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 settheoretic 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 settheoretic 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.
 sSet, the category of simplicial sets.
 "weak" instead of "strict" is given the default status; e.g., "ncategory" means "weak ncategory", not the strict one, by default.
 By an ∞category, we mean a quasicategory, the most popular model, unless other models are being discussed.
 The number zero 0 is a natural number.
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Category Theory Foundations, Lecture 1

Category Theory Lulz  Ken Scambler

Abstract Algebra: The definition of a Group

Sets, logic and graphs  Math Terminology  NJ Wildberger

(Abstract Algebra 1) Definition of a Group
Transcription
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 κaccessible (or κcompact or κpresentable) if commutes with κfiltered colimits.
 2. Given a regular cardinal κ, a category is κaccessible if it has κfiltered colimits and there exists a small set S of κcompact 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 preadditive 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
 ;
 algebra for a monad
 Given a monad T in a category X, an algebra for T or a Talgebra is an object in X with a monoid action of T ("algebra" is misleading and "Tobject" is perhaps a better term.) For example, given a group G that determines a monad T in Set in the standard way, a Talgebra 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 2category.
 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.
 bimonoidal
 A bimonoidal category is a category with two monoidal structures, one distributing over the other.
 bimorphism
 A bimorphism is a morphism that is both an epimorphism and a monomorphism.
 Bousfield localization
 See Bousfield localization.
C
 calculus of functors
 The calculus of functors is a technique of studying functors in the manner similar to the way a function is studied via its Taylor series expansion; whence, the term "calculus".
 cartesian closed
 A category is cartesian closed if it has a terminal object and that any two objects have a product and exponential.
 cartesian functor
 Given relative categories 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 πcartesian 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 πcoCartesian 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 πcartesian 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 settheoretic concepts with categories and categorytheoretic 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 .
 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 oplimit in the sense that it is a limit in the opposite category. But there might be a distinction; for example, an opfibration 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
 .
 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 ,
 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
 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 setvalued 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
 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
 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 isomorphismdense) 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 homset.
 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 0simplex (vertex) , a morphism is a homotopy class of a 1simplex (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 πcartesian 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 πfiltrant 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 Rmodule 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 homset.
 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 structurepreserving 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,
 ,
 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.
 Galois category
 1. In SGA 1, Exposé V (Definition 5.1.), a category is called a Galois category if it is equivalent to the category of finite Gsets for some profinite group G.
 2. For technical reasons, some authors (e.g., Stacks project^{[7]} or ^{[8]}) use slightly different definitions.
 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 categorytheoretic generalization of Galois theory; see Grothendieck's Galois theory.
 Grothendieck category
 A Grothendieck category is a certain wellbehaved 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 tstructure (, ) 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 ncategories and ∞categories.
 homological dimension
 The homological dimension of an abelian category with enough injectives is the least nonnegativer 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 ngroupoid can be used as a homotopy ntype.)
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 .
 indlimit
 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 nsimplex
 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 Rlinear 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.^{[9]} 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 Talgebras (called Eilenberg–Moore category) that consists of free Talgebras.
L
 lax
 A lax functor is a generalisation of a pseudofunctor, in which the structural transformations associated to composition and identities are not required to be invertible.
 length
 An object in an abelian category is said to have finite length if it has a composition series. The maximum number of proper subobjects in any such composition series is called the length of A.^{[10]}
 limit
 1. The limit (or projective limit) of a functor is
M
 MittagLeffler condition
 An inverse system is said to satisfy the MittagLeffler 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
 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 in Rmod is an associative algebra over a commutative ring 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.^{[11]}
N
 ncategory
 1. A strict ncategory is defined inductively: a strict 0category is a set and a strict ncategory is a category whose Hom sets are strict (n1)categories. Precisely, a strict ncategory is a category enriched over strict (n1)categories. For example, a strict 1category is an ordinary category.
 2. The notion of a weak ncategory 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 ncategories. Conversely, if one has the notion of a (weak) ∞category (say a quasicategory) in the beginning, then a weak ncategory 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
 2. A natural isomorphism is a natural transformation that is an isomorphism (i.e., admits the inverse).
 nerve
 The nerve functor N is the functor from Cat to sSet given by . For example, if is a functor in (called a 2simplex), 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 ncategory 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 ncategories 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 ncategories is equivalent to mine—but whose definition of weak (n + 1)category are we using here... ?
Tom Leinster, A survey of definitions of ncategory
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 Γobject is a pointed contravariant functor from Γ (roughly the pointed category of pointed finite sets) to C provided C is pointed.
 opfibration
 A functor π:C → D is an opfibration if, for each object x in C and each morphism g : π(x) → y in D, there is at least one πcoCartesian 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 finitedimensional 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 Rlinear if it is enriched over the monoidal category of Rmodules, for R a commutative ring.
 presentable
 Given a regular cardinal κ, a category is κpresentable if it admits all small colimits and is κaccessible. A category is presentable if it is κpresentable 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 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 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 homsets are ; the morphisms are composed componentwise. 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 2categoryversion of an algebra for a monad (with a monad replaced by a 2monad).
Q
 Q
 Qcategory.
 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 setvalued 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.
 rig
 A rig category is a category with two monoidal structures, one distributing over the other.
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 klinear category C over a field k, a Serre functor is an autoequivalence 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 orderpreserving 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 orderpreserving functions. One writes and an element of the set is called an nsimplex. For example, is a simplicial set called the standard nsimplex. 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.^{[12]} (NB: some authors use the term "quasicategory" with a different meaning.^{[13]})
 2. An object in a category is said to be small if it is κcompact 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 ncategory
 A strict 0category is a set and for any integer n > 0, a strict ncategory is a category enriched over strict (n1)categories. For example, a strict 1category is an ordinary category. Note: the term "ncategory" typically refers to "weak ncategory"; 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.^{[14]} 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
 tstructure
 A tstructure is an additional structure on a triangulated category (more generally stable ∞category) that axiomatizes the notions of complexes whose cohomology concentrated in nonnegative degrees or nonpositive 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).^{[15]}^{[16]}
 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:
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
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 nonisomorphic subobjects.
Y
 Yoneda
 1.
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.
where Nat means the set of natural transformations. In particular, the functor
 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..
 ^ "Day convolution in nLab".
 ^ Hinich, V. (20131117). "DwyerKan localization revisited". arXiv:1311.4128 [math.QA].
 ^ Definition 3.6. in https://stacks.math.columbia.edu/download/pione.pdf#nameddest=0BQ6
 ^ Definition 7.2.1. in Bhatt, Bhargav; Scholze, Peter (2015), "The proétale topology for schemes", Astérisque: 99–201, arXiv:1309.1198, Bibcode:2013arXiv1309.1198B, MR 3379634
 ^ "Universal Homology Equivalences (Lecture 11)" (PDF). www.math.harvard.edu.
 ^ Kashiwara & Schapira 2006, exercise 8.20
 ^ "Multicategory in nLab".
 ^ 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 0471609226.
 ^ Joyal, A. (2002). "Quasicategories and Kan complexes". Journal of Pure and Applied Algebra. 175 (1–3): 207–222. doi:10.1016/S00224049(02)001354.
 ^ Vistoli 2004, Definition 2.57.
 ^ Jacob Lurie. Tannaka duality for geometric stacks. http://math.harvard.edu/~lurie/, 2004.
 ^ Bhatt, Bhargav (20140429). "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; JeanLouis Verdier (eds.). Théorie des Topos et Cohomologie Etale des Schémas. Lecture Notes in Mathematics (in French). Vol. 269. Berlin; New York: SpringerVerlag. xix+525. doi:10.1007/BFb0081551. ISBN 9783540058960.
 Grothendieck, Alexandre (1971). Revêtements Etales et Groupe Fondamental. Lecture Notes in Mathematics (in French). Vol. 224. Berlin; New York: SpringerVerlag. xxii+447. doi:10.1007/BFb0058656. ISBN 9783540056140. MR 0354651.
 Kashiwara, Masaki; Schapira, Pierre (2006). Categories and sheaves.
 A. Joyal, The theory of quasicategories 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. Vol. 5 (2nd ed.). New York, NY: SpringerVerlag. ISBN 0387984038. 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. Vol. 97. Cambridge: Cambridge University Press. ISBN 0521834147. Zbl 1034.18001.
 Vistoli, Angelo (20041228). "Notes on Grothendieck topologies, fibered categories and descent theory". arXiv:math/0412512.
Further reading
 Groth, M., A Short Course on ∞categories Archived 20160303 at the Wayback Machine
 Cisinski's notes
 History of topos theory
 http://plato.stanford.edu/entries/categorytheory/
 Leinster, Tom (2014). Basic Category Theory. Cambridge Studies in Advanced Mathematics. Vol. 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 2category)