Glossary of category theory

This is a glossary of properties and concepts in category theory in mathematics. (see also Outline of category theory.)

• 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.^[1] 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 ${\displaystyle i\to j\Leftrightarrow i\leq j}$.)
• 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.
• sSet, 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

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.

additive

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.^[2]

adjunction

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.)

algebra for a monad

Given a monad T in a category X, 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

algebras for a given monad

.

bicategory

A

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.

bimonoidal

A bimonoidal category is a category with two monoidal structures, one distributing over the other.

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 ${\displaystyle p:F\to C,q:G\to C}$ over the same base category C, a functor ${\displaystyle f:F\to G}$ over C is cartesian if it sends cartesian morphisms to cartesian morphisms.

cartesian morphism

1.  Given a functor π: C → D (e.g., a

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

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.

categorical logic

Categorical logic is an approach to mathematical logic that uses category theory.

categorification

Categorification is 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)

   ${\displaystyle \circ :\operatorname {Hom} (Y,Z)\times \operatorname {Hom} (X,Y)\to \operatorname {Hom} (X,Z),\,(g,f)\mapsto g\circ f}$,

4. For each object X, an identity morphism ${\displaystyle \operatorname {id} _{X}\in \operatorname {Hom} (X,X)}$

subject to the conditions: for any morphisms ${\displaystyle f:X\to Y}$, ${\displaystyle g:Y\to Z}$ and ${\displaystyle h:Z\to W}$,

• ${\displaystyle (h\circ g)\circ f=h\circ (g\circ f)}$ and ${\displaystyle \operatorname {id} _{Y}\circ f=f\circ \operatorname {id} _{X}=f}$.

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 ${\displaystyle x\to y}$ if and only if ${\displaystyle x\leq y}$; 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 ${\displaystyle F:C^{\text{op}}\times C\to X}$ is the dual of the end of F and is denoted by

${\displaystyle \int ^{c\in C}F(c,c)}$.

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

${\displaystyle M\otimes _{R}N=\int ^{R}M\otimes _{\mathbb {Z} }N}$

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 ${\displaystyle f,g:A\to B}$ is the colimit of the pair. It is the dual of an equalizer.

coherence theorem

A

coherence theorem

is a theorem of a form that states a weak structure is equivalent to a strict structure.

coimage

The coimage of a morphism f: X → Y is the coequalizer of ${\displaystyle X\times _{Y}X\rightrightarrows X}$.

colored operad

Another term for multicategory, a generalized category where a morphism can have several domains. The notion of "colored operad" is more primitive than that of operad: in fact, an operad can be defined as a colored operad with a single object.

comma

Given functors ${\displaystyle f:C\to B,g:D\to B}$, the comma category ${\displaystyle (f\downarrow g)}$ 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 ${\displaystyle c\to c'}$ and ${\displaystyle d\to d'}$ 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.

comonad

A

comonoid

in the monoidal category of endofunctors of X.

compact

Probably synonymous with #accessible.

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 ${\displaystyle f:C\to D,\,g:D\to E}$ are functors, then the composition ${\displaystyle g\circ f}$ or ${\displaystyle gf}$ is the functor defined by: for an object x and a morphism u in C, ${\displaystyle (g\circ f)(x)=g(f(x)),\,(g\circ f)(u)=g(f(u))}$.

3.  Natural transformations are composed pointwise: if ${\displaystyle \varphi :f\to g,\,\psi :g\to h}$ 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

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^[3] that the colimit ${\displaystyle \varinjlim }$ is the left adjoint to the diagonal functor ${\displaystyle \Delta :C\to \operatorname {Fct} (I,C)}$, which sends an object X to the constant functor with value X; that is, for any X and any functor ${\displaystyle f:I\to C}$,

${\displaystyle \operatorname {Hom} (\varinjlim f,X)\simeq \operatorname {Hom} (f,\Delta _{X}),}$

provided the colimit in question exists. The right-hand side is then the set of cones with vertex X.^[4]

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 ${\displaystyle f:C\to D}$ is constant if it factors as: ${\displaystyle C\to \{A\}{\overset {i}{\to }}D}$ 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 ${\displaystyle f:S\to T}$, 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 ${\displaystyle I\to C,\,i\mapsto X_{i}}$, 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.

core

The core of a category is the maximal groupoid contained in the category.

D

Day convolution

Given a group or monoid M, the Day convolution is the tensor product in ${\displaystyle \mathbf {Fct} (M,\mathbf {Set} )}$.^[5]

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

${\displaystyle \Delta :C\to \mathbf {Fct} (I,C),\,A\mapsto \Delta _{A}}$

that sends each object A to the constant functor with value A and each morphism ${\displaystyle f:A\to B}$ to the natural transformation ${\displaystyle \Delta _{f,i}:\Delta _{A}(i)=A\to \Delta _{B}(i)=B}$ that is f at each i.

diagram

Given a category C, a diagram in C is a functor ${\displaystyle f:I\to C}$ from a small category I.

differential graded category

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.

direct limit

A

direct system

.

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.^[6]

E

Eilenberg–Moore category

Another name for the category of

algebras for a given monad

.

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.

end

The end of a functor ${\displaystyle F:C^{\text{op}}\times C\to X}$ is the limit

${\displaystyle \int _{c\in C}F(c,c)=\varprojlim (F^{\#}:C^{\#}\to X)}$

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 ${\displaystyle u:b\to c}$ and where ${\displaystyle F^{\#}}$ is induced by F so that ${\displaystyle c^{\#}}$ would go to ${\displaystyle F(c,c)}$ and ${\displaystyle u^{\#},u:b\to c}$ would go to ${\displaystyle F(b,c)}$. For example, for functors ${\displaystyle F,G:C\to X}$,

${\displaystyle \int _{c\in C}\operatorname {Hom} (F(c),G(c))}$

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.

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 ${\displaystyle 1_{X}:1\to \operatorname {Map} _{D}(X,X)}$ 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.

epimorphism

A morphism f is an epimorphism if ${\displaystyle g=h}$ whenever ${\displaystyle g\circ f=h\circ f}$. In other words, f is the dual of a monomorphism.

equalizer

The

equalizer

of a pair of morphisms ${\displaystyle f,g:A\to B}$ 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

${\displaystyle \mathbf {Fct} (C,D)\to D,\,\,F\mapsto F(A).}$

For example, the Eilenberg–Steenrod axioms give an instance when the functor is an equivalence.

F

faithful

A functor is

hom-set

.

fundamental category

The fundamental category functor ${\displaystyle \tau _{1}:s\mathbf {Set} \to \mathbf {Cat} }$ 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

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 ${\displaystyle X\times _{S}Y}$ 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.

finitary monad

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

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) ${\displaystyle F(f\circ g)=F(f)\circ F(g)}$ whenever ${\displaystyle f\circ g}$ is defined and (2) ${\displaystyle F(\operatorname {id} _{x})=\operatorname {id} _{F(x)}}$. For example,

${\displaystyle {\mathfrak {P}}:\mathbf {Set} \to \mathbf {Set} ,\,S\mapsto {\mathfrak {P}}(S)}$,

where ${\displaystyle {\mathfrak {P}}(S)}$ is the power set of S is a functor if we define: for each function ${\displaystyle f:S\to T}$, ${\displaystyle {\mathfrak {P}}(f):{\mathfrak {P}}(S)\to {\mathfrak {P}}(T)}$ by ${\displaystyle {\mathfrak {P}}(f)(A)=f(A)}$.

functor category

The functor category Fct(C, D) or ${\displaystyle D^{C}}$ 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

quotient

of the category of modules.

Galois category

1.  In

SGA 1, Exposé V (Definition 5.1.), a category is called a Galois category

if it is equivalent to the category of finite G-sets for some profinite group G.

2.  For technical reasons, some authors (e.g.,

Stacks project^[7] or ^[8]

) use slightly different definitions.

generator

In a category C, a family of objects ${\displaystyle G_{i},i\in I}$ is a system of generators of C if the functor ${\displaystyle X\mapsto \prod _{i\in I}\operatorname {Hom} (G_{i},X)}$ is conservative. Its dual is called a system of cogenerators.

Grothendieck's Galois theory

A category-theoretic generalization of Galois theory; see Grothendieck's Galois theory.

Grothendieck category

A Grothendieck category is a certain well-behaved kind of an abelian category.

Grothendieck construction

Given a functor ${\displaystyle U:C\to \mathbf {Cat} }$, 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

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

∞-categories

.

homological dimension

The homological dimension of an abelian category with enough injectives is the least non-negativer integer 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 Mod_R 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, ${\displaystyle g\circ f=g}$ and ${\displaystyle f\circ h=h}$.

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 ${\displaystyle Y\rightrightarrows Y\sqcup _{X}Y}$.

ind-limit

A colimit (or inductive limit) in ${\displaystyle \mathbf {Fct} (C^{\text{op}},\mathbf {Set} )}$.

inductive limit

Another name for

colimit

.

∞-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 Δⁿ is the standard n-simplex and ${\displaystyle \Lambda _{i}^{n}}$ is obtained from Δⁿ by removing the i-th face and the interior (see

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

1.  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.

2.  The term “injective limit” is another name for a direct limit.

internal Hom

Given a

internal Hom

is a functor ${\displaystyle [-,-]:C^{\text{op}}\times C\to C}$ 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

1.  A morphism f is an inverse to a morphism g if ${\displaystyle g\circ f}$ is defined and is equal to the identity morphism on the codomain of g, and ${\displaystyle f\circ g}$ 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 ${\displaystyle f\circ g}$ is defined and is equal to the identity morphism on the domain of g, and similarly for a right inverse.

2.  An

inverse system

.

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 ${\displaystyle f:I\to J}$ 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 ${\displaystyle \alpha :I\to C}$, the functor ${\displaystyle f_{!}\alpha :J\to C}$ is called the left Kan extension of α along f.^[9] 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 ${\displaystyle f^{*}}$.

Ken Brown's lemma

Ken Brown's lemma is a lemma in the theory of model categories.

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

A lax functor is a generalisation of a pseudo-functor, in which the structural transformations associated to composition and identities are not required to be invertible.

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.^[10]

limit

1.  The

projective limit

) of a functor ${\displaystyle f:I^{\text{op}}\to \mathbf {Set} }$ is

${\displaystyle \varprojlim _{i\in I}f(i)=\{(x_{i}|i)\in \prod _{i}f(i)|f(s)(x_{j})=x_{i}{\text{ for any }}s:i\to j\}.}$

2.  The limit ${\displaystyle \varprojlim _{i\in I}f(i)}$ 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, ${\displaystyle \operatorname {Hom} (X,\varprojlim _{i\in I}f(i))=\varprojlim _{i\in I}\operatorname {Hom} (X,f(i))}$; i.e., it is an object representing the functor ${\displaystyle X\mapsto \varprojlim _{i}\operatorname {Hom} (X,f(i)).}$

3.  The

inductive limit

) ${\displaystyle \varinjlim _{i\in I}f(i)}$ is the dual of a limit; i.e., given a functor ${\displaystyle f:I\to C}$, 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 ${\displaystyle i\to j}$, ${\displaystyle f(i)\to X}$ 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 ${\displaystyle L=\varinjlim _{X\in C}f(X)}$ exists; then the identity morphism on L corresponds to a compatible family of morphisms ${\displaystyle \alpha _{X}:X\to L}$ such that ${\displaystyle \alpha _{L}}$ is the identity. If ${\displaystyle f:X\to L}$ is any morphism, then ${\displaystyle f=\alpha _{L}\circ f=\alpha _{X}}$; i.e., L is a final object of C.

M

Mittag-Leffler condition

An

inverse system

${\displaystyle \cdots \to X_{2}\to X_{1}\to X_{0}}$ is said to satisfy the

Mittag-Leffler condition

if for each integer ${\displaystyle n\geq 0}$, there is an integer ${\displaystyle m\geq n}$ such that for each ${\displaystyle l\geq m}$, the images of ${\displaystyle X_{m}\to X_{n}}$ and ${\displaystyle X_{l}\to X_{n}}$ are the same.

monad

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 ${\displaystyle T(X)=G\times X}$. Then define the multiplication μ on T as the natural transformation ${\displaystyle \mu :T\circ T\to T}$ given by

${\displaystyle \mu _{X}:G\times (G\times X)\to G\times X,\,\,(g,(h,x))\mapsto (gh,x)}$

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 ${\displaystyle F:X\rightleftarrows A:G}$ determines a monad in X; namely, one takes ${\displaystyle T=G\circ F}$, the identity map η on T to be a unit of the adjunction and also defines μ using the adjunction.

monadic

1.  An adjunction is said to be

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

bifunctor

${\displaystyle \otimes :C\times C\to C}$, (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 ${\displaystyle f\circ g=f\circ h}$; e.g., an injection in Set. In other words, f is the dual of an epimorphism.

multicategory

A

colored operad.^[11]

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, A survey of definitions of n-category

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

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

${\displaystyle \{\phi _{x}:F(x)\to G(x)\mid x\in \operatorname {Ob} (C)\}}$

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, ${\displaystyle R\mapsto R^{*}}$ 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 sSet 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 ${\displaystyle x_{i}=\varphi (i),\,0\leq i\leq 2}$. Then ${\displaystyle \varphi (0\to 1)}$ is a morphism ${\displaystyle f:x_{0}\to x_{1}}$ in C and also ${\displaystyle \varphi (1\to 2)=g:x_{1}\to x_{2}}$ for some g in C. Since ${\displaystyle 0\to 2}$ is ${\displaystyle 0\to 1}$ followed by ${\displaystyle 1\to 2}$ and since ${\displaystyle \varphi }$ is a functor, ${\displaystyle \varphi (0\to 2)=g\circ f}$. In other words, ${\displaystyle \varphi }$ encodes f, g and their compositions.

normal

A monomorphism is normal if it is the kernel of some morphism, and an epimorphism is conormal if it is the cokernel of some morphism. A category is

normal

if every monomorphism 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

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.

preadditive

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

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 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 X_i in a category C indexed by a set I is the projective limit ${\displaystyle \varprojlim }$ of the functor ${\displaystyle I\to C,\,i\mapsto X_{i}}$, 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 ${\displaystyle D^{\text{op}}\times C\to \mathbf {Set} }$.

projective

1.  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.

2.  The term “projective limit” is another name for an inverse limit.

PROP

A PROP is a symmetric strict monoidal category whose objects are natural numbers and whose tensor product addition of natural numbers.

pseudoalgebra

A pseudoalgebra is a 2-category-version of an algebra for a monad (with a monad replaced by a 2-monad).

Q

Q

Q-category.

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 isomorphisms if 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

Yoneda embedding

${\displaystyle C\to \mathbf {Fct} (C^{\text{op}},\mathbf {Set} )}$; i.e., ${\displaystyle F\simeq \operatorname {Hom} _{C}(-,Z)}$ 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.

rig

A rig category is a category with two monoidal structures, one distributing over the other.

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

(∞, 1)-categories

.

semisimple

An abelian category is

semisimple

if and only if the category of modules over it is semisimple.

Serre functor

Given a k-linear category C over a field k, a Serre functor ${\displaystyle f:C\to C}$ is an auto-equivalence such that ${\displaystyle \operatorname {Hom} (A,B)\simeq \operatorname {Hom} (B,f(A))^{*}}$ for any objects A, B.

simple object

A simple object in an abelian category is an object A that is not isomorphic to the zero object and whose every subobject is isomorphic to zero or to A. For example, a simple module is precisely a simple object in the category of (say left) modules.

simplex category

The simplex category Δ is the category where an object is a set [n] = { 0, 1, …, n }, n ≥ 0, totally ordered in the standard way and a morphism is an order-preserving function.

simplicial category

A category enriched over simplicial sets.

Simplicial localization

Simplicial localization is a method of localizing a category.

simplicial object

A

simplicial object

in a category C is roughly a sequence of objects ${\displaystyle X_{0},X_{1},X_{2},\dots }$ in C that forms a simplicial set. In other words, it is a covariant or contravariant functor Δ → C. For example, a simplicial presheaf is a simplicial object in the category of presheaves.

simplicial set

A simplicial set is a contravariant functor from Δ to Set, where Δ is the simplex category, a category whose objects are the sets [n] = { 0, 1, …, n } and whose morphisms are order-preserving functions. One writes ${\displaystyle X_{n}=X([n])}$ and an element of the set ${\displaystyle X_{n}}$ is called an n-simplex. For example, ${\displaystyle \Delta ^{n}=\operatorname {Hom} _{\Delta }(-,[n])}$ is a simplicial set called the standard n-simplex. By Yoneda's lemma, ${\displaystyle X_{n}\simeq \operatorname {Nat} (\Delta ^{n},X)}$.

site

A category equipped with a Grothendieck topology.

skeletal

1.  A category is skeletal if isomorphic objects are necessarily identical.

2.  A (not unique) skeleton of a category is a full subcategory that is skeletal.

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 ${\displaystyle p_{X}:X\to A}$ to ${\displaystyle p_{Y}:Y\to A}$, then ${\displaystyle p_{Y}\circ f=p_{X}}$ in C and whose composition is that of C.

small

1.  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.^[12] (NB: some authors use the term "quasicategory" with a different meaning.^[13]

)

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)

species

A

symmetric sequence

.

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 ${\displaystyle \operatorname {Coim} (f)\to \operatorname {Im} (f)}$ 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.^[14] 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

Given an object A in a category, a subobject of A is an equivalence class of monomorphisms to A; two monomorphisms f, g are considered equivalent if f factors through g and g factors through f.

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.

symmetric sequence

A

symmetric sequence is a sequence of objects with actions of symmetric groups. It is categorically equivalent to a (combinatorial) species

.

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 ${\displaystyle f:X\to Y}$ 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).^[15][16]

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 ${\displaystyle G:C\to B}$ is the coend:

${\displaystyle F\otimes _{C}G=\int ^{c\in C}F(c)\otimes G(c).}$

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.

U

universal

1.  Given a functor ${\displaystyle f:C\to D}$ and an object X in D, a

universal morphism from X to f is an initial object in the comma category

${\displaystyle (X\downarrow f)}$. (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 ${\displaystyle (X\downarrow f)}$ is a function ${\displaystyle j:X\to f(V_{X})}$. That it is initial means that if ${\displaystyle k:X\to f(W)}$ 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 ${\displaystyle X\to f(u_{X})}$ in D is universal if and only if the natural map

${\displaystyle \operatorname {Hom} _{C}(u_{X},c)\to \operatorname {Hom} _{D}(X,f(c)),\,\alpha \mapsto (X\to f(u_{x}){\overset {f(\alpha )}{\to }}f(c))}$

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 ${\displaystyle \operatorname {Hom} _{D}(X,f(-))}$.

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, Thinking about Grothendieck

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

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

is fully faithful and is called the Yoneda embedding.^[17]

2.  If ${\displaystyle F:C\to D}$ 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

.