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.^{[1]}Like those expositions, this glossary also generally ignores the set-theoretic issues, except when they are relevant (e.g., the discussion on accessibility.)

Especially for higher categories, the concepts from algebraic topology are also used in the category theory. For that see also glossary of algebraic topology.

The notations and the conventions used throughout the article are:

- [
*n*] = {0, 1, 2, …,*n*}, which is viewed as a category (by writing .) **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.*s***Set**, 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

- 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 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.^{[2]} - 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- ;

*F*is said to be left adjoint to*G*and*G*to right adjoint to*F*. Here, "natural" means there is a natural isomorphism 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 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*, is a bifunctor from*C*^{op}and*C*to**Set**. - 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 over the same base category
*C*, a functor 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.
- categorification
- Categorification is a process of replacing sets and set-theoretic concepts with categories and category-theoretic concepts in some nontrivial way to capture categoric flavors. Decategorification is the reverse of categorification.
- category
- A category consists of the following data
- A class of objects,
- For each pair of objects
*X*,*Y*, a set , whose elements are called morphisms from*X*to*Y*, - For each triple of objects
*X*,*Y*,*Z*, a map (called composition)- ,

- For each object
*X*, an identity morphism

- and .

*x*,*y*, there is a unique morphism if and only if ; the associativity of composition means transitivity. - category of categories
- 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. - 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.
- coend
- The coend of a functor is the dual of the end of
*F*and is denoted by- .

*R*is a ring,*M*a right*R*-module and*N*a left*R*-module, then the tensor product of*M*and*N*is*R*is viewed as a category with one object whose morphisms are the elements of*R*. - coequalizer
- The coequalizer of a pair of morphisms 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.
- coimage
- The coimage of a morphism
*f*:*X*→*Y*is the coequalizer of . - 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 , the comma category is a category where (1) the objects are morphisms and (2) a morphism from to consists of and such that is 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.
- composition
- 1. A composition of morphisms in a category is part of the datum defining the category.
- 2. If are functors, then the composition or is the functor defined by: for an object
*x*and a morphism*u*in*C*, . - 3. Natural transformations are composed pointwise: if are natural transformations, then is the natural transformation given by .
- 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
^{[3]}that the colimit is the left adjoint to the diagonal functor , which sends an object*X*to the constant functor with value*X*; that is, for any*X*and any functor ,*X*.^{[4]} - connected
- A category is connected if, for each pair of objects
*x*,*y*, there exists a finite sequence of objects*z*_{i}such that and either or 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 is constant if it factors as: 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*^{op}to*D*. It is sometimes also called a presheaf especially when*D*is**Set**or the variants. For example, for each set*S*, let be the power set of*S*and for each function , define*A*of*T*to the pre-image . With this, 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 of the functor , 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 .^{[5]} - 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*A*to the constant functor with value*A*and each morphism to the natural transformation that is*f*at each*i*. - diagram
- Given a category
*C*, a diagram in*C*is a functor 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.
^{[6]}

## E

- 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 is the limit
*C*) whose objects are symbols for all objects*c*and all morphisms*u*in*C*and whose morphisms are and if and where is induced by*F*so that would go to and would go to . For example, for functors ,*F*to*G*. For more examples, see this mathoverflow thread. The dual of an end is a coend. - endofunctor
- A functor between the same category.
- enriched category
- Given a monoidal category (
*C*, ⊗, 1), a category enriched over*C*is, informally, a category whose Hom sets are in*C*. More precisely, a category*D*enriched over*C*is a data consisting of- A class of objects,
- For each pair of objects
*X*,*Y*in*D*, an object in*C*, called the mapping object from*X*to*Y*, - For each triple of objects
*X*,*Y*,*Z*in*D*, a morphism in*C*,- ,
- called the composition,

- For each object
*X*in*D*, a morphism in*C*, called the unit morphism of*X*

- epimorphism
- A morphism
*f*is an epimorphism if whenever . In other words,*f*is the dual of a monomorphism. - equalizer
- The equalizer of a pair of morphisms is the limit of the pair. It is the dual of a coequalizer.
- equivalence
- 1. A functor is an equivalence if it is faithful, full and essentially surjective.
- 2. A morphism in an ∞-category
*C*is an equivalence if it gives an isomorphism in the homotopy category of*C*. - equivalent
- A category is equivalent to another category if there is an equivalence between them.
- essentially surjective
- A functor
*F*is called essentially surjective (or isomorphism-dense) if for every object*B*there exists an object*A*such that*F*(*A*) is isomorphic to*B*. - evaluation
- Given categories
*C*,*D*and an object*A*in*C*, the evaluation at*A*is the functor

## F

- faithful
- A functor is faithful if it is injective when restricted to each hom-set.
- fundamental category
- The fundamental category functor is the left adjoint to the nerve functor
*N*. For every category*C*, . - fundamental groupoid
- The fundamental groupoid of a Kan complex
*X*is the category where an object is a 0-simplex (vertex) , a morphism is a homotopy class of a 1-simplex (path) 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 of*C*over*S*(provided there are ). The fiber product of two objects*X*and*Y*over an object*S*is denoted by 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 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 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 that sends a group to its underlying set and a group homomorphism to itself is a forgetful functor.
- 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) whenever is defined and (2) . For example,- ,

*S*is a functor if we define: for each function , by . - functor category
- The functor category
**Fct**(*C*,*D*) or 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.
- generator
- In a category
*C*, a family of objects is a system of generators of*C*if the functor is conservative. Its dual is called a system of cogenerators. - 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 , 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 (, ) on a triangulated category is the intersection . 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

- 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*, and . - 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 . - ind-limit
- A colimit (or inductive limit) in .
- inductive limit
- Another name for colimit.
- ∞-category
- An ∞-category
*C*is a simplicial set satisfying the following condition: for each 0 <*i*<*n*,- every map of simplicial sets extends to an
*n*-simplex

^{n}is the standard*n*-simplex and is obtained from Δ^{n}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. - every map of simplicial sets extends to an
- 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 is contractible for each object*B*in*C*. - injective
- 1. An object
*A*in an abelian category is injective if the functor 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 such that is the right adjoint to for each object*Y*in*C*. For example, the category of modules over a commutative ring*R*has the internal Hom given as , the set of*R*-linear maps. - inverse
- 1. A morphism
*f*is an inverse to a morphism*g*if is defined and is equal to the identity morphism on the codomain of*g*, and 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 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.
- 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 is the left adjoint (if it exists) to and is denoted by . For any , the functor is called the left Kan extension of α along*f*.^{[7]}One can show: - 2. The right Kan extension functor is the right adjoint (if it exists) to .
- 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
- The term "lax functor" is essentially synonymous with "pseudo-functor".
- 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*.^{[8]} - limit
- 1. The limit (or projective limit) of a functor is

*C*that satisfies: for any object

*X*in

*C*, ; i.e., it is an object representing the functor

*X*, . Explicitly, to give is to give a family of morphisms such that for any , is . Perhaps the simplest example of a colimit is a coequalizer. For another example, take

*f*to be the identity functor on

*C*and suppose exists; then the identity morphism on

*L*corresponds to a compatible family of morphisms such that is the identity. If is any morphism, then ; i.e.,

*L*is a final object of

*C*.

## M

- Mittag-Leffler condition
- An inverse system is said to satisfy the Mittag-Leffler condition if for each integer , there is an integer such that for each , the images of and 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 . Then define the multiplication*μ*on*T*as the natural transformation given by*η*in the analogous fashion. Then (*T*,*μ*,*η*) constitutes a monad in**Set**. More substantially, an adjunction between functors determines a monad in*X*; namely, one takes , 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 , (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 inis an associative algebra over a commutative ring*R*-mod*R*. - monomorphism
- A morphism
*f*is a monomorphism (also called monic) if whenever ; 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.
^{[9]}

## N

*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**f*:*x*→*y*in*C*, . For example, writing for the group of invertible*n*-by-*n*matrices with coefficients in a commutative ring*R*, we can view as a functor from the category**CRing**of commutative rings to the category**Grp**of groups. Similarly, is a functor from**CRing**to**Grp**. Then the determinant det is a natural transformation from 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*s***Set**given by . For example, if is a functor in (called a 2-simplex), let . Then is a morphism in*C*and also for some*g*in*C*. Since is followed by and since is a functor, . In other words, 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.

[T]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

## O

- object
- 1. An object is part of a data defining a category.
- 2. 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. - op-fibration
- 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. - 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

- perfect
- Sometimes synonymous with "compact". See perfect complex.
- pointed
- A category (or ∞-category) is called pointed if it has a zero object.
- polynomial
- 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*,*F*: Hom(*V*,*W*) → Hom(*F*(*V*),*F*(*W*)) is a polynomial map between the vector spaces. A Schur functor is a basic example. - preadditive
- 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. - presentable
- 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. - presheaf
- Another term for a contravariant functor: a functor from a category
*C*^{op}to**Set**is a presheaf of sets on*C*and a functor from*C*^{op}to*s***Set**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
*X*_{i}in a category*C*indexed by a set*I*is the projective limit of the functor , where*I*is viewed as a discrete category. It is denoted by and is the dual of the coproduct of the family. - 2. The product of a family of categories
*C*_{i}'s indexed by a set*I*is the category denoted by whose class of objects is the product of the classes of objects of*C*_{i}'s and whose hom-sets are ; the morphisms are composed component-wise. It is the dual of the disjoint union. - profunctor
- Given categories
*C*and*D*, a profunctor (or a distributor) from*C*to*D*is a functor of the form . - projective
- 1. An object
*A*in an abelian category is projective if the functor is exact. It is the dual of an injective object. - 2. The term “projective limit” is another name for an inverse limit.
- PROP
- A PROP is a symmetric strict monoidal category whose objects are natural numbers and whose tensor product addition of natural numbers.
- pseudoalgebra
- A pseudoalgebra is a 2-category-version of an algebra for a monad (with a monad replaced by a 2-monad).

## Q

- Quillen
- Quillen’s theorem A provides a criterion for a functor to be a weak equivalence.

## 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 isomorphisms if 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 ; i.e., for some object*Z*. The object*Z*is said to be the representing object of*F*. - retraction
- 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.
- semisimple
- 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.
- Serre functor
- Given a
*k*-linear category*C*over a field*k*, a Serre functor is an auto-equivalence such that for any objects*A*,*B*. - simple 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. - simplex category
- 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. - simplicial category
- A category enriched over simplicial sets.
- Simplicial localization
- Simplicial localization is a method of localizing a category.
- simplicial object
- A simplicial object in a category
*C*is roughly a sequence of objects 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. - simplicial set
- 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 and an element of the set is called an*n*-simplex. For example, is a simplicial set called the standard*n*-simplex. By Yoneda's lemma, . - site
- A category equipped with a Grothendieck topology.
- skeletal
- 1. A category is skeletal if isomorphic objects are necessarily identical.
- 2. A (not unique) skeleton of a category is a full subcategory that is skeletal.
- 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 to , then in*C*and whose composition is that of*C*. - small
- 1. 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.^{[10]}(NB: some authors use the term "quasicategory" with a different meaning.^{[11]}) - 2. 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 (cf. https://ncatlab.org/nlab/show/small+object+argument)
- species
- A (combinatorial) species is an endofunctor on the groupoid of finite sets with bijections. It is categorically equivalent to a symmetric sequence.
- 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 is an isomorphism. - strict
*n*-category - 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. - subcanonical
- A topology on a category is subcanonical if every representable contravariant functor on
*C*is a sheaf with respect to that topology.^{[12]}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
- 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*. - subquotient
- A subquotient is a quotient of a subobject.
- subterminal object
- A subterminal object is an object
*X*such that every object has at most one morphism into*X*. - symmetric monoidal category
- A symmetric monoidal category is a monoidal category (i.e., a category with ⊗) that has maximally symmetric braiding.
- symmetric sequence
- A symmetric sequence is a sequence of objects with actions of symmetric groups. It is categorically equivalent to a (combinatorial) species.

## T

- t-structure
- 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.
- Tannakian duality
- The Tannakian duality states that, in an appropriate setup, to give a morphism is to give a pullback functor along it. In other words, the Hom set can be identified with the functor category , perhaps in the derived sense, where is the category associated to
*X*(e.g., the derived category).^{[13]}^{[14]} - tensor category
- Usually synonymous with monoidal category (though some authors distinguish between the two concepts.)
- tensor triangulated category
- 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.
- tensor product
- Given a monoidal category
*B*, the**tensor product of functors**and is the coend:

*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*in an ∞-category

*C*is terminal if is contractible for every object

*B*in

*C*.

## U

- universal
- 1. Given a functor and an object
*X*in*D*, a universal morphism from*X*to*f*is an initial object in the comma category . (Its dual is also called a universal morphism.) For example, take*f*to be the forgetful functor and*X*a set. An initial object of is a function . That it is initial means that if is another morphism, then there is a unique morphism from*j*to*k*, which consists of a linear map that extends*k*via*j*; that is to say, is the free vector space generated by*X*. - 2. Stated more explicitly, given
*f*as above, a morphism in*D*is universal if and only if the natural map*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 .

## 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
- 1. The Yoneda lemma says: for each set-valued contravariant functorYoneda’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.
*F*on*C*and an object*X*in*C*, there is a natural bijectionwhere Nat means the set of natural transformations. In particular, the functor

^{[15]} - 2. If 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*.

## Z

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

## Notes

**^**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.**^**Remark 2.7. of https://ncatlab.org/nlab/show/additive+category**^**Kashiwara & Schapira 2006, Ch. 2, Exercise 2.8.**^**Mac Lane 1998, Ch. III, § 3..**^**http://ncatlab.org/nlab/show/Day+convolution**^**Hinich, V. (2013-11-17). "Dwyer-Kan localization revisited". arXiv:1311.4128 [math.QA].**^**http://www.math.harvard.edu/~lurie/282ynotes/LectureXI-Homological.pdf**^**Kashiwara & Schapira 2006, exercise 8.20**^**https://ncatlab.org/nlab/show/multicategory**^**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.**^**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.**^**Vistoli 2004, Definition 2.57.**^**Jacob Lurie. Tannaka duality for geometric stacks. http://math.harvard.edu/~lurie/, 2004.**^**Bhatt, Bhargav (2014-04-29). "Algebraization and Tannaka duality". arXiv:1404.7483 [math.AG].**^**Technical note: the lemma implicitly involves a choice of**Set**; i.e., a choice of universe.

## References

- Artin, Michael (1972). Alexandre Grothendieck; Jean-Louis Verdier (eds.).
*Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 1*. Lecture Notes in Mathematics (in French).**269**. Berlin; New York: Springer-Verlag. xix+525. doi:10.1007/BFb0081551. ISBN 978-3-540-05896-0. - Kashiwara, Masaki; Schapira, Pierre (2006).
*Categories and sheaves*. - A. Joyal, The theory of quasi-categories II (Volume I is missing??)
- Lurie, J.,
*Higher Algebra* - Lurie, J.,
*Higher Topos Theory* - Mac Lane, Saunders (1998).
*Categories for the Working Mathematician*. Graduate Texts in Mathematics.**5**(2nd ed.). New York, NY: Springer-Verlag. ISBN 0-387-98403-8. Zbl 0906.18001. - Pedicchio, Maria Cristina; Tholen, Walter, eds. (2004).
*Categorical foundations. Special topics in order, topology, algebra, and sheaf theory*. Encyclopedia of Mathematics and Its Applications.**97**. Cambridge: Cambridge University Press. ISBN 0-521-83414-7. Zbl 1034.18001. - Vistoli, Angelo (2004-12-28). "Notes on Grothendieck topologies, fibered categories and descent theory". arXiv:math/0412512.

## Further reading

- Groth, M., A Short Course on ∞-categories
- Cisinski's notes
- History of topos theory
- http://plato.stanford.edu/entries/category-theory/
- Leinster, Tom (2014).
*Basic Category Theory*. Cambridge Studies in Advanced Mathematics.**143**. Cambridge University Press. arXiv:1612.09375. Bibcode:2016arXiv161209375L. - Emily Riehl, A leisurely introduction to simplicial sets
- Categorical Logic lecture notes by Steve Awodey
- Street, Ross (20 Mar 2003). "Categorical and combinatorial aspects of descent theory". arXiv:math/0303175. (a detailed discussion of a 2-category)