Glossary of category theory

From Wikipedia, the free encyclopedia
  (Redirected from Locally small)
Jump to: navigation, search

This is a glossary of properties and concepts in category theory in mathematics.[1]

The notations used throughout the article are:


A category is abelian if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal.
A category is additive if it is preadditive and admits all finitary biproducts.
An adjunction (also called an adjoint pair) is a pair of functors F: CD, G: DC 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.)
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 category is balanced if every bimorphism is an isomorphism.
A bifunctor from a pair of categories C and D to a category E is a functor C × DE. For example, for any category C, \operatorname{Hom}(-, -) is a bifunctor from Cop and C to Set.
A bimorphism is a morphism that is both an epimorphism and a monomorphism.


cartesian closed
A category is cartesian closed if it has a terminal object and that any two objects have a product and exponential.
Cartesian square
A commutative diagram that is isomorphic to the diagram given as a fiber product.
A category consists of the following data
  1. A class of objects,
  2. For each pair of objects X, Y, a set \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)
    \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 \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.
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 coequalizer of a pair of morphisms f, g: A \to B is the colimit of the pair. It is the dual of an equalizer.
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 category is complete if all small limits exist.
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 cone is a way to express the universal property of a colimit (or dually a limit). One can show[2] 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.[3]
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 }.
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 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.
The coproduct of a family of objects Xi 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.


diagonal functor
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 category is discrete if each morphism is an identity morphism (of some object). For example, a set can be viewed as a discrete 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 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.
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.
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.
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.


A functor is faithful if it is injective when restricted to each hom-set.
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} 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.
forgetful functor
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.
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 a function f: S \to T, \mathfrak{P}(f): \mathfrak{P}(S) \to \mathfrak{P}(T) by \mathfrak{P}(f)(A) = f(A).
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.


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.
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.)


homological dimension
The homological dimension of an abelian category with enough injectives is the least non-negative intege n such that every object in the category admits an injective resolution of length at most n. The dimension is ∞ if no such integer exists. For example, the homological dimension of ModR with a principal ideal domain R is at most one.
homotopy category
See homotopy category. It is closely related to a localization of a category.


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, g\circ f=g and 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.
A colimit (or inductive limit) in \mathbf{Fct}(C^{\text{op}}, \mathbf{Set}).
An ∞-category C is a simplicial set satisfying the following condition: for each 0 < i < n,
  • every map of simplicial sets f: \Lambda^n_i \to C extends to an n-simplex f: \Delta^n \to C
where Δn 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.
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 \operatorname{Map}_C(A, B) is contractible for each object B in C.
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.
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. 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.
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.
A morphism f is an isomorphism if there exists an inverse of 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.[4]
1.  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 \}.
2.  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)).
3.  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.
localization of a category
See localization of a category.


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.


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
\{ \phi_x: F(x) \to G(x) | x \in \operatorname{Ob}(C) \}
satisfying the condition: for each morphism f: xy 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 -*.
2.  A natural isomorphism is a natural transformation that is an isomorphism (i.e., admits the inverse).
The composition is encoded as a 2-simplex.
The nerve functor N is the functor from Cat to sSet given by N(C)_n = \operatorname{Hom}_{\mathbf{Cat}}([n], C).
A category is normal if every monic is normal.[citation needed]


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.


Given a cardinal number π, an object X in a category is π-accessible if \operatorname{Hom}(X, -) commutes with π-filtrant inductive limits.
An ∞-category is pointed if it has a zero object.
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.
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).
1.  The product of a family of objects Xi 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.
2.  The product of a family of categories Ci'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 Ci'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.
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.


1.  A functor is said to reflect identities if it has the property: if F(k) is an identity then k is an identity as well.
2.  A functor is said to reflect isomorphismsif it has the property: F(k) is an isomorphism then k is an isomorphism as well.
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.
f is a retraction of g. g is a section of f.
A morphism is a retraction if it has 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.
Segal space
Segal spaces were certain simplicial spaces, introduced as models for (∞, 1)-categories.
An object in an abelian category is simple if it is not isomorphic to the zero object and any subobject of A is isomorphic to zero or to A.
Simplicial localization
Simplicial localization is a method of localizing a category.
simplicial set
A simplicial set is a contravariant functor from Δ to Set, where Δ is the category whose objects are the sets [n] = { 0, 1, …, n } and whose morphisms are order-preserving functions.
A category is skeletal if isomorphic objects are necessarily identical.
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.
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.[5] (NB other authors use the term "quasicategory" with a different meaning.[6]
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.
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.
A topology on a category is subcanonical if every representable contravariant functor on C is a sheaf with respect to that topology.[7] Generally speaking, some flat topology may fail to be subcanonical; but flat topologies appearing in practice tend to be subcanonical.
A category A is a subcategory of a category B if there is an inclusion functor from A to B.
See subobject. For example, a subgroup is a subobject of a group.
A subquotient is a quotient of a subobject.
symmetric monoidal category
A symmetric monoidal category is a monoidal category (i.e., a category with ⊗) that has maximally symmetric braiding.


tensor category
Usually synonymous with monoidal category (though some authors distinguish between the two concepts.)
1.  An object A is terminal (also called final) if there is exactly one morphism from each object to A; e.g., singletons in Set. It is the dual of an initial object.
2.  An object A in an ∞-category C is terminal if \operatorname{Map}_C(B, A) is contractible for every object B in C.
A thin is a category where there is at most one morphism between any pair of objects.


1.  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.
2.  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 uX 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(-)).


Waldhausen category
A Waldhausen category is, roughly, a category with families of cofibrations and weak equivalences.
A category is wellpowered if for each object there is only a set of pairwise non-isomorphic subobjects.


Yoneda lemma
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{Hom}(\operatorname{Hom}_C(-, X), F);
in particular, the functor
C \to \mathbf{Fct}(C^{\text{op}}, \mathbf{Set}), \, X \mapsto \operatorname{Hom}_C(-, X)
is fully faithful.[8]


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


  1. ^ 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. If one believes in the existence of strongly inaccessible cardinals, then there can be a rigorous theory where statements and constructions have references to Grothendieck universes; this approach is taken, for example, in Lurie's Higher Topos Theory.
  2. ^ Kashiwara–Schapira 2006, Ch. 2, Exercise 2.8.
  3. ^ Mac Lane 1998, Ch. III, § 3..
  4. ^ Kashiwara & Schapira 2006, exercise 8.20
  5. ^ Adámek, Jiří; Herrlich, Horst; Strecker, George E (2004) [1990]. Abstract and Concrete Categories (The Joy of Cats) (PDF). New York: Wiley & Sons. p. 40. ISBN 0-471-60922-6. 
  6. ^ Joyal, A. (2002). "Quasi-categories and Kan complexes". Journal of Pure and Applied Algebra 175 (1-3): 207–222. doi:10.1016/S0022-4049(02)00135-4. 
  7. ^ Vistoli, Definition 2.57.
  8. ^ Technical note: the lemma implicitly involves a choice of Set; i.e., a choice of universe.


Further reading[edit]