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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. 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:

• = {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
• B
• C
• D
• E
• F
• G
• H
• I
• J
• K
• L
• M
• N
• O
• P
• Q
• R
• S
• T
• U
• V
• W
• XYZ
• See also
• References

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.

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.

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 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 (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, ${\displaystyle \operatorname {Hom} (-,-)}$ is a bifunctor from C 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 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.

The theory of categories originated ... with the need to guide complicated calculations involving passage to the limit in the study of the qualitative leap from spaces to homotopical/homological objects. ... But category theory does not rest content with mere classification in the spirit of Wolffian metaphysics (although a few of its practitioners may do so); rather it is the mutability of mathematically precise structures (by morphisms) which is the essential content of category theory.

William Lawvere,

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

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 ${\displaystyle F:C^{\text{op}}\times C\to X}$ 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.

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

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 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 .

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 ${\displaystyle (\psi \circ \varphi )_{x}=\psi _{x}\circ \varphi _{x}}$.

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 ${\displaystyle \varinjlim }$ 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.

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 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 ${\displaystyle \mathbf {Fct} (M,\mathbf {Set} )}$.

Dendroidal

Dendroidal set.

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 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".

Dwyer–Kan equivalence

A Dwyer–Kan equivalence is a generalization of an equivalence of categories to the simplicial context.

E

Elementary Theory of the Category of Sets

The Elementary Theory of the Category of Sets. The link is a redirect; for now, see https://ncatlab.org/nlab/show/ETCS.

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.

exact

1.  An exact sequence is typically a sequence (from arbitrary negative integers to arbitrary positive integers) of maps

${\displaystyle \cdots \to E_{1}{\overset {f_{1}}{\to }}E_{2}{\overset {f_{2}}{\to }}E_{3}\to \cdots }$

such that the image of ${\displaystyle f_{i}}$ is the kernel of ${\displaystyle f_{i+1}}$. The notion can be generalized in various ways.

2.  A short exact sequence is a sequence of the form ${\displaystyle 0\to E\to F\to G\to 0}$.

F

faithful

A functor is faithful if it is injective when restricted to each hom-set.

fundamental category

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

free category

free completion

free completion, free cocompletion.

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) ${\displaystyle F(f\circ g)=F(f)\circ F(g)}$ 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 Gabriel–Popescu theorem says an abelian category is a 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 or ) 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.

generalized

generalized metric space.

Gray

1.  A Gray tensor product is a lax analog of a Cartesian product.

2.  A Gray category is a certain semi-strict 3-category; see https://ncatlab.org/nlab/show/Gray-category

gros topos

The notion of a gros topos (of topological spaces) is due to Jean Giraud.

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 ∞-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-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

idempotent

An endomorphism f is idempotent if ${\displaystyle f\circ f=f}$.

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 is obtained from a category by replacing the class/set of objects and morphisms by the spaces of objects and morphisms. Precisely, 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 is the standard n-simplex and ${\displaystyle \Lambda _{i}^{n}}$ is obtained from Δ by removing the i-th face and the interior (see Kan fibration#Definitions). For example, the nerve of a category satisfies the condition and thus can be considered as an ∞-category.

(∞, n)-category

An (∞, n)-category is obtained from an ∞-category by replacing the space of morphisms by the (∞, n - 1)-category of morphisms.

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 monoidal category (C, ⊗), the internal Hom is a functor ${\displaystyle :C^{\text{op}}\times C\to C}$ such that ${\displaystyle }$ 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 ${\displaystyle =\operatorname {Hom} _{R}(M,N)}$, 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. 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 limit is the limit of an inverse system.

Isbell

1.  Isbell duality/Isbell conjugacy

2.  Isbell completion.

3.  Isbell envelop.

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. 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.

limit

1.  The limit (or 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 colimit (or 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 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 ${\displaystyle T(X)=G\times X}$. 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 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 ${\displaystyle \otimes :C\times C\to C}$, (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 multicategory is a generalization of a category in which a morphism is allowed to have more than one domain. It is the same thing as a colored operad.

N

n-category

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 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

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

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 ${\displaystyle N(C)_{n}=\operatorname {Hom} _{\mathbf {Cat} }(,C)}$. 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.