This is a glossary of properties and concepts in category theory in mathematics, including those in topos theory. (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. 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 i → j ⇔ i ≤ j {\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.
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2-category 1. A 2-category is a generalization of a category where there are also 2-morphisms between morphisms. 2. A (2, 1)-category is a 2-category in which every 2-morphism is invertible.
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 Hom ( X , − ) {\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.
adequate An adequate subcategory is a subcategory on which the Yoneda embedding of the ambient category is still fully faithful.
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
Hom D ( F ( X ) , Y ) ≃ Hom C ( X , G ( Y ) ) {\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 Hom D ( F ( − ) , − ) ≃ Hom C ( − , G ( − ) ) {\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 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.
algebraic An algebraic category is a category that is monadic over Set.
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.
anodyne anodyne extension
B
balanced A category is balanced if every bimorphism (i.e., both mono and epi) 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, Hom ( − , − ) {\displaystyle \operatorname {Hom} (-,-)} is a bifunctor from Cop 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".
calculus of fractions calculus of fractions.
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 p : F → C , q : G → C {\displaystyle p:F\to C,q:G\to C} over the same base category C, a functor f : F → G {\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 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.
categorical logic Categorical logic is an approach to mathematical logic that uses category theory.
categorical probability categorical probability
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 A class of objects, For each pair of objects X, Y, a set Hom ( X , Y ) {\displaystyle \operatorname {Hom} (X,Y)} , whose elements are called morphisms from X to Y, For each triple of objects X, Y, Z, a map (called composition)
∘ : Hom ( Y , Z ) × Hom ( X , Y ) → Hom ( X , Z ) , ( g , f ) ↦ g ∘ f {\displaystyle \circ :\operatorname {Hom} (Y,Z)\times \operatorname {Hom} (X,Y)\to \operatorname {Hom} (X,Z),\,(g,f)\mapsto g\circ f} , For each object X, an identity morphism id X ∈ Hom ( X , X ) {\displaystyle \operatorname {id} _{X}\in \operatorname {Hom} (X,X)}
subject to the conditions: for any morphisms f : X → Y {\displaystyle f:X\to Y} , g : Y → Z {\displaystyle g:Y\to Z} and h : Z → W {\displaystyle h:Z\to W} ,
( h ∘ g ) ∘ f = h ∘ ( g ∘ f ) {\displaystyle (h\circ g)\circ f=h\circ (g\circ f)} and id Y ∘ f = f ∘ id X = f {\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 x → y {\displaystyle x\to y} if and only if x ≤ y {\displaystyle x\leq y} ; the associativity of composition means transitivity.
category of 1. 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. 2. Category of modules, Category of topological spaces, Category of groups, Category of metric spaces, etc.
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.
codensity monad Codensity monad.
coend The coend of a functor F : C op × C → X {\displaystyle F:C^{\text{op}}\times C\to X} is the dual of the end of F and is denoted by
∫ c ∈ C F ( c , c ) {\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
M ⊗ R N = ∫ R M ⊗ Z N {\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 f , g : A → B {\displaystyle f,g:A\to B} is the colimit of the pair. It is the dual of an equalizer.
coherator coherator
coherence theorem A coherence theorem is a theorem of a form that states a weak structure is equivalent to a strict structure.
coherent 1. A coherent category (for now, see https://ncatlab.org/nlab/show/coherent+category). 2. A coherent topos.
cohesive cohesive category.
coimage The coimage of a morphism f: X → Y is the coequalizer of X × Y X ⇉ X {\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 f : C → B , g : D → B {\displaystyle f:C\to B,g:D\to B} , the comma category ( f ↓ g ) {\displaystyle (f\downarrow g)} is a category where (1) the objects are morphisms f ( c ) → g ( d ) {\displaystyle f(c)\to g(d)} and (2) a morphism from α : f ( c ) → g ( d ) {\displaystyle \alpha :f(c)\to g(d)} to β : f ( c ′ ) → g ( d ′ ) {\displaystyle \beta :f(c')\to g(d')} consists of c → c ′ {\displaystyle c\to c'} and d → d ′ {\displaystyle d\to d'} such that f ( c ) → f ( c ′ ) → β g ( d ′ ) {\displaystyle f(c)\to f(c'){\overset {\beta }{\to }}g(d')} is f ( c ) → α g ( d ) → g ( d ′ ) . {\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 comonad in a category X is 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.
completeness Deligne's completeness theorem; see [1].
composition 1. A composition of morphisms in a category is part of the datum defining the category. 2. If f : C → D , g : D → E {\displaystyle f:C\to D,\,g:D\to E} are functors, then the composition g ∘ f {\displaystyle g\circ f} or g f {\displaystyle gf} is the functor defined by: for an object x and a morphism u in C, ( g ∘ f ) ( x ) = g ( f ( x ) ) , ( g ∘ f ) ( u ) = g ( f ( u ) ) {\displaystyle (g\circ f)(x)=g(f(x)),\,(g\circ f)(u)=g(f(u))} . 3. Natural transformations are composed pointwise: if φ : f → g , ψ : g → h {\displaystyle \varphi :f\to g,\,\psi :g\to h} are natural transformations, then ψ ∘ φ {\displaystyle \psi \circ \varphi } is the natural transformation given by ( ψ ∘ φ ) x = ψ x ∘ φ x {\displaystyle (\psi \circ \varphi )_{x}=\psi _{x}\circ \varphi _{x}} .
computad computad.
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 lim → {\displaystyle \varinjlim } is the left adjoint to the diagonal functor Δ : C → Fct ( I , C ) {\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 f : I → C {\displaystyle f:I\to C} ,
Hom ( lim → f , X ) ≃ Hom ( f , Δ X ) , {\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 zi such that z 0 = x , z n = y {\displaystyle z_{0}=x,z_{n}=y} and either Hom ( z i , z i + 1 ) {\displaystyle \operatorname {Hom} (z_{i},z_{i+1})} or Hom ( z i + 1 , z i ) {\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 f : C → D {\displaystyle f:C\to D} is constant if it factors as: C → { A } → i D {\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 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 P ( S ) {\displaystyle {\mathfrak {P}}(S)} be the power set of S and for each function f : S → T {\displaystyle f:S\to T} , define
P ( f ) : P ( T ) → P ( S ) {\displaystyle {\mathfrak {P}}(f):{\mathfrak {P}}(T)\to {\mathfrak {P}}(S)}
by sending a subset A of T to the pre-image f − 1 ( A ) {\displaystyle f^{-1}(A)} . With this, P : S e t → S e t {\displaystyle {\mathfrak {P}}:\mathbf {Set} \to \mathbf {Set} } 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 lim → {\displaystyle \varinjlim } of the functor I → C , i ↦ X i {\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.
cubical A cubical set is an alternative for a simplicial set; a simplex is replaced by a cube.
D
Dagger compact category Dagger compact category.
Day convolution Given a group or monoid M, the Day convolution is the tensor product in F c t ( M , S e t ) {\displaystyle \mathbf {Fct} (M,\mathbf {Set} )} .
Dendroidal Dendroidal set.
dense A dense subcategory is another name for an adequate subcategory.
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 1. Given categories I, C, the diagonal functor is the functor
Δ : C → F c t ( I , C ) , A ↦ Δ A {\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 f : A → B {\displaystyle f:A\to B} to the natural transformation Δ f , i : Δ A ( i ) = A → Δ B ( i ) = B {\displaystyle \Delta _{f,i}:\Delta _{A}(i)=A\to \Delta _{B}(i)=B} that is f at each i.
diagram 1. Given a category C, a diagram in C is a functor f : I → C {\displaystyle f:I\to C} from a category I. For example, if I = N {\displaystyle I=\mathbb {N} } with no morphisms other than the identities, a diagram simply amounts to a sequence of objects. For a general I, typically there is a morphism between the images of objects in I under f (whence the term diagram). 2. simplicial diagram, a diagram from the opposite Δ op {\displaystyle \Delta ^{\textrm {op}}} of the simplex category.
diagrammatic set A diagrammatic set is an alternative to a simplicial set or a cubical set.
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 limit is the colimit of 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".
double double category.
Dwyer–Kan equivalence A Dwyer–Kan equivalence is a generalization of an equivalence of categories to the simplicial context.
E
Elementary 1. The Elementary Theory of Abstract Categories. 2. The Elementary Theory of the Category of Sets. 3. The Elementary Theory of the Category of Categories.
Eilenberg–Moore category Another name for the category of algebras for a given monad.
Eilenberg–Zilber category Eilenberg–Zilber 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.
end The end of a functor F : C op × C → X {\displaystyle F:C^{\text{op}}\times C\to X} is the limit
∫ c ∈ C F ( c , c ) = lim ← ( F # : C # → X ) {\displaystyle \int _{c\in C}F(c,c)=\varprojlim (F^{\#}:C^{\#}\to X)}
where C # {\displaystyle C^{\#}} is the category (called the subdivision category of C) whose objects are symbols c # , u # {\displaystyle c^{\#},u^{\#}} for all objects c and all morphisms u in C and whose morphisms are b # → u # {\displaystyle b^{\#}\to u^{\#}} and u # → c # {\displaystyle u^{\#}\to c^{\#}} if u : b → c {\displaystyle u:b\to c} and where F # {\displaystyle F^{\#}} is induced by F so that c # {\displaystyle c^{\#}} would go to F ( c , c ) {\displaystyle F(c,c)} and u # , u : b → c {\displaystyle u^{\#},u:b\to c} would go to F ( b , c ) {\displaystyle F(b,c)} . For example, for functors F , G : C → X {\displaystyle F,G:C\to X} ,
∫ c ∈ C Hom ( F ( c ) , G ( c ) ) {\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 A class of objects, For each pair of objects X, Y in D, an object Map D ( X , Y ) {\displaystyle \operatorname {Map} _{D}(X,Y)} in C, called the mapping object from X to Y, For each triple of objects X, Y, Z in D, a morphism in C,
∘ : Map D ( Y , Z ) ⊗ Map D ( X , Y ) → Map D ( X , Z ) {\displaystyle \circ :\operatorname {Map} _{D}(Y,Z)\otimes \operatorname {Map} _{D}(X,Y)\to \operatorname {Map} _{D}(X,Z)} , called the composition, For each object X in D, a morphism 1 X : 1 → Map D ( X , X ) {\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 g = h {\displaystyle g=h} whenever g ∘ f = h ∘ f {\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 f , g : A → B {\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
F c t ( C , D ) → D , F ↦ F ( A ) . {\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.
exact 1. An exact sequence is typically a sequence (from arbitrary negative integers to arbitrary positive integers) of maps
⋯ → E 1 → f 1 E 2 → f 2 E 3 → ⋯ {\displaystyle \cdots \to E_{1}{\overset {f_{1}}{\to }}E_{2}{\overset {f_{2}}{\to }}E_{3}\to \cdots }
such that the image of f i {\displaystyle f_{i}} is the kernel of f i + 1 {\displaystyle f_{i+1}} . The notion can be generalized in various ways. 2. A short exact sequence is a sequence of the form 0 → E → F → G → 0 {\displaystyle 0\to E\to F\to G\to 0} . 3. A functor (for example, between abelian categories) is said to be exact if it takes short exact sequences to short exact sequences. 4. An exact category is roughly a category where there is the notion of a short exact sequence. 5. An exact category in the sense of Barr.
exit exit-path category
F
faithful A functor is faithful if it is injective when restricted to each hom-set.
fundamental category The fundamental category functor τ 1 : s S e t → C a t {\displaystyle \tau _{1}:s\mathbf {Set} \to \mathbf {Cat} } is the left adjoint to the nerve functor N. For every category C, τ 1 N C = C {\displaystyle \tau _{1}NC=C} .
fundamental groupoid 1. The fundamental groupoid of a topological space X is a category where the objects are the points on X and the morphisms the homotopy classes of paths. 2. The fundamental groupoid of a Kan complex X is the category where an object is a 0-simplex (vertex) Δ 0 → X {\displaystyle \Delta ^{0}\to X} , a morphism is a homotopy class of a 1-simplex (path) Δ 1 → X {\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 Xi in C indexed by I is the product of the family in the slice category C / S {\displaystyle C_{/S}} of C over S (provided there are X i → S {\displaystyle X_{i}\to S} ). The fiber product of two objects X and Y over an object S is denoted by X × S Y {\displaystyle X\times _{S}Y} and is also called a Cartesian square.
fibrant An object is fibrant if the unique morphism from it to the final object is a fibration, when there is a notion of a fibration.
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 lim ← Hom ( f ( j ) , i ) {\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 lim ← Hom ( f ( j ) , i ) {\displaystyle \varprojlim \operatorname {Hom} (f(j),i)} is nonempty for some object i in I.
final Synonymous with terminal
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 G r p → S e t {\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 category The free category generated by a graph G {\displaystyle G} is a category together with the map G → U C {\displaystyle G\to UC} = the underlying graph of C {\displaystyle C} such that (1) the objects are exactly the vertices of G {\displaystyle G} and (2) G → U