrations also play an important role in categorical semantics of type theory, and in particular that of dependent type theories.

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

In mathematics, specifically in category theory, hom-sets, i.e. sets of morphisms between objects, give rise to important functors to the category of sets. These functors are called **hom-functors** and have numerous applications in category theory and other branches of mathematics.

In category theory, the notion of a **projective object** generalizes the notion of a projective module. Projective objects in abelian categories are used in homological algebra. The dual notion of a projective object is that of an injective object.

In mathematics, a **quotient category** is a category obtained from another one by identifying sets of morphisms. Formally, it is a quotient object in the category of categories, analogous to a quotient group or quotient space, but in the categorical setting.

In mathematics, a **Grothendieck category** is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957 in order to develop the machinery of homological algebra for modules and for sheaves in a unified manner. The theory of these categories was further developed in Pierre Gabriel's seminal thesis in 1962.

- ↑ Jaap van Oosten. "Basic category theory" (PDF).
- ↑ Freyd, Peter (1991). "Algebraically complete categories".
*Proceedings of the International Conference on Category Theory, Como, Italy (CT 1990)*. Lecture Notes in Mathematics.**1488**. Springer. pp. 95–104. doi:10.1007/BFb0084215. ISBN 978-3-540-54706-8.CS1 maint: discouraged parameter (link) - ↑ Wide subcategory in
*nLab*

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