# Glossary of category theory

(Redirected from Locally small)

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

The notations 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 natural transformations.
• 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.

## 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.
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.)
amnestic
A functor is amnestic if it has the property: if k is an isomorphism and F(k) is an identity, then k is an identity.

## B

balanced
A category is balanced if every bimorphism is an isomorphism.
bifunctor
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.
bimorphism
A bimorphism is a morphism that is both an epimorphism and a monomorphism.

## C

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.
category
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.
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.
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.
comma
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.
complete
A category is complete if all small limits exist.
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[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]
constant
A functor is constant if it maps every object in a category to the same object A and every morphism to the identity on A. Put in another way, a functor $f: C \to D$ 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.
coproduct
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.

## D

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.
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.

## E

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.
epimorphism
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.
equalizer
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.
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.

## F

faithful
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.
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 $\mathfrak{P}(f)(A) = f(A)$.
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.

## G

generator
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.
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

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.

## I

identity
1.  The identity morphism f of an object A is a morphism from A to A such that for any morphisms g with domain A and h with codomain A, $g\circ f=g$ 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.
ind-limit
A colimit (or inductive limit) in $\mathbf{Fct}(C^{\text{op}}, \mathbf{Set})$.
∞-category
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.
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 $\operatorname{Map}_C(A, B)$ is contractible for each object B in C.
injective
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.
inverse
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.
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.

## L

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.[4]
limit
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.

## M

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.

## N

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
$\{ \phi_x: F(x) \to G(x) | x \in \operatorname{Ob}(C) \}$
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.
nerve
The nerve functor N is the functor from Cat to sSet given by $N(C)_n = \operatorname{Hom}_{\mathbf{Cat}}([n], C)$.
normal
A category is normal if every monic is normal.[citation needed]

## O

opposite
The opposite category of a category is obtained by reversing the arrows. For example, if a partially ordered set is viewed as a category, taking its opposite amounts to reversing the ordering.

## P

π-accessible
Given a cardinal number π, an object X in a category is π-accessible if $\operatorname{Hom}(X, -)$ commutes with π-filtrant inductive limits.
pointed
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.
presheaf
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).
product
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.
projective
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.

## R

reflect
1.  A functor is said to reflect identities if it has the property: if F(k) is an identity then k is an identity as well.
2.  A functor is said to reflect isomorphismsif it has the property: F(k) is an isomorphism then k is an isomorphism as well.
representable
A set-valued contravariant functor F on a category C is said to be 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.
retraction
f is a retraction of g. g is a section of f.
A morphism is a retraction if it has a right inverse.

## S

section
A morphism is a section if it has a left inverse. For example, the axiom of choice says that any surjective function admits a section.
Segal space
Segal spaces were certain simplicial spaces, introduced as models for (∞, 1)-categories.
simple
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.
skeletal
A category is skeletal if isomorphic objects are necessarily identical.
slice
Given a category C and an object A in it, the slice category C/A of C over A is the category whose objects are all the morphisms in C with codomain A, whose morphisms are morphisms in C such that if f is a morphism from $p_X: X \to A$ to $p_Y: Y \to A$, then $p_Y \circ f = p_X$ in C and whose composition is that of C.
small
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]
stable
An ∞-category is stable if (1) it has a zero object, (2) every morphism in it admits a fiber and a cofiber and (3) a triangle in it is a fiber sequence if and only if it is a cofiber sequence.
strict
A morphism f in a category admitting finite limits and finite colimits is strict if the natural morphism $\operatorname{Coim}(f) \to \operatorname{Im}(f)$ is an isomorphism.
subcanonical
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.
subcategory
A category A is a subcategory of a category B if there is an inclusion functor from A to B.
subobject
See subobject. For example, a subgroup is a subobject of a group.
subquotient
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.

## T

tensor category
Usually synonymous with monoidal category (though some authors distinguish between the two concepts.)
terminal
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.
thin
A thin is a category where there is at most one morphism between any pair of objects.

## U

universal
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 $\operatorname{Hom}_D(X, f(-))$.

## W

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

## Y

Yoneda 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]

## Z

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

## Notes

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.