Math 4 Wisdom. "Mathematics for Wisdom" by Andrius Kulikauskas.

Investigation: Organize the concepts in category theory to reveal underlying themes

Look for the structure that is needed for each of Grothendieck's six operations.
Organize the glossary in terms of canonical examples. Provide additional examples.
Think through in what sense Product means "and" and Coproduct means "or".

This is a glossary of properties and concepts in category theory in mathematics. It is based especially on the following sources:

Wikipedia: Glossary of category theory

The notations and the conventions used throughout the article are:

[n] = { 0, 1, 2, …, n }, which is viewed as a category (by writing {$i \to j \Leftrightarrow i \le j$}.)
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.

Types of object

An object is part of a data defining a category.
An object is isomorphic to another object if there is an isomorphism between them.
An object A is initial if there is exactly one morphism from A to each object; e.g., empty set in Set.
An object A is terminal (also called final) if there is exactly one morphism from each object to A; e.g., singletons in Set. It is the dual of an initial object.
A zero object is an object that is both initial and terminal, such as a trivial group in Grp.
A subterminal object is an object X such that every object has at most one morphism into X.
In a category C, a family of objects {$G_i, i \in I$} is a system of generators of C if the functor {$X \mapsto \prod_{i \in I} \operatorname{Hom}(G_i, X)$} is conservative. Its dual is called a system of cogenerators.
An object A in an abelian category is injective if the functor {$\operatorname{Hom}(-, A)$} is exact. It is the dual of a projective object.
An object A in an abelian category is projective if the functor {$\operatorname{Hom}(A, -)$} is exact. It is the dual of an injective 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.
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.
Given a monad T in a category X, an algebra for a 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.
A Kan complex is a fibrant object in the category of simplicial sets.
Given a cardinal number κ, an object X in a category is κ-accessible (or κ-compact or κ-presentable) if {$\operatorname{Hom}(X, -)$} commutes with κ-filtered colimits.
compact Probably synonymous with #accessible.
perfect Sometimes synonymous with "compact". See perfect complex.
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 n-lab: small object argument.

Variations of objects

An object A in an ∞-category C is initial if {$\operatorname{Map}_C(A, B)$} is contractible for each object B in C.
An object A in an ∞-category C is terminal if {$\operatorname{Map}_C(B, A)$} is contractible for every object B in C.
In a category, a sieve is a set S of objects having the property: if f is a morphism with the codomain in S, then the domain of f is in S.
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.
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.
A subquotient is a quotient of a subobject.
A symmetric sequence is a sequence of objects with actions of symmetric groups. It is categorically equivalent to a (combinatorial) species.
Hall algebra of a category. See Ringel–Hall algebra. (based on equivalence classes of objects)

Types of morphism

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, {$g\circ f=g$} and {$f\circ h=h$}.
A morphism f is an epimorphism if {$g=h$} whenever {$g\circ f=h\circ f$}. In other words, f is the dual of a monomorphism.
A morphism f is a monomorphism (also called monic) if {$g=h$} whenever {$f\circ g=f\circ h$}; e.g., an injection in Set. In other words, f is the dual of an epimorphism.
A bimorphism is a morphism that is both an epimorphism and a monomorphism.
A monomorphism is normal if it is the kernel of some morphism.
An epimorphism is conormal if it is the cokernel of some morphism.
A morphism f is an inverse to a morphism g if {$g\circ f$} is defined and is equal to the identity morphism on the codomain of g, and {$f\circ g$} is defined and equal to the identity morphism on the domain of g. The inverse of g is unique and is denoted by {$g^{-1}$}.
A morphism f is an isomorphism if there exists an inverse of f.
f is a left inverse to g if {$f\circ g$} is defined and is equal to the identity morphism on the domain of g, and similarly for a right inverse.
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.
A morphism is a retraction if it has a right inverse.
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.
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.)
A morphism f in a category admitting finite limits and finite colimits is strict if the natural morphism {$\operatorname{Coim}(f) \to \operatorname{Im}(f)$} is an isomorphism.
Given a functor {$f: C \to D$} and an object X in D, a universal morphism from X to f is an initial object in the comma category {$(X \downarrow f)$}. (Its dual is also called a universal morphism.) For example, take f to be the forgetful functor {$\mathbf{Vec}_k \to \mathbf{Set}$} and X a set. An initial object of {$(X \downarrow f)$} is a function {$j: X \to f(V_X)$}. That it is initial means that if {$k: X \to f(W)$} is another morphism, then there is a unique morphism from j to k, which consists of a linear map {$V_X \to W$} that extends k via j; that is to say, {$V_X$} is the free vector space generated by X.
Stated more explicitly, given f as above, a morphism {$X \to f(u_X)$} in D is universal if and only if the natural map {$\operatorname{Hom}_C(u_X, c) \to \operatorname{Hom}_D(X, f(c)), \, \alpha \mapsto (X \to f(u_x) \overset{f(\alpha)}\to f(c))$} is bijective. In particular, if {$\operatorname{Hom}_C(u_X, -) \simeq \operatorname{Hom}_D(X, f(-))$}, 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 {$\operatorname{Hom}_D(X, f(-))$}.

Types of category

A category consists of the following data

A class of objects,
For each pair of objects X, Y, a set {$\operatorname{Hom}(X, Y)$}, whose elements are called morphisms from X to Y,
For each triple of objects X, Y, Z, a map (called composition)
:{$\circ: \operatorname{Hom}(Y, Z) \times \operatorname{Hom}(X, Y) \to \operatorname{Hom}(X, Z), \, (g, f) \mapsto g \circ f$},
For each object X, an identity morphism {$\operatorname{id}_X \in \operatorname{Hom}(X, X)$} subject to the conditions: for any morphisms {$f: X \to Y$}, {$g: Y \to Z$} and {$h: Z \to W$}, {$(h \circ g) \circ f = h \circ (g \circ f)$} and {$\operatorname{id}_Y \circ f = f \circ \operatorname{id}_X = f$}.

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 {$x \to y$} if and only if {$x \le y$}; the associativity of composition means transitivity.
composition A composition of morphisms in a category is part of the datum defining the category.
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.
A small category is a category in which the class of all morphisms is a set (i.e., not a proper class); otherwise large. A category is locally small if the morphisms between every pair of objects A and B form a set. Some authors assume a foundation in which the collection of all classes forms a "conglomerate", in which case a quasicategory is a category whose objects and morphisms merely form a conglomerate.
A category is wellpowered if for each object there is only a set of pairwise non-isomorphic subobjects.
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.
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.
The functor category Fct(C, D) from a category C to a category D is the category where the objects are all the functors from C to D and the morphisms are all the natural transformations between the functors.
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.
A category is isomorphic to another category if there is an isomorphism between them.
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.
A category is equivalent to another category if there is an equivalence between them.
A category A is a subcategory of a category B if there is an inclusion functor from A to B.
A category A is a full subcategory of a category B if the inclusion functor from A to B is full.
A category (or ∞-category) is called pointed if it has a zero object.
A category is skeletal if isomorphic objects are necessarily identical.
A category is finite if it has only finitely many morphisms.
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.
A category is thin (or posetal) if there is at most one morphism between any pair of objects.
A category is balanced if every bimorphism is an isomorphism.
A category is normal if every monomorphism is normal.
A category is called a groupoid if every morphism in it is an isomorphism.
The core of a category is the maximal groupoid contained in the category.
The fundamental groupoid {$\Pi_1 X$} of a Kan complex X is the category where an object is a 0-simplex (vertex) {$\Delta^0 \to X$}, a morphism is a homotopy class of a 1-simplex (path) {$\Delta^1 \to X$} and a composition is determined by the Kan property.
A concrete category C is a category such that there is a faithful functor from C to Set; e.g., Vec, Grp and Top.
A category is closed if it has an internal Hom functor.
A category is cartesian closed if it has a terminal object and that any two objects have a product and exponential.
A category is closed monoidal if it is both monoidal and closed in a compatible way. It is the most general framework which allows currying and uncurrying.
A category is compact closed if it is a monoidal closed category that supports dual objects, as in the case of a finite dimensional vector space.
The product of a family of categories {$C_i$}'s indexed by a set I is the category denoted by {$\prod_i C_i$} whose class of objects is the product of the classes of objects of {$C_i$}'s and whose hom-sets are {$\prod_i \operatorname{Hom}_{\operatorname{C_i}}(X_i, Y_i)$}; the morphisms are composed component-wise. It is the dual of the disjoint union.
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 {$p_X: X \to A$} to {$p_Y: Y \to A$}, then {$p_Y \circ f = p_X$} in C and whose composition is that of C.
Given functors {$f: C \to B, g: D \to B$}, the comma category {$(f \downarrow g)$} is a category where (1) the objects are morphisms {$f(c) \to g(d)$} and (2) a morphism from {$\alpha: f(c) \to g(d)$} to {$\beta: f(c') \to g(d')$} consists of {$c \to c'$} and {$d \to d'$} such that {$f(c) \to f(c') \overset{\beta}\to g(d')$} is {$f(c) \overset{\alpha}\to g(d) \to g(d').$} For example, if f is the identity functor and g is the constant functor with a value b, then it is the slice category of B over an object b.
A 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 {$\varprojlim \operatorname{Hom}(f(j), i)$} is nonempty for some object i in I.
A category is complete if all small limits exist.
A finitely complete category is a category C which admits all finite limits.
A regular category is a finitely complete category which admits a good notion of image factorization. A primary raison d’être behind regular categories C is to have a decently behaved calculus of relations in C.
A coherent category (also called a pre-logos) is a regular category in which the subobject posets Sub(X) all have finite unions which are preserved by the base change functors {$f^*:Sub(Y)\to Sub(X)$}.
A geometric category is a regular category in which the subobject posets Sub(X) have all small unions which are stable under pullback.
Well-powered.
A monoidal category, also called a tensor category, is a category C equipped with (1) a bifunctor {$\otimes: C \times C \to C$}, (2) an identity object and (3) natural isomorphisms that make ⊗ associative and the identity object an identity for ⊗, subject to certain coherence conditions.
tensor category Usually synonymous with monoidal category (though some authors distinguish between the two concepts.)
A braided monoidal category is a monoidal category C equipped with a braiding—that is, a commutativity constraint γ {$A\otimes B\cong B\otimes A$} that satisfies axioms including the hexagon identities.
A symmetric monoidal category is a monoidal category (i.e., a category with ⊗) that has maximally symmetric braiding.
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.
A PROP is a symmetric strict monoidal category whose objects are natural numbers and whose tensor product addition of natural numbers.
A category is abelian if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal.
An abelian category is semisimple if every short exact sequence splits. For example, a ring is semisimple if and only if the category of modules over it is semisimple.
A full subcategory of an abelian category is thick if it is closed under extensions.
A Grothendieck category is a certain well-behaved kind of an abelian category.
A triangulated category is a category where one can talk about distinguished triangles, generalization of exact sequences. An abelian category is a prototypical example of a triangulated category.
The heart of a t-structure ({$D^{\ge 0}$}, {$D^{\le 0}$}) on a triangulated category is the intersection {$D^{\ge 0} \cap D^{\le 0}$}. It is an abelian category.
A derived category is a triangulated category that is not necessary an abelian category.
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.
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.
An exact category is a particular kind of additive category consisting of "short exact sequences".
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.
The homological dimension of an abelian category with enough injectives is the least non-negative 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.
A category is connected if, for each pair of objects x, y, there exists a finite sequence of objects {$z_i$} such that {$z_0 = x, z_n = y$} and either {$\operatorname{Hom}(z_i, z_{i+1})$} or {$\operatorname{Hom}(z_{i+1}, z_i)$} is nonempty for any i.
A differential graded category is a category whose Hom sets are equipped with structures of differential graded modules. In particular, if the category has only one object, it is the same as a differential graded module.
Eilenberg–Moore category. Another name for the category of algebras for a given monad.
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 {$\operatorname{Map}_D(X, Y)$} in C, called the mapping object from X to Y,
For each triple of objects X, Y, Z in D, a morphism in C,
:{$\circ: \operatorname{Map}_D(Y, Z) \otimes \operatorname{Map}_D(X, Y) \to \operatorname{Map}_D(X, Z)$},
:called the composition,
For each object X in D, a morphism {$1_X: 1 \to \operatorname{Map}_D(X, X)$} 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.
simplicial category A category enriched over simplicial sets.
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.
A Fukaya category is a certain kind of category of Lagrangian submanifolds of a symplectic manifold.
Grothendieck fibration A fibered category, which is used for a general framework of descent theory, to discuss vector bundles, principal bundles and sheaves over topological spaces.
site A category equipped with a Grothendieck topology, which makes its objects act like open sets of a topological space.
A Waldhausen category is, roughly, a category C with families of cofibrations and weak equivalences, which makes it possible to calculate the K-spectrum of C.
homotopy category. It is closely related to a localization of a category.
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.
Given a regular cardinal κ, a category is κ-presentable if it admits all small colimits and is κ-accessible. A category is presentable if it is κ-presentable for some regular cardinal κ (hence presentable for any larger cardinal). Note: Some authors call a presentable category a locally presentable category.
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 {$\varprojlim \operatorname{Hom}(f(j), i)$} is nonempty for some object i in I.

Variation of categories

A topology on a category is subcanonical if every representable contravariant functor on C is a sheaf with respect to that topology. Generally speaking, some flat topology may fail to be subcanonical; but flat topologies appearing in practice tend to be subcanonical.
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.

Generalizations of a category

A bicategory is a model of a weak 2-category.
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.
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.
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.
An ∞-category is called an ∞-groupoid if every morphism in it is an equivalence (or equivalently if it is a Kan complex.
1=∞-category An ∞-category C is a simplicial set satisfying the following condition: for each 0 n is the standard n-simplex and {$\Lambda^n_i$} is obtained from Δn by removing the i-th face and the interior (see Kan fibration#Definition). For example, the nerve of a category satisfies the condition and thus can be considered as an ∞-category.
A morphism in an ∞-category C is an equivalence if it gives an isomorphism in the homotopy category of C.
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.
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.
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.
A Dwyer–Kan equivalence is a generalization of an equivalence of categories to the simplicial context.

Types of 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) {$F(f \circ g) = F(f) \circ F(g)$} whenever {$f \circ g$} is defined and (2) {$F(\operatorname{id}_x) = \operatorname{id}_{F(x)}$}.
For example, {$\mathfrak{P}: \mathbf{Set} \to \mathbf{Set}, \, S \mapsto \mathfrak{P}(S)$}, where {$\mathfrak{P}(S)$} is the power set of S is a functor if we define: for each function {$f: S \to T$}, {$\mathfrak{P}(f): \mathfrak{P}(S) \to \mathfrak{P}(T)$} by {$\mathfrak{P}(f)(A) = f(A)$}.
The identity functor on a category C is a functor from C to C that sends objects and morphisms to themselves.
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 {$f: C \to D$} is constant if it factors as: {$C \to \{ A \} \overset{i}\to D$} for some object A in D, where i is the inclusion of the discrete category { A }.
Given categories I, C, the diagonal functor is the functor:{$\Delta: C \to \mathbf{Fct}(I, C), \, A \mapsto \Delta_A$} that sends each object A to the constant functor with value A and each morphism {$f: A \to B$} to the natural transformation {$\Delta_{f, i}: \Delta_A(i) = A \to \Delta_B(i) =B$} that is f at each i.
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.
A functor is said to reflect isomorphisms if it has the property: F(k) is an isomorphism then k is an isomorphism as well.
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.
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.
A functor is faithful if it is injective when restricted to each hom-set.
A functor is full if it is surjective when restricted to each hom-set.
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.
A functor is an equivalence if it is faithful, full and essentially surjective.
Given relative categories {$p: F \to C, q: G \to C$} over the same base category C, a functor {$f: F \to G$} over C is cartesian if it sends cartesian morphisms to cartesian morphisms.
A functor π:C → D is an op-fibration if, for each object x in C and each morphism g : π(x) → y in D, there is at least one π-coCartesian morphism f: x → y in C such that π(f) = g. In other words, π is the dual of a Grothendieck fibration.
endofunctor. A functor between the same category.
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 {$\mathfrak{P}(S)$} be the power set of S and for each function {$f: S \to T$}, define:{$\mathfrak{P}(f): \mathfrak{P}(T) \to \mathfrak{P}(S)$} by sending a subset A of T to the pre-image {$f^{-1}(A)$}. With this, {$\mathfrak{P}: \mathbf{Set} \to \mathbf{Set}$} is a contravariant functor.
Another term for a contravariant functor: a functor from a category Cop to Set is a presheaf of sets on C and a functor from Cop to sSet is a presheaf of simplicial sets or simplicial presheaf, etc. A topology on C, if any, tells which presheaf is a sheaf (with respect to that topology).
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, {$\operatorname{Hom}(-, -)$} is a bifunctor from Cop and C to Set.
A simplicial object in a category C is roughly a sequence of objects {$X_0, X_1, X_2, \dots$} 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.
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 {$X_n = X([n])$} and an element of the set {$X_n$} is called an n-simplex. For example, {$\Delta^n = \operatorname{Hom}_{\Delta}(-, [n])$} is a simplicial set called the standard n-simplex. By Yoneda's lemma, {$X_n \simeq \operatorname{Nat}(\Delta^n, X)$}.
The functor {$y: C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set}), \, X \mapsto \operatorname{Hom}_C(-, X)$} is fully faithful and is called the Yoneda embedding. Technical note: the lemma implicitly involves a choice of Set; i.e., a choice of universe.
A set-valued contravariant functor F on a category C is said to be representable if it belongs to the essential image of the Yoneda embedding {$C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$}; i.e., {$F \simeq \operatorname{Hom}_C(-, Z)$} for some object Z. The object Z is said to be the representing object of F.
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.
Given categories C and D, a profunctor (or a distributor) from C to D is a functor of the form {$D^{\text{op}} \times C \to \mathbf{Set}$}.
distributor. Another term for "profunctor".
If {$f: C \to D, \, g: D \to E$} are functors, then the composition {$g \circ f$} or {$gf$} is the functor defined by: for an object x and a morphism u in C, {$(g \circ f)(x) = g(f(x)), \, (g \circ f)(u) = g(f(u))$}.
Given categories C, D and an object A in C, the evaluation at A is the functor:{$\mathbf{Fct}(C, D) \to D, \,\, F \mapsto F(A).$} For example, the Eilenberg–Steenrod axioms give an instance when the functor is an equivalence.
A (combinatorial) species is an endofunctor on the groupoid of finite sets with bijections. It is categorically equivalent to a symmetric sequence.
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, {{nowrap|F: Hom(V, W) → Hom(F(V), F(W))}} is a polynomial map between the vector spaces. A Schur functor is a basic example.
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:{$\operatorname{Hom}_D (F(X), Y) \simeq \operatorname{Hom}_C (X, G(Y))$};F is said to be left adjoint to G and G to right adjoint to F. Here, "natural" means there is a natural isomorphism {$\operatorname{Hom}_D (F(-), -) \simeq \operatorname{Hom}_C (-, G(-))$} of bifunctors (which are contravariant in the first variable.)
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).
A functor is said to be monadic if it is a constituent of a monadic adjunction.
The forgetful functor is, roughly, a functor that loses some of data of the objects; for example, the functor {$\mathbf{Grp} \to \mathbf{Set}$} that sends a group to its underlying set and a group homomorphism to itself is a forgetful 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).
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<nowiki>'</nowiki> → 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).
Given a monoidal category (C, ⊗), the internal Hom is a functor {$[-, -]: C^{\text{op}} \times C \to C$} such that {$[Y, -]$} is the right adjoint to {$- \otimes Y$} for each object Y in C. For example, the category of modules over a commutative ring R has the internal Hom given as {$[M, N] = \operatorname{Hom}_R(M, N)$}, the set of R-linear maps.
Given a category C, the left Kan extension functor along a functor {$f: I \to J$} is the left adjoint (if it exists) to {$f^* = - \circ f: \operatorname{Fct}(J, C) \to \operatorname{Fct}(I, C)$} and is denoted by {$f_!$}. For any {$\alpha: I \to C$}, the functor {$f_! \alpha: J \to C$} is called the left Kan extension of α along f.reference. One can show: {$(f_! \alpha)(j) = \varinjlim_{f(i) \to j} \alpha(i)$} where the colimit runs over all objects {$f(i) \to j$} in the comma category. The right Kan extension functor is the right adjoint (if it exists) to {$f^*$}.
Given a group or monoid M, the Day convolution is the tensor product in {$\mathbf{Fct}(M, \mathbf{Set})$}. Day convolution is equivalently a left Kan extension.
If {$F: C \to D$} 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.
The nerve functor N is the functor from Cat to sSet given by {$N(C)_n = \operatorname{Hom}_{\mathbf{Cat}}([n], C)$}. For example, if {$\varphi$} is a functor in {$N(C)_2$} (called a 2-simplex), let {$x_i = \varphi(i), \, 0 \le i \le 2$}. Then {$\varphi(0 \to 1)$} is a morphism {$f: x_0 \to x_1$} in C and also {$\varphi(1 \to 2) = g: x_1 \to x_2$} for some g in C. Since {$0 \to 2$} is {$0 \to 1$} followed by {$1 \to 2$} and since {$\varphi$} is a functor, {$\varphi(0 \to 2) = g \circ f$}. In other words, {$\varphi$} encodes f, g and their compositions.
The fundamental category functor {$\tau_1: s\mathbf{Set} \to \mathbf{Cat}$} is the left adjoint to the nerve functor N. For every category C, {$\tau_1 NC = C$}.
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 {$T(X) = G \times X$}. Then define the multiplication μ on T as the natural transformation {$\mu: T \circ T \to T$} given by {$\mu_X: G \times (G \times X) \to G \times X, \,\, (g, (h, x)) \mapsto (gh, x)$} 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 {$F: X \rightleftarrows A : G$} determines a monad in X; namely, one takes {$T = G \circ F$}, the identity map η on T to be a unit of the adjunction and also defines μ using the adjunction.
A finitary monad or an algebraic monad is a monad on Set whose underlying endofunctor commutes with filtered colimits.
A comonad in a category X is a comonid in the monoidal category of endofunctors of X.
Given a k-linear category C over a field k, a Serre functor {$f: C \to C$} is an auto-equivalence such that {$\operatorname{Hom}(A, B) \simeq \operatorname{Hom}(B,f(A))^*$} for any objects A, B.

Generalizations of a functor

The term "lax functor" is essentially synonymous with "pseudo-functor".

Natural transformations

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:{$\{ \phi_x: F(x) \to G(x) \mid x \in \operatorname{Ob}(C) \}$} satisfying the condition: for each morphism f: x → y in C, {$\phi_y \circ F(f) = G(f) \circ \phi_x$}. For example, writing {$GL_n(R)$} for the group of invertible n'by'n matrices with coefficients in a commutative ring R, we can view {$GL_n$} as a functor from the category CRing of commutative rings to the category Grp of groups. Similarly, {$R \mapsto R^*$} is a functor from CRing to Grp. Then the determinant det is a natural transformation from {$GL_n$} to -*.
Natural transformations are composed pointwise: if {$\varphi: f \to g, \, \psi: g \to h$} are natural transformations, then {$\psi \circ \varphi$} is the natural transformation given by {$(\psi \circ \varphi)_x = \psi_x \circ \varphi_x$}.
A natural isomorphism is a natural transformation that is an isomorphism (i.e., admits the inverse).
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.

Types of diagram

Given a category C, a diagram in C is a functor {$f: I \to C$} from a small category I.
Cartesian square A commutative diagram that is isomorphic to the diagram given as a fiber product.
A cone is a way to express the universal property of a colimit (or dually a limit). One can show that the colimit {$\varinjlim$} is the left adjoint to the diagonal functor {$\Delta: C \to \operatorname{Fct}(I, C)$}, which sends an object X to the constant functor with value X; that is, for any X and any functor {$f: I \to C$}, {$\operatorname{Hom}(\varinjlim f, X) \simeq \operatorname{Hom}(f, \Delta_X),$} provided the colimit in question exists. The right-hand side is then the set of cones with vertex X.
The limit (or projective limit) of a functor {$f: I^{\text{op}} \to \mathbf{Set}$} is {$\varprojlim_{i \in I} f(i) = \{ (x_i|i) \in \prod_{i} f(i) | f(s)(x_j) = x_i \text{ for any } s: i \to j \}.$}
The limit {$\varprojlim_{i \in I} f(i)$} of a functor {$f: I^{\text{op}} \to C$} is an object, if any, in C that satisfies: for any object X in C, {$\operatorname{Hom}(X, \varprojlim_{i \in I} f(i)) = \varprojlim_{i \in I} \operatorname{Hom}(X, f(i))$}; i.e., it is an object representing the functor {$X \mapsto \varprojlim_i \operatorname{Hom}(X, f(i)).$}
The colimit (or inductive limit) {$\varinjlim_{i \in I} f(i)$} is the dual of a limit; i.e., given a functor {$f: I \to C$}, it satisfies: for any X, {$\operatorname{Hom}(\varinjlim f(i), X) = \varprojlim \operatorname{Hom}(f(i), X)$}. Explicitly, to give {$\varinjlim f(i) \to X$} is to give a family of morphisms {$f(i) \to X$} such that for any {$i \to j$}, {$f(i) \to X$} is {$f(i) \to f(j) \to X$}. Perhaps the simplest example of a colimit is a coequalizer. For another example, take f to be the identity functor on C and suppose {$L = \varinjlim_{X \in C} f(X)$} exists; then the identity morphism on L corresponds to a compatible family of morphisms {$\alpha_X: X \to L$} such that {$\alpha_L$} is the identity. If {$f: X \to L$} is any morphism, then {$f = \alpha_L \circ f = \alpha_X$}; i.e., L is a final object of C.
ind-limit A colimit (or inductive limit) in {$\mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$}.
The equalizer of a pair of morphisms {$f, g: A \to B$} is the limit of the pair. It is the dual of a coequalizer.
The coequalizer of a pair of morphisms {$f, g: A \to B$} is the colimit of the pair. It is the dual of an equalizer.
The image of a morphism f: X → Y is the equalizer of {$Y \rightrightarrows Y \sqcup_X Y$}.
The coimage of a morphism f: X → Y is the coequalizer of {$X \times_Y X \rightrightarrows X$}.
If f : X → Y is an arbitrary morphism in C, then a kernel of f is an equaliser of f and the zero morphism from X to Y.
The dual concept to that of kernel is that of cokernel. The cokernel of a morphism is its kernel in the opposite category.
The product of a family of objects {$X_i$} in a category C indexed by a set I is the projective limit {$\varprojlim$} of the functor {$I \to C, \, i \mapsto X_i$}, where I is viewed as a discrete category. It is denoted by {$\prod_i X_i$} and is the dual of the coproduct of the family.
The coproduct of a family of objects {$X_i$} in a category C indexed by a set I is the inductive limit {$\varinjlim$} of the functor {$I \to C, \, i \mapsto X_i$}, 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.
Given a monoidal category B, the tensor product of functors {$F: C^{\text{op}} \to B$} and {$G: C \to B$} is the coend: {$F \otimes_C G = \int^{c \in C} F(c) \otimes G(c).$}
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 {$C_{/S}$} of C over S (provided there are {$X_i \to S$}). The fiber product of two objects X and Y over an object S is denoted by {$X \times_S Y$} and is also called a Cartesian square.
The direct limit of algebraic objects is a colimit.
The direct limit in an arbitrary category is a colimit.
The inverse limit of algebraic objects is a limit.
The inverse limit in an arbitrary category is a limit.
The end of a functor {$F: C^{\text{op}} \times C \to X$} is the limit {$\int_{c \in C} F(c, c) = \varprojlim (F^{\#}: C^{\#} \to X)$} where {$C^{\#}$} is the category (called the subdivision category of C) whose objects are symbols {$c^{\#}, u^{\#}$} for all objects c and all morphisms u in C and whose morphisms are {$b^{\#} \to u^{\#}$} and {$u^{\#} \to c^{\#}$} if {$u: b \to c$} and where {$F^{\#}$} is induced by F so that {$c^{\#}$} would go to {$F(c, c)$} and {$u^{\#}, u: b \to c$} would go to {$F(b, c)$}. For example, for functors {$F, G: C \to X$}, {$\int_{c \in C} \operatorname{Hom}(F(c), G(c))$} 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.
The coend of a functor {$F: C^{\text{op}} \times C \to X$} is the dual of the end of F and is denoted by {$\int^{c \in C} F(c, c)$}. 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 {$M \otimes_R N = \int^{R} M \otimes_{\mathbb{Z}} N$} where R is viewed as a category with one object whose morphisms are the elements of R.

Spaces

The classifying space of a category C is the geometric realization of the nerve of C.
Segal spaces were certain simplicial spaces, introduced as models for (∞, 1)-categories.

Theorems

Beck's theorem characterizes the category of algebras for a given monad.
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.
The homotopy hypothesis states an ∞-groupoid is a space (less equivocally, an n-groupoid can be used as a homotopy n-type.)
Quillen’s theorem A provides a criterion for a functor to be a weak equivalence.
The Gabriel–Popescu theorem says an abelian category is a quotient of the category of modules.
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.
The Yoneda lemma says: for each set-valued contravariant functor F on C and an object X in C, there is a natural bijection {$F(X) \simeq \operatorname{Nat}(\operatorname{Hom}_C(-, X), F)$} where Nat means the set of natural transformations.

Techniques

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".
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.
Grothendieck construction Given a functor {$U: C \to \mathbf{Cat}$}, 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.
localization of a category
Bousfield localization
Simplicial localization is a method of localizing a category.

Dualities

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.
The Tannakian duality states that, in an appropriate setup, to give a morphism {$f: X \to Y$} is to give a pullback functor {$f^*$} along it. In other words, the Hom set {$\operatorname{Hom}(X, Y)$} can be identified with the functor category {$\operatorname{Fct}(D(Y), D(X))$}, perhaps in the derived sense, where {$D(X)$} is the category associated to X (e.g., the derived category).

Areas of math

Categorical logic is an approach to mathematical logic that uses category theory.
Grothendieck's Galois theory A category-theoretic generalization of Galois theory.
Higher category theory is a subfield of category theory that concerns the study of n-categories and ∞-categories.