# Glossary of category theory

This is a glossary of properties and concepts in category theory in mathematics.[1]

Especially for higher categories, the concepts from algebraic topology are also used in the category theory. For that see also glossary of algebraic topology.

The notations used throughout the article are:

• [n] = { 0, 1, 2, …, n }, which is viewed as a category (by writing .)
• Cat, the category of (small) categories, where the objects are categories (which are small with respect to some universe) and the morphisms functors.
• Fct(C, D), the functor category: the category of functors from a category C to a category D.
• Set, the category of (small) sets.
• sSet, the category of simplicial sets.
Contents :

## A

abelian
A category is abelian if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal.
accessible
1.  Given a cardinal number κ, an object X in a category is κ-accessible (or κ-compact or κ-presentable) if commutes with κ-filtered colimits.
2.  Given a regular cardinal κ, a category is κ-accessible if it has κ-filtered colimits and there exists a small set S of κ-compact objects that generates the category under colimits, meaning every object can be written as a colimit of diagrams of objects in S.
An adjunction (also called an adjoint pair) is a pair of functors F: CD, G: DC such that there is a "natural" bijection
;
F is said to be left adjoint to G and G to right adjoint to F. Here, "natural" means there is a natural isomorphism of bifunctors (which are contravariant in the first variable.)
amnestic
A functor is amnestic if it has the property: if k is an isomorphism and F(k) is an identity, then k is an identity.

## B

balanced
A category is balanced if every bimorphism is an isomorphism.
bifunctor
A bifunctor from a pair of categories C and D to a category E is a functor C × DE. For example, for any category C, is a bifunctor from Cop and C to Set.
bimorphism
A bimorphism is a morphism that is both an epimorphism and a monomorphism.
Bousfield localization
See Bousfield localization.

## C

calculus of functors
The calculus of functors is a technique of studying functors in the manner similar to the way a function is studied via its Taylor series expansion; whence, the term "calculus".
cartesian closed
A category is cartesian closed if it has a terminal object and that any two objects have a product and exponential.
cartesian morphism
1.  Given a functor π: CD (e.g., a prestack over schemes), a morphism f: xy in C is π-cartesian if, for each object z in C, each morphism g: zy in C and each morphism v: π(z) → π(x) in D such that π(g) = π(f) ∘ v, there exists a unique morphism u: zx such that π(u) = v and g = fu.
2.  Given a functor π: CD (e.g., a prestack over rings), a morphism f: xy in C is π-coCartesian if, for each object z in C, each morphism g: xz in C and each morphism v: π(y) → π(z) in D such that π(g) = v ∘ π(f), there exists a unique morphism u: yz such that π(u) = v and g = uf. (In short, f is the dual of a π-cartesian morphism.)
Cartesian square
A commutative diagram that is isomorphic to the diagram given as a fiber product.
categorification
The term "categorification" is an informal term referring to a process of replacing sets and set-theoretic concepts with categories and category-theoretic concepts in some nontrivial way to capture categoric flavors. Decategorification is the reverse of categorification.
category
A category consists of the following data
1. A class of objects,
2. For each pair of objects X, Y, a set , whose elements are called morphisms from X to Y,
3. For each triple of objects X, Y, Z, a map (called composition)
,
4. For each object X, an identity morphism

subject to the conditions: for any morphisms , and ,

• and .
For example, a partially ordered set can be viewed as a category: the objects are the elements of the set and for each pair of objects x, y, there is a unique morphism if and only if ; the associativity of composition means transitivity.
category of categories
The category of (small) categories, denoted by Cat, is a category where the objects are all the categories which are small with respect to some fixed universe and the morphisms are all the functors.
classifying space
The classifying space of a category C is the geometric realization of the nerve of C.
co-
Often used synonymous with op-; for example, a colimit refers to an op-limit in the sense that it is a limit in the opposite category. But there might be a distinction; for example, an op-fibration is not the same thing as a cofibration.
coend
The coend of a functor is the dual of the end of F and is denoted by
.

For example, if R is a ring, M a right R-module and N a left R-module, then the tensor product of M and N is

where R is viewed as a category with one object whose morphisms are the elements of R.
coequalizer
The coequalizer of a pair of morphisms is the colimit of the pair. It is the dual of an equalizer.
coimage
The coimage of a morphism f: XY is the coequalizer of .
comma
Given functors , the comma category is a category where (1) the objects are morphisms and (2) a morphism from to consists of and such that is For example, if f is the identity functor and g is the constant functor with a value b, then it is the slice category of B over an object b.
compact
Probably synonymous with #accessible.
complete
A category is complete if all small limits exist.
concrete
A concrete category C is a category such that there is a faithful functor from C to Set; e.g., Vec, Grp and Top.
cone
A cone is a way to express the universal property of a colimit (or dually a limit). One can show[2] that the colimit is the left adjoint to the diagonal functor , which sends an object X to the constant functor with value X; that is, for any X and any functor ,
provided the colimit in question exists. The right-hand side is then the set of cones with vertex X.[3]
connected
A category is connected if, for each pair of objects x, y, there exists a finite sequence of objects zi such that and either or is nonempty for any i.
conservative functor
A conservative functor is a functor that reflects isomorphisms. Many forgetful functors are conservative, but the forgetful functor from Top to Set is not conservative.
constant
A functor is constant if it maps every object in a category to the same object A and every morphism to the identity on A. Put in another way, a functor is constant if it factors as: for some object A in D, where i is the inclusion of the discrete category { A }.
contravariant functor
A contravariant functor F from a category C to a category D is a (covariant) functor from Cop to D. It is sometimes also called a presheaf especially when D is Set or the variants. For example, for each set S, let be the power set of S and for each function , define
by sending a subset A of T to the pre-image . With this, is a contravariant functor.
coproduct
The coproduct of a family of objects Xi in a category C indexed by a set I is the inductive limit of the functor , where I is viewed as a discrete category. It is the dual of the product of the family. For example, a coproduct in Grp is a free product.

## D

Day convolution
Given a group or monoid M, the Day convolution is the tensor product in .[4]
diagonal functor
Given categories I, C, the diagonal functor is the functor
that sends each object A to the constant functor with value A and each morphism to the natural transformation that is f at each i.
A differential graded category is a category whose Hom sets are equipped with structures of differential graded modules. In particular, if the category has only one object, it is the same as a differential graded module.
discrete
A category is discrete if each morphism is an identity morphism (of some object). For example, a set can be viewed as a discrete category.

## E

end
The end of a functor is the limit

where is the category (called the subdivision category of C) whose objects are symbols for all objects c and all morphisms u in C and whose morphisms are and if and where is induced by F so that would go to and would go to . For example, for functors ,

is the set of natural transformations from F to G. For more examples, see this mathoverflow thread. The dual of an end is a coend.
enriched category
Given a monoidal category (C, ⊗, 1), a category enriched over C is, informally, a category whose Hom sets are in C. More precisely, a category D enriched over C is a data consisting of
1. A class of objects,
2. For each pair of objects X, Y in D, an object in C, called the mapping object from X to Y,
3. For each triple of objects X, Y, Z in D, a morphism in C,
,
called the composition,
4. For each object X in D, a morphism in C, called the unit morphism of X

subject to the conditions that (roughly) the compositions are associative and the unit morphisms act as the multiplicative identity.

For example, a category enriched over sets is an ordinary category.
empty
The empty category is a category with no object. It is the same thing as the empty set when the empty set is viewed as a discrete category.
epimorphism
A morphism f is an epimorphism if whenever . In other words, f is the dual of a monomorphism.
equalizer
The equalizer of a pair of morphisms is the limit of the pair. It is the dual of a coequalizer.
equivalence
1.  A functor is an equivalence if it is faithful, full and essentially surjective.
2.  A morphism in an ∞-category C is an equivalence if it gives an isomorphism in the homotopy category of C.
equivalent
A category is equivalent to another category if there is an equivalence between them.
essentially surjective
A functor F is called essentially surjective (or isomorphism-dense) if for every object B there exists an object A such that F(A) is isomorphic to B.

## F

faithful
A functor is faithful if it is injective when restricted to each hom-set.
fibered category
A functor π: CD is said to exhibit C as a category fibered over D if, for each morphism g: x → π(y) in D, there exists a π-cartesian morphism f: x'y in C such that π(f) = g. If D is the category of affine schemes (say of finite type over some field), then π is more commonly called a prestack. Note: π is often a forgetful functor and in fact the Grothendieck construction implies that every fibered category can be taken to be that form (up to equivalences in a suitable sense).
fiber product
Given a category C and a set I, the fiber product over an object S of a family of objects Xi in C indexed by I is the product of the family in the slice category of C over S (provided there are ). The fiber product of two objects X and Y over an object S is denoted by and is also called a Cartesian square.
filtered
1.  A filtered category (also called a filtrant category) is a nonempty category with the properties (1) given objects i and j, there are an object k and morphisms ik and jk and (2) given morphisms u, v: ij, there are an object k and a morphism w: jk such that wu = wv. A category I is filtered if and only if, for each finite category J and functor f: JI, the set is nonempty for some object i in I.
2.  Given a cardinal number π, a category is said to be π-filtrant if, for each category J whose set of morphisms has cardinal number strictly less than π, the set is nonempty for some object i in I.
finite
A category is finite if it has only finitely many morphisms.
forgetful functor
The forgetful functor is, roughly, a functor that loses some of data of the objects; for example, the functor that sends a group to its underlying set and a group homomorphism to itself is a forgetful functor.
Fukaya category
See Fukaya category.
full
1.  A functor is full if it is surjective when restricted to each hom-set.
2.  A category A is a full subcategory of a category B if the inclusion functor from A to B is full.
functor
Given categories C, D, a functor F from C to D is a structure-preserving map from C to D; i.e., it consists of an object F(x) in D for each object x in C and a morphism F(f) in D for each morphism f in C satisfying the conditions: (1) whenever is defined and (2) . For example,
,
where is the power set of S is a functor if we define: for each function , by .
functor category
The functor category Fct(C, D) from a category C to a category D is the category where the objects are all the functors from C to D and the morphisms are all the natural transformations between the functors.

## G

Gabriel–Popescu theorem
The Gabriel–Popescu theorem says an abelian category is a quotient of the category of modules.
generator
In a category C, a family of objects is a system of generators of C if the functor is conservative. Its dual is called a system of cogenerators.
Grothendieck category
A Grothendieck category is a certain well-behaved kind of an abelian category.
Grothendieck construction
Given a functor , let DU be the category where the objects are pairs (x, u) consisting of an object x in C and an object u in the category U(x) and a morphism from (x, u) to (y, v) is a pair consisting of a morphism f: xy in C and a morphism U(f)(u) → v in U(y). The passage from U to DU is then called the Grothendieck construction.
Grothendieck fibration
A fibered category.
groupoid
1.  A category is called a groupoid if every morphism in it is an isomorphism.
2.  An ∞-category is called an ∞-groupoid if every morphism in it is an equivalence (or equivalently if it is a Kan complex.)

## H

Hall algebra of a category
See Ringel–Hall algebra.
heart
The heart of a t-structure (, ) on a triangulated category is the intersection . It is an abelian category.
homological dimension
The homological dimension of an abelian category with enough injectives is the least non-negative intege n such that every object in the category admits an injective resolution of length at most n. The dimension is ∞ if no such integer exists. For example, the homological dimension of ModR with a principal ideal domain R is at most one.
homotopy category
See homotopy category. It is closely related to a localization of a category.

## I

identity
1.  The identity morphism f of an object A is a morphism from A to A such that for any morphisms g with domain A and h with codomain A, and .
2.  The identity functor on a category C is a functor from C to C that sends objects and morphisms to themselves.
3.  Given a functor F: CD, the identity natural transformation from F to F is a natural transformation consisting of the identity morphisms of F(X) in D for the objects X in C.
image
The image of a morphism f: XY is the equalizer of .
ind-limit
A colimit (or inductive limit) in .
∞-category
An ∞-category C is a simplicial set satisfying the following condition: for each 0 < i < n,
• every map of simplicial sets extends to an n-simplex
where Δn is the standard n-simplex and is obtained from Δn by removing the i-th face and the interior (see Kan fibration#Definition). For example, the nerve of a category satisfies the condition and thus can be considered as an ∞-category.
initial
1.  An object A is initial if there is exactly one morphism from A to each object; e.g., empty set in Set.
2.  An object A in an ∞-category C is initial if is contractible for each object B in C.
injective
An object A in an abelian category is injective if the functor is exact. It is the dual of a projective object.
internal Hom
Given a monoidal category (C, ⊗), the internal Hom is a functor such that is the right adjoint to for each object Y in C. For example, the category of modules over a commutative ring R has the internal Hom given as , the set of R-linear maps.
inverse
A morphism f is an inverse to a morphism g if is defined and is equal to the identity morphism on the codomain of g, and is defined and equal to the identity morphism on the domain of g. The inverse of g is unique and is denoted by g−1. f is a left inverse to g if is defined and is equal to the identity morphism on the domain of g, and similarly for a right inverse.
isomorphic
1.  An object is isomorphic to another object if there is an isomorphism between them.
2.  A category is isomorphic to another category if there is an isomorphism between them.
isomorphism
A morphism f is an isomorphism if there exists an inverse of f.

## L

length
An object in an abelian category is said to have finite length if it has a composition series. The maximum number of proper subobjects in any such composition series is called the length of A.[5]
limit
1.  The limit (or projective limit) of a functor is
2.  The limit of a functor is an object, if any, in C that satisfies: for any object X in C, ; i.e., it is an object representing the functor
3.  The colimit (or inductive limit) is the dual of a limit; i.e., given a functor , it satisfies: for any X, . Explicitly, to give is to give a family of morphisms such that for any , is . Perhaps the simplest example of a colimit is a coequalizer. For another example, take f to be the identity functor on C and suppose exists; then the identity morphism on L corresponds to a compatible family of morphisms such that is the identity. If is any morphism, then ; i.e., L is a final object of C.
localization of a category
See localization of a category.

## M

monomorphism
A morphism f is a monomorphism (also called monic) if whenever ; e.g., an injection in Set. In other words, f is the dual of an epimorphism.

## N

natural
1.  A natural transformation is, roughly, a map between functors. Precisely, given a pair of functors F, G from a category C to category D, a natural transformation φ from F to G is a set of morphisms in D
satisfying the condition: for each morphism f: xy in C, . For example, writing for the group of invertible n-by-n matrices with coefficients in a commutative ring R, we can view as a functor from the category CRing of commutative rings to the category Grp of groups. Similarly, is a functor from CRing to Grp. Then the determinant det is a natural transformation from to -*.
2.  A natural isomorphism is a natural transformation that is an isomorphism (i.e., admits the inverse).
The composition is encoded as a 2-simplex.
nerve
The nerve functor N is the functor from Cat to sSet given by .
normal
A category is normal if every monic is normal.

## O

object
1.  An object is part of a data defining a category.
2.  An [adjective] object in a category C is a contravariant functor (or presheaf) from some fixed category corresponding to the "adjective" to C. For example, a simplicial object in C is a contravariant functor from the simplicial category to C and a Γ-object is a pointed contravariant functor from Γ (roughly the pointed category of pointed finite sets) to C provided C is pointed.
op-fibration
A functor π:CD is an op-fibration if, for each object x in C and each morphism g : π(x) → y in D, there is at least one π-coCartesian morphism f: xy' in C such that π(f) = g. In other words, π is the dual of a Grothendieck fibration.
opposite
The opposite category of a category is obtained by reversing the arrows. For example, if a partially ordered set is viewed as a category, taking its opposite amounts to reversing the ordering.

## P

perfect
Sometimes synonymous with "compact". See perfect complex.
pointed
A category (or ∞-category) is called pointed if it has a zero object.
polynomial
A functor from the category of finite-dimensional vector spaces to itself is called a polynomial functor if, for each pair of vector spaces V, W, F: Hom(V, W) → Hom(F(V), F(W)) is a polynomial map between the vector spaces. A Schur functor is a basic example.
A category is preadditive if it is enriched over the monoidal category of abelian groups. More generally, it is R-linear if it is enriched over the monoidal category of R-modules, for R a commutative ring.
presentable
Given a regular cardinal κ, a category is κ-presentable if it admits all small colimits and is κ-accessible. A category is presentable if it is κ-presentable for some regular cardinal κ (hence presentable for any larger cardinal). Note: Some authors call a presentable category a locally presentable category.
presheaf
Another term for a contravariant functor: a functor from a category Cop to Set is a presheaf of sets on C and a functor from Cop to sSet is a presheaf of simplicial sets or simplicial presheaf, etc. A topology on C, if any, tells which presheaf is a sheaf (with respect to that topology).
product
1.  The product of a family of objects Xi in a category C indexed by a set I is the projective limit of the functor , where I is viewed as a discrete category. It is denoted by and is the dual of the coproduct of the family.
2.  The product of a family of categories Ci's indexed by a set I is the category denoted by whose class of objects is the product of the classes of objects of Ci's and whose hom-sets are ; the morphisms are composed component-wise. It is the dual of the disjoint union.
projective
An object A in an abelian category is projective if the functor is exact. It is the dual of an injective object.

## Q

Quillen
Quillen’s theorem A provides a criterion for a functor to be a weak equivalence.

## R

reflect
1.  A functor is said to reflect identities if it has the property: if F(k) is an identity then k is an identity as well.
2.  A functor is said to reflect isomorphismsif it has the property: F(k) is an isomorphism then k is an isomorphism as well.
representable
A set-valued contravariant functor F on a category C is said to be representable if it belongs to the essential image of the Yoneda embedding ; i.e., for some object Z. The object Z is said to be the representing object of F.
retraction
f is a retraction of g. g is a section of f.
A morphism is a retraction if it has a right inverse.

## S

section
A morphism is a section if it has a left inverse. For example, the axiom of choice says that any surjective function admits a section.
Segal space
Segal spaces were certain simplicial spaces, introduced as models for (∞, 1)-categories.
semisimple
An abelian category is semisimple if every short exact sequence splits. For example, a ring is semisimple if and only if the category of modules over it is semisimple.
simple
An object in an abelian category is simple if it is not isomorphic to the zero object and any subobject of A is isomorphic to zero or to A.
Simplicial localization
Simplicial localization is a method of localizing a category.
simplicial set
A simplicial set is a contravariant functor from Δ to Set, where Δ is the category whose objects are the sets [n] = { 0, 1, …, n } and whose morphisms are order-preserving functions. A morphism between simplicial sets is a natural transformation.
skeletal
A category is skeletal if isomorphic objects are necessarily identical.
slice
Given a category C and an object A in it, the slice category C/A of C over A is the category whose objects are all the morphisms in C with codomain A, whose morphisms are morphisms in C such that if f is a morphism from to , then in C and whose composition is that of C.
small
1.  A small category is a category in which the class of all morphisms is a set (i.e., not a proper class); otherwise large. A category is locally small if the morphisms between every pair of objects A and B form a set. Some authors assume a foundation in which the collection of all classes forms a "conglomerate", in which case a quasicategory is a category whose objects and morphisms merely form a conglomerate.[6] (NB: some authors use the term "quasicategory" with a different meaning.[7])
2.  An object in a category is said to be small if it is κ-compact for some regular cardinal κ. The notion prominently appears in Quiilen's small object argument (cf. https://ncatlab.org/nlab/show/small+object+argument)
stable
An ∞-category is stable if (1) it has a zero object, (2) every morphism in it admits a fiber and a cofiber and (3) a triangle in it is a fiber sequence if and only if it is a cofiber sequence.
strict
A morphism f in a category admitting finite limits and finite colimits is strict if the natural morphism is an isomorphism.
strict n-category
A strict 0-category is a set and for any integer n > 0, a strict n-category is a category enriched over strict (n-1)-categories. For example, a strict 1-category is an ordinary category. Note: the term "n-category" typically refers to "weak n-category"; not strict one.
subcanonical
A topology on a category is subcanonical if every representable contravariant functor on C is a sheaf with respect to that topology.[8] Generally speaking, some flat topology may fail to be subcanonical; but flat topologies appearing in practice tend to be subcanonical.
subcategory
A category A is a subcategory of a category B if there is an inclusion functor from A to B.
subobject
See subobject. For example, a subgroup is a subobject of a group.
subquotient
A subquotient is a quotient of a subobject.
subterminal object
A subterminal object is an object X such that every object has at most one morphism into X.
symmetric monoidal category
A symmetric monoidal category is a monoidal category (i.e., a category with ⊗) that has maximally symmetric braiding.

## T

t-structure
A t-structure is an additional structure on a triangulated category (more generally stable ∞-category) that axiomatizes the notions of complexes whose cohomology concentrated in non-negative degrees or non-positive degrees.
tensor category
Usually synonymous with monoidal category (though some authors distinguish between the two concepts.)
tensor product
Given a monoidal category B, the tensor product of functors and is the coend:
.
terminal
1.  An object A is terminal (also called final) if there is exactly one morphism from each object to A; e.g., singletons in Set. It is the dual of an initial object.
2.  An object A in an ∞-category C is terminal if is contractible for every object B in C.
thick subcategory
A full subcategory of an abelian category is thick if it is closed under extensions.
thin
A thin is a category where there is at most one morphism between any pair of objects.
triangulated category
A triangulated category is a category where one can talk about distinguished triangles, generalization of exact sequences. An abelian category is a prototypical example of a triangulated category. A derived category is a triangulated category that is not necessary an abelian category.

## U

universal
1.  Given a functor and an object X in D, a universal morphism from X to f is an initial object in the comma category . (Its dual is also called a universal morphism.) For example, take f to be the forgetful functor and X a set. An initial object of is a function . That it is initial means that if is another morphism, then there is a unique morphism from j to k, which consists of a linear map that extends k via j; that is to say, is the free vector space generated by X.
2.  Stated more explicitly, given f as above, a morphism in D is universal if and only if the natural map
is bijective. In particular, if , then taking c to be uX one gets a universal morphism by sending the identity morphism. In other words, having a universal morphism is equivalent to the representability of the functor .

## W

Waldhausen category
A Waldhausen category is, roughly, a category with families of cofibrations and weak equivalences.
wellpowered
A category is wellpowered if for each object there is only a set of pairwise non-isomorphic subobjects.

## Y

Yoneda lemma
The Yoneda lemma says: For each set-valued contravariant functor F on C and an object X in C, there is a natural bijection
;

in particular, the functor

is fully faithful.[9]

## Z

zero
A zero object is an object that is both initial and terminal, such as a trivial group in Grp.

## Notes

1. Notes on foundations: In many expositions (e.g., Vistoli), the set-theoretic issues are ignored; this means, for instance, that one does not distinguish between small and large categories and that one can arbitrarily form a localization of a category. If one believes in the existence of strongly inaccessible cardinals, then there can be a rigorous theory where statements and constructions have references to Grothendieck universes; this approach is taken, for example, in Lurie's Higher Topos Theory.
2. Kashiwara & Schapira 2006, Ch. 2, Exercise 2.8.
3. Mac Lane 1998, Ch. III, § 3..
4. http://ncatlab.org/nlab/show/Day+convolution
5. Kashiwara & Schapira 2006, exercise 8.20
6. Adámek, Jiří; Herrlich, Horst; Strecker, George E (2004) [1990]. Abstract and Concrete Categories (The Joy of Cats) (PDF). New York: Wiley & Sons. p. 40. ISBN 0-471-60922-6.
7. Joyal, A. (2002). "Quasi-categories and Kan complexes". Journal of Pure and Applied Algebra. 175 (1-3): 207–222. doi:10.1016/S0022-4049(02)00135-4.
8. Vistoli 2004, Definition 2.57.
9. Technical note: the lemma implicitly involves a choice of Set; i.e., a choice of universe.