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

• 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. Like those expositions, this glossary also generally ignores the set-theoretic issues, except when they are relevant (e.g., the discussion on accessibility.)

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 and the conventions used throughout the article are:

• [ n ] = { 0, 1, 2, …, n }, which is viewed as a category (by writing ${displaystyle ito jLeftrightarrow ileq j}$ .)
• 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.
• s Set , the category of simplicial sets .
• "weak" instead of "strict" is given the default status; e.g., " n -category" means "weak n -category", not the strict one, by default.
• By an ∞-category , we mean a quasi-category , the most popular model, unless other models are being discussed.
• The number zero 0 is a natural number.

## 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 ${displaystyle operatorname {Hom} (X,-)}$ 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 .
A category is additive if it is preadditive (to be precise, has some pre-additive structure) and admits all finite coproducts . Although "preadditive" is an additional structure, one can show "additive" is a property of a category; i.e., one can ask whether a given category is additive or not. [7]
An adjunction (also called an adjoint pair) is a pair of functors F : C D , G : D C such that there is a "natural" bijection
${displaystyle operatorname {Hom} _{D}(F(X),Y)simeq operatorname {Hom} _{C}(X,G(Y))}$ ;
F is said to be left adjoint to G and G to right adjoint to F . Here, "natural" means there is a natural isomorphism ${displaystyle operatorname {Hom} _{D}(F(-),-)simeq operatorname {Hom} _{C}(-,G(-))}$ of bifunctors (which are contravariant in the first variable.)
Given a monad T in a category X , an algebra for T or a T -algebra is an object in X with a monoid action of T ("algebra" is misleading and " T -object" is perhaps a better term.) For example, given a group G that determines a monad T in Set in the standard way, a T -algebra is a set with an action of G .
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.
Beck's theorem
Beck's theorem characterizes the category of algebras for a given monad .
bicategory
A bicategory is a model of a weak 2-category .
bifunctor
A bifunctor from a pair of categories C and D to a category E is a functor C × D E . For example, for any category C , ${displaystyle operatorname {Hom} (-,-)}$ is a bifunctor from C op 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 functor
Given relative categories ${displaystyle p:Fto C,q:Gto C}$ over the same base category C , a functor ${displaystyle f:Fto G}$ over C is cartesian if it sends cartesian morphisms to cartesian morphisms.
cartesian morphism
1. Given a functor π: C D (e.g., a prestack over schemes), a morphism f : x y in C is π-cartesian if, for each object z in C , each morphism g : z y 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 : z x such that π( u ) = v and g = f u .
2. Given a functor π: C D (e.g., a prestack over rings), a morphism f : x y in C is π-coCartesian if, for each object z in C , each morphism g : x z 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 : y z such that π( u ) = v and g = u f . (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 ${displaystyle operatorname {Hom} (X,Y)}$ , whose elements are called morphisms from X to Y ,
3. For each triple of objects X , Y , Z , a map (called composition)
${displaystyle circ :operatorname {Hom} (Y,Z)times operatorname {Hom} (X,Y)to operatorname {Hom} (X,Z),,(g,f)mapsto gcirc f}$ ,
4. For each object X , an identity morphism ${displaystyle operatorname {id} _{X}in operatorname {Hom} (X,X)}$
subject to the conditions: for any morphisms ${displaystyle f:Xto Y}$ , ${displaystyle g:Yto Z}$ and ${displaystyle h:Zto W}$ ,
• ${displaystyle (hcirc g)circ f=hcirc (gcirc f)}$ and ${displaystyle operatorname {id} _{Y}circ f=fcirc operatorname {id} _{X}=f}$ .
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 ${displaystyle xto y}$ if and only if ${displaystyle xleq y}$ ; 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 ${displaystyle F:C^{text{op}}times Cto X}$ is the dual of the end of F and is denoted by
${displaystyle int ^{cin C}F(c,c)}$ .
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
${displaystyle Motimes _{R}N=int ^{R}Motimes _{mathbb {Z} }N}$
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 ${displaystyle f,g:Ato B}$ is the colimit of the pair. It is the dual of an equalizer.
coimage
The coimage of a morphism f : X Y is the coequalizer of ${displaystyle Xtimes _{Y}Xrightrightarrows X}$ .
comma
Given functors ${displaystyle f:Cto B,g:Dto B}$ , the comma category ${displaystyle (fdownarrow g)}$ is a category where (1) the objects are morphisms ${displaystyle f(c)to g(d)}$ and (2) a morphism from ${displaystyle alpha :f(c)to g(d)}$ to ${displaystyle beta :f(c')to g(d')}$ consists of ${displaystyle cto c'}$ and ${displaystyle dto d'}$ such that ${displaystyle f(c)to f(c'){overset {beta }{to }}g(d')}$ is ${displaystyle f(c){overset {alpha }{to }}g(d)to g(d').}$ 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 .
A comonad in a category X is a comonid in the monoidal category of endofunctors of X .
compact
Probably synonymous with .
complete
A category is complete if all small limits exist.
composition
1. A composition of morphisms in a category is part of the datum defining the category.
2. If ${displaystyle f:Cto D,,g:Dto E}$ are functors, then the composition ${displaystyle gcirc f}$ or ${displaystyle gf}$ is the functor defined by: for an object x and a morphism u in C , ${displaystyle (gcirc f)(x)=g(f(x)),,(gcirc f)(u)=g(f(u))}$ .
3. Natural transformations are composed pointwise: if ${displaystyle varphi :fto g,,psi :gto h}$ are natural transformations, then ${displaystyle psi circ varphi }$ is the natural transformation given by ${displaystyle (psi circ varphi )_{x}=psi _{x}circ varphi _{x}}$ .
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 that the colimit ${displaystyle varinjlim }$ is the left adjoint to the diagonal functor ${displaystyle Delta :Cto operatorname {Fct} (I,C)}$ , which sends an object X to the constant functor with value X ; that is, for any X and any functor ${displaystyle f:Ito C}$ ,
${displaystyle operatorname {Hom} (varinjlim f,X)simeq operatorname {Hom} (f,Delta _{X}),}$
provided the colimit in question exists. The right-hand side is then the set of cones with vertex X .
connected
A category is connected if, for each pair of objects x , y , there exists a finite sequence of objects z i such that ${displaystyle z_{0}=x,z_{n}=y}$ and either ${displaystyle operatorname {Hom} (z_{i},z_{i+1})}$ or ${displaystyle operatorname {Hom} (z_{i+1},z_{i})}$ 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 ${displaystyle f:Cto D}$ is constant if it factors as: ${displaystyle Cto {A}{overset {i}{to }}D}$ 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 C op to D . It is sometimes also called a presheaf especially when D is Set or the variants. For example, for each set S , let ${displaystyle {mathfrak {P}}(S)}$ be the power set of S and for each function ${displaystyle f:Sto T}$ , define
${displaystyle {mathfrak {P}}(f):{mathfrak {P}}(T)to {mathfrak {P}}(S)}$
by sending a subset A of T to the pre-image ${displaystyle f^{-1}(A)}$ . With this, ${displaystyle {mathfrak {P}}:mathbf {Set} to mathbf {Set} }$ is a contravariant functor.
coproduct
The coproduct of a family of objects X i in a category C indexed by a set I is the inductive limit ${displaystyle varinjlim }$ of the functor ${displaystyle Ito C,,imapsto X_{i}}$ , 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 ${displaystyle mathbf {Fct} (M,mathbf {Set} )}$ . [2]
density theorem
The density theorem states that every presheaf (a set-valued contravariant functor) is a colimit of representable presheaves. Yoneda's lemma embeds a category C into the category of presheaves on C . The density theorem then says the image is "dense", so to say. The name "density" is because of the analogy with the Jacobson density theorem (or other variants) in abstract algebra.
diagonal functor
Given categories I , C , the diagonal functor is the functor
${displaystyle Delta :Cto mathbf {Fct} (I,C),,Amapsto Delta _{A}}$
that sends each object A to the constant functor with value A and each morphism ${displaystyle f:Ato B}$ to the natural transformation ${displaystyle Delta _{f,i}:Delta _{A}(i)=Ato Delta _{B}(i)=B}$ that is f at each i .
diagram
Given a category C , a diagram in C is a functor ${displaystyle f:Ito C}$ from a small category 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.
distributor
Another term for "profunctor".
Dwyer–Kan equivalence
A Dwyer–Kan equivalence is a generalization of an equivalence of categories to the simplicial context. [9]

## E

Eilenberg–Moore category
Another name for the category of algebras for a given monad .
end
The end of a functor ${displaystyle F:C^{text{op}}times Cto X}$ is the limit
${displaystyle int _{cin C}F(c,c)=varprojlim (F^{#}:C^{#}to X)}$
where ${displaystyle C^{#}}$ is the category (called the subdivision category of C ) whose objects are symbols ${displaystyle c^{#},u^{#}}$ for all objects c and all morphisms u in C and whose morphisms are ${displaystyle b^{#}to u^{#}}$ and ${displaystyle u^{#}to c^{#}}$ if ${displaystyle u:bto c}$ and where ${displaystyle F^{#}}$ is induced by F so that ${displaystyle c^{#}}$ would go to ${displaystyle F(c,c)}$ and ${displaystyle u^{#},u:bto c}$ would go to ${displaystyle F(b,c)}$ . For example, for functors ${displaystyle F,G:Cto X}$ ,
${displaystyle int _{cin C}operatorname {Hom} (F(c),G(c))}$
is the set of natural transformations from F to G . For more examples, see . The dual of an end is a coend.
endofunctor
A functor between the same category.
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 ${displaystyle operatorname {Map} _{D}(X,Y)}$ 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 ,
${displaystyle circ :operatorname {Map} _{D}(Y,Z)otimes operatorname {Map} _{D}(X,Y)to operatorname {Map} _{D}(X,Z)}$ ,
called the composition,
4. For each object X in D , a morphism ${displaystyle 1_{X}:1to operatorname {Map} _{D}(X,X)}$ 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 ${displaystyle g=h}$ whenever ${displaystyle gcirc f=hcirc f}$ . In other words, f is the dual of a monomorphism.
equalizer
The equalizer of a pair of morphisms ${displaystyle f,g:Ato B}$ 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 .
evaluation
Given categories C , D and an object A in C , the evaluation at A is the functor
${displaystyle mathbf {Fct} (C,D)to D,,,Fmapsto F(A).}$
For example, the Eilenberg–Steenrod axioms give an instance when the functor is an equivalence.

## F

faithful
A functor is faithful if it is injective when restricted to each hom-set .
fundamental category
The fundamental category functor ${displaystyle tau _{1}:smathbf {Set} to mathbf {Cat} }$ is the left adjoint to the nerve functor N . For every category C , ${displaystyle tau _{1}NC=C}$ .
fundamental groupoid
The fundamental groupoid ${displaystyle Pi _{1}X}$ of a Kan complex X is the category where an object is a 0-simplex (vertex) ${displaystyle Delta ^{0}to X}$ , a morphism is a homotopy class of a 1-simplex (path) ${displaystyle Delta ^{1}to X}$ and a composition is determined by the Kan property.
fibered category
A functor π: C D 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 X i in C indexed by I is the product of the family in the slice category ${displaystyle C_{/S}}$ of C over S (provided there are ${displaystyle X_{i}to S}$ ). The fiber product of two objects X and Y over an object S is denoted by ${displaystyle Xtimes _{S}Y}$ 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 i k and j k and (2) given morphisms u , v : i j , there are an object k and a morphism w : j k such that w u = w v . A category I is filtered if and only if, for each finite category J and functor f : J I , the set ${displaystyle varprojlim operatorname {Hom} (f(j),i)}$ 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 ${displaystyle varprojlim operatorname {Hom} (f(j),i)}$ is nonempty for some object i in I .
A finitary monad or an algebraic monad is a monad on Set whose underlying endofunctor commutes with filtered colimits.
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 ${displaystyle mathbf {Grp} to mathbf {Set} }$ that sends a group to its underlying set and a group homomorphism to itself is a forgetful functor.
free functor
A free functor is a left adjoint to a forgetful functor. For example, for a ring R , the functor that sends a set X to the free R-module generated by X is a free functor (whence the name).
Frobenius category
A Frobenius category is an exact category that has enough injectives and enough projectives and such that the class of injective objects coincides with that of projective objects.
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) ${displaystyle F(fcirc g)=F(f)circ F(g)}$ whenever ${displaystyle fcirc g}$ is defined and (2) ${displaystyle F(operatorname {id} _{x})=operatorname {id} _{F(x)}}$ . For example,
${displaystyle {mathfrak {P}}:mathbf {Set} to mathbf {Set} ,,Smapsto {mathfrak {P}}(S)}$ ,
where ${displaystyle {mathfrak {P}}(S)}$ is the power set of S is a functor if we define: for each function ${displaystyle f:Sto T}$ , ${displaystyle {mathfrak {P}}(f):{mathfrak {P}}(S)to {mathfrak {P}}(T)}$ by ${displaystyle {mathfrak {P}}(f)(A)=f(A)}$ .
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 ${displaystyle G_{i},iin I}$ is a system of generators of C if the functor ${displaystyle Xmapsto prod _{iin I}operatorname {Hom} (G_{i},X)}$ 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 ${displaystyle U:Cto mathbf {Cat} }$ , let D U 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 : x y in C and a morphism U ( f )( u ) → v in U ( y ). The passage from U to D U 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 ( ${displaystyle D^{geq 0}}$ , ${displaystyle D^{leq 0}}$ ) on a triangulated category is the intersection ${displaystyle D^{geq 0}cap D^{leq 0}}$ . It is an abelian category.
Higher category theory
Higher category theory is a subfield of category theory that concerns the study of n-categories and ∞-categories .
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 .
homotopy hypothesis
The homotopy hypothesis states an ∞-groupoid is a space (less equivocally, an n -groupoid can be used as a homotopy n -type.)

## 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 , ${displaystyle gcirc f=g}$ and ${displaystyle fcirc h=h}$ .
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 : C D , 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 : X Y is the equalizer of ${displaystyle Yrightrightarrows Ysqcup _{X}Y}$ .
ind-limit
A colimit (or inductive limit) in ${displaystyle mathbf {Fct} (C^{text{op}},mathbf {Set} )}$ .
∞-category
An ∞-category C is a simplicial set satisfying the following condition: for each 0 < i < n ,
• every map of simplicial sets ${displaystyle f:Lambda _{i}^{n}to C}$ extends to an n -simplex ${displaystyle f:Delta ^{n}to C}$
where Δ n is the standard n -simplex and ${displaystyle Lambda _{i}^{n}}$ is obtained from Δ n 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 ${displaystyle operatorname {Map} _{C}(A,B)}$ is contractible for each object B in C .
injective
An object A in an abelian category is injective if the functor ${displaystyle operatorname {Hom} (-,A)}$ is exact. It is the dual of a projective object.
internal Hom
Given a monoidal category ( C , ⊗), the internal Hom is a functor ${displaystyle [-,-]:C^{text{op}}times Cto C}$ such that ${displaystyle [Y,-]}$ is the right adjoint to ${displaystyle -otimes Y}$ for each object Y in C . For example, the category of modules over a commutative ring R has the internal Hom given as ${displaystyle [M,N]=operatorname {Hom} _{R}(M,N)}$ , the set of R -linear maps.
inverse
A morphism f is an inverse to a morphism g if ${displaystyle gcirc f}$ is defined and is equal to the identity morphism on the codomain of g , and ${displaystyle fcirc g}$ 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 ${displaystyle fcirc g}$ 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 .

## K

Kan complex
A Kan complex is a fibrant object in the category of simplicial sets.
Kan extension
1. Given a category C , the left Kan extension functor along a functor ${displaystyle f:Ito J}$ is the left adjoint (if it exists) to ${displaystyle f^{*}=-circ f:operatorname {Fct} (J,C)to operatorname {Fct} (I,C)}$ and is denoted by ${displaystyle f_{!}}$ . For any ${displaystyle alpha :Ito C}$ , the functor ${displaystyle f_{!}alpha :Jto C}$ is called the left Kan extension of α along f . [10] One can show:
${displaystyle (f_{!}alpha )(j)=varinjlim _{f(i)to j}alpha (i)}$
where the colimit runs over all objects ${displaystyle f(i)to j}$ in the comma category.
2. The right Kan extension functor is the right adjoint (if it exists) to ${displaystyle f^{*}}$ .
Kleisli category
Given a monad T , the Kleisli category of T is the full subcategory of the category of T -algebras (called Eilenberg–Moore category) that consists of free T -algebras.

## L

lax
The term " lax functor " is essentially synonymous with " pseudo-functor ".
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 .
limit
1. The limit (or projective limit ) of a functor ${displaystyle f:I^{text{op}}to mathbf {Set} }$ is
${displaystyle varprojlim _{iin I}f(i)={(x_{i}|i)in prod _{i}f(i)|f(s)(x_{j})=x_{i}{text{ for any }}s:ito j}.}$
2. The limit ${displaystyle varprojlim _{iin I}f(i)}$ of a functor ${displaystyle f:I^{text{op}}to C}$ is an object, if any, in C that satisfies: for any object X in C , ${displaystyle operatorname {Hom} (X,varprojlim _{iin I}f(i))=varprojlim _{iin I}operatorname {Hom} (X,f(i))}$ ; i.e., it is an object representing the functor ${displaystyle Xmapsto varprojlim _{i}operatorname {Hom} (X,f(i)).}$
3. The colimit (or inductive limit ) ${displaystyle varinjlim _{iin I}f(i)}$ is the dual of a limit; i.e., given a functor ${displaystyle f:Ito C}$ , it satisfies: for any X , ${displaystyle operatorname {Hom} (varinjlim f(i),X)=varprojlim operatorname {Hom} (f(i),X)}$ . Explicitly, to give ${displaystyle varinjlim f(i)to X}$ is to give a family of morphisms ${displaystyle f(i)to X}$ such that for any ${displaystyle ito j}$ , ${displaystyle f(i)to X}$ is ${displaystyle f(i)to f(j)to X}$ . Perhaps the simplest example of a colimit is a coequalizer . For another example, take f to be the identity functor on C and suppose ${displaystyle L=varinjlim _{Xin C}f(X)}$ exists; then the identity morphism on L corresponds to a compatible family of morphisms ${displaystyle alpha _{X}:Xto L}$ such that ${displaystyle alpha _{L}}$ is the identity. If ${displaystyle f:Xto L}$ is any morphism, then ${displaystyle f=alpha _{L}circ f=alpha _{X}}$ ; i.e., L is a final object of C .
localization of a category
See localization of a category .

## M

A monad in a category X is a monoid object in the monoidal category of endofunctors of X with the monoidal structure given by composition. For example, given a group G , define an endofunctor T on Set by ${displaystyle T(X)=Gtimes X}$ . Then define the multiplication μ on T as the natural transformation ${displaystyle mu :Tcirc Tto T}$ given by
${displaystyle mu _{X}:Gtimes (Gtimes X)to Gtimes X,,,(g,(h,x))mapsto (gh,x)}$
and also define the identity map η in the analogous fashion. Then ( T , μ , η ) constitutes a monad in Set . More substantially, an adjunction between functors ${displaystyle F:Xrightleftarrows A:G}$ determines a monad in X ; namely, one takes ${displaystyle T=Gcirc F}$ , the identity map η on T to be a unit of the adjunction and also defines μ using the adjunction.
1. An adjunction is said to be monadic if it comes from the monad that it determines by means of the Eilenberg–Moore category (the category of algebras for the monad).
2. A functor is said to be monadic if it is a constituent of a monadic adjunction.
monoidal category
A monoidal category , also called a tensor category, is a category C equipped with (1) a bifunctor ${displaystyle otimes :Ctimes Cto C}$ , (2) an identity object and (3) natural isomorphisms that make ⊗ associative and the identity object an identity for ⊗, subject to certain coherence conditions.
monoid object
A monoid object in a monoidal category is an object together with the multiplication map and the identity map that satisfy the expected conditions like associativity. For example, a monoid object in Set is a usual monoid (unital semigroup) and a monoid object in R-mod is an associative algebra over a commutative ring R .
monomorphism
A morphism f is a monomorphism (also called monic) if ${displaystyle g=h}$ whenever ${displaystyle fcirc g=fcirc h}$ ; e.g., an injection in Set . In other words, f is the dual of an epimorphism.

## N

n -category
[T]he issue of comparing definitions of weak n -category is a slippery one, as it is hard to say what it even means for two such definitions to be equivalent. [...] It is widely held that the structure formed by weak n -categories and the functors, transformations, ... between them should be a weak ( n + 1)-category; and if this is the case then the question is whether your weak ( n + 1)-category of weak n- categories is equivalent to mine—but whose definition of weak ( n + 1)-category are we using here... ?
Tom Leinster,
1. A strict n-category is defined inductively: a strict 0-category is a set and a strict n -category is a category whose Hom sets are strict ( n -1)-categories. Precisely, a strict n -category is a category enriched over strict ( n -1)-categories. For example, a strict 1-category is an ordinary category.
2. The notion of a weak n-category is obtained from the strict one by weakening the conditions like associativity of composition to hold only up to coherent isomorphisms in the weak sense.
3. One can define an ∞-category as a kind of a colim of n -categories. Conversely, if one has the notion of a (weak) ∞-category (say a quasi-category ) in the beginning, then a weak n -category can be defined as a type of a truncated ∞-category.
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
${displaystyle {phi _{x}:F(x)to G(x)mid xin operatorname {Ob} (C)}}$

satisfying the condition: for each morphism f : x y in C , ${displaystyle phi _{y}circ F(f)=G(f)circ phi _{x}}$ . For example, writing ${displaystyle GL_{n}(R)}$ for the group of invertible n -by- n matrices with coefficients in a commutative ring R , we can view ${displaystyle GL_{n}}$ as a functor from the category CRing of commutative rings to the category Grp of groups. Similarly, ${displaystyle Rmapsto R^{*}}$ is a functor from CRing to Grp . Then the determinant det is a natural transformation from ${displaystyle GL_{n}}$ to - * .

2. A natural isomorphism is a natural transformation that is an isomorphism (i.e., admits the inverse).
nerve
The nerve functor N is the functor from Cat to s Set given by ${displaystyle N(C)_{n}=operatorname {Hom} _{mathbf {Cat} }([n],C)}$ . For example, if ${displaystyle varphi }$ is a functor in ${displaystyle N(C)_{2}}$ (called a 2-simplex), let ${displaystyle x_{i}=varphi (i),,0leq ileq 2}$ . Then ${displaystyle varphi (0to 1)}$ is a morphism ${displaystyle f:x_{0}to x_{1}}$ in C and also ${displaystyle varphi (1to 2)=g:x_{1}to x_{2}}$ for some g in C . Since ${displaystyle 0to 2}$ is ${displaystyle 0to 1}$ followed by ${displaystyle 1to 2}$ and since ${displaystyle varphi }$ is a functor, ${displaystyle varphi (0to 2)=gcirc f}$ . In other words, ${displaystyle varphi }$ encodes f , g and their compositions.
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: C D 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 : x y' 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 . 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 C op to Set is a presheaf of sets on C and a functor from C op to s Set 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 X i in a category C indexed by a set I is the projective limit ${displaystyle varprojlim }$ of the functor ${displaystyle Ito C,,imapsto X_{i}}$ , where I is viewed as a discrete category. It is denoted by ${displaystyle prod _{i}X_{i}}$ and is the dual of the coproduct of the family.
2. The product of a family of categories C i 's indexed by a set I is the category denoted by ${displaystyle prod _{i}C_{i}}$ whose class of objects is the product of the classes of objects of C i 's and whose hom-sets are ${displaystyle prod _{i}operatorname {Hom} _{operatorname {C_{i}} }(X_{i},Y_{i})}$ ; the morphisms are composed component-wise. It is the dual of the disjoint union.
profunctor
Given categories C and D , a profunctor (or a distributor) from C to D is a functor of the form ${displaystyle D^{text{op}}times Cto mathbf {Set} }$ .
projective
An object A in an abelian category is projective if the functor ${displaystyle operatorname {Hom} (A,-)}$ 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 ${displaystyle Cto mathbf {Fct} (C^{text{op}},mathbf {Set} )}$ ; i.e., ${displaystyle Fsimeq operatorname {Hom} _{C}(-,Z)}$ for some object Z . The object Z is said to be the representing object of F .
retraction
A morphism is a retraction if it has a right inverse.

## 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.
Tannakian duality
The Tannakian duality states that, in an appropriate setup, to give a morphism ${displaystyle f:Xto Y}$ is to give a pullback functor ${displaystyle f^{*}}$ along it. In other words, the Hom set ${displaystyle operatorname {Hom} (X,Y)}$ can be identified with the functor category ${displaystyle operatorname {Fct} (D(Y),D(X))}$ , perhaps in the derived sense , where ${displaystyle D(X)}$ is the category associated to X (e.g., the derived category). [13]
tensor category
Usually synonymous with monoidal category (though some authors distinguish between the two concepts.)
tensor triangulated category
A tensor triangulated category is a category that carries the structure of a symmetric monoidal category and that of a triangulated category in a compatible way.
tensor product
Given a monoidal category B , the tensor product of functors ${displaystyle F:C^{text{op}}to B}$ and ${displaystyle G:Cto B}$ is the coend:
${displaystyle Fotimes _{C}G=int ^{cin C}F(c)otimes G(c).}$
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 ${displaystyle operatorname {Map} _{C}(B,A)}$ 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 ${displaystyle f:Cto D}$ and an object X in D , a universal morphism from X to f is an initial object in the comma category ${displaystyle (Xdownarrow f)}$ . (Its dual is also called a universal morphism.) For example, take f to be the forgetful functor ${displaystyle mathbf {Vec} _{k}to mathbf {Set} }$ and X a set. An initial object of ${displaystyle (Xdownarrow f)}$ is a function ${displaystyle j:Xto f(V_{X})}$ . That it is initial means that if ${displaystyle k:Xto f(W)}$ is another morphism, then there is a unique morphism from j to k , which consists of a linear map ${displaystyle V_{X}to W}$ that extends k via j ; that is to say, ${displaystyle V_{X}}$ is the free vector space generated by X .
2. Stated more explicitly, given f as above, a morphism ${displaystyle Xto f(u_{X})}$ in D is universal if and only if the natural map
${displaystyle operatorname {Hom} _{C}(u_{X},c)to operatorname {Hom} _{D}(X,f(c)),,alpha mapsto (Xto f(u_{x}){overset {f(alpha )}{to }}f(c))}$
is bijective. In particular, if ${displaystyle operatorname {Hom} _{C}(u_{X},-)simeq operatorname {Hom} _{D}(X,f(-))}$ , then taking c to be u X 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 ${displaystyle operatorname {Hom} _{D}(X,f(-))}$ .

## 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
1.
Yoneda’s Lemma asserts ... in more evocative terms, a mathematical object X is best thought of in the context of a category surrounding it, and is determined by the network of relations it enjoys with all the objects of that category. Moreover, to understand X it might be more germane to deal directly with the functor representing it. This is reminiscent of Wittgenstein’s ’language game’; i.e., that the meaning of a word is—in essence—determined by, in fact is nothing more than, its relations to all the utterances in a language.
Barry Mazur ,
The Yoneda lemma says: for each set-valued contravariant functor F on C and an object X in C , there is a natural bijection
${displaystyle F(X)simeq operatorname {Nat} (operatorname {Hom} _{C}(-,X),F)}$

where Nat means the set of natural transformations. In particular, the functor

${displaystyle y:Cto mathbf {Fct} (C^{text{op}},mathbf {Set} ),,Xmapsto operatorname {Hom} _{C}(-,X)}$

is fully faithful and is called the Yoneda embedding.

2. If ${displaystyle F:Cto D}$ is a functor and y is the Yoneda embedding of C , then the Yoneda extension of F is the left Kan extension of F along y .

## Z

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

## Notes

1. 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 .
2. Remark 2.7. of
3. , Ch. 2, Exercise 2.8.
4. , Ch. III, § 3..
5. Hinich, V. (2013-11-17). "Dwyer-Kan localization revisited". arXiv : [].
6. , exercise 8.20
7. Adámek, Jiří; Herrlich, Horst; Strecker, George E (2004) [1990]. (PDF) . New York: Wiley & Sons. p. 40. ISBN 0-471-60922-6.
8. Joyal, A. (2002). "Quasi-categories and Kan complexes". Journal of Pure and Applied Algebra . 175 (1–3): 207–222. doi :.
9. , Definition 2.57.
10. Jacob Lurie. Tannaka duality for geometric stacks. , 2004.
11. Bhatt, Bhargav (2014-04-29). "Algebraization and Tannaka duality". arXiv : [].
12. Technical note: the lemma implicitly involves a choice of Set ; i.e., a choice of universe.