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Special objects used in (mathematical) category theory
a terminal object (also called terminal element): T is terminal if for every object X in C there exists exactly one morphism X → T. Initial objects are
Initial_and_terminal_objects
Characterizing property of mathematical constructions
described more concisely as initial and terminal objects in a comma category (i.e. one where morphisms are seen as objects in their own right). Let F :
Universal_property
Topics referred to by the same term
0o, or zero object, a mathematics term for a simultaneously initial and terminal object 0O, also ZO, an abbreviation for zero order Zero-order hold,
0O
Mathematically obvious
any other zeros are considered to be non-trivial. Degeneracy Initial and terminal objects List of mathematical jargon Pathological Trivialism Trivial measure
Triviality_(mathematics)
Concept in mathematics
supremum and essential infimum Initial and terminal objects Maximal and minimal elements Limit superior and limit inferior (infimum limit) Upper and lower
Greatest element and least element
Greatest_element_and_least_element
Object in category theory
with a terminal object 1 and binary coproducts (denoted by +), an NNO can be defined as the initial algebra of the endofunctor that acts on objects by X
Natural_numbers_object
Category whose objects are rings and whose morphisms are ring homomorphisms
colimits. The zero ring serves as both an initial and terminal object in Rng (that is, it is a zero object). It follows that Rng, like Grp but unlike
Category_of_rings
Type of category in category theory
following three properties: It has a terminal object. Any two objects X and Y of C have a product X×Y in C. Any two objects Y and Z of C have an exponential ZY
Cartesian_closed_category
Relationship between two functors abstracting many common constructions
For each object Y in D, choose an initial morphism (F(Y), ηY) from Y to G, so that ηY : Y → G(F(Y)). We have the map of F on objects and the family
Adjoint_functors
Relation of categories in category theory
one object and only its identity morphism (in fact, 1 is the terminal category), and C is any category, then the functor category C1, with objects functors
Isomorphism_of_categories
Family of quantum invariants
represented by certain decorated framed tangle diagrams, where the initial and terminal objects are represented by the boundary components of the tangle. In
Reshetikhin–Turaev_invariant
Generalized object in category theory
groups or rings, and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism
Product_(category_theory)
Algebraic structure with only one element
by definition, must be a terminal object, which means that a morphism A → {0} must exist and be unique for an arbitrary object A. This morphism maps any
Zero_object_(algebra)
Construction in category theory
"glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Inverse limits can be defined in
Inverse_limit
Special case of colimit in category theory
construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector
Direct_limit
Mathematical object
In mathematics, an initial algebra is an initial object in the category of F-algebras for a given endofunctor F. This initiality provides a general framework
Initial_algebra
In mathematics, invertible homomorphism
that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures
Isomorphism
by +), and binary products (denoted by ×), a list object over A can be defined as the initial algebra of the endofunctor that acts on objects by X ↦ 1
List_object
Category whose objects are small categories and whose morphisms are functors
2-morphisms. The initial object of Cat is the empty category 0, which is the category of no objects and no morphisms. The terminal object is the terminal category
Category_of_small_categories
Mathematical concept
factorization is possible for every cone. Limits may also be characterized as terminal objects in the category of cones to F. It is possible that a diagram does not
Limit_(category_theory)
History of maths
This is a timeline of category theory and related mathematics. Its scope ("related mathematics") is taken as: Categories of abstract algebraic structures
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
Category in which all small limits exist
has pullbacks and pushouts, but not products, coproducts, equalizers, coequalizers, terminal objects, or initial objects. Abstract and Concrete Categories
Complete_category
Category-theoretic construction
{\displaystyle C} be a category and let X 1 {\displaystyle X_{1}} and X 2 {\displaystyle X_{2}} be objects of C . {\displaystyle C.} An object is called the coproduct
Coproduct
Most general completion of a commutative square given two morphisms with same codomain
specializing Z to be the terminal object, when it exists. f and g are then uniquely determined and thus carry no information, and the pullback of this cospan
Pullback_(category_theory)
Category theory constructs
1 {\displaystyle \mathbf {1} } (the category with one object and one arrow, a terminal object in C a t {\displaystyle \mathbf {Cat} } ). The colimit
Kan_extension
General theory of mathematical structures
formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the source and the target of the morphism
Category_theory
Embedding of categories into functor categories
embedded category of representable functors and their natural transformations relates to the other objects in the larger functor category. It is an important
Yoneda_lemma
Overview of and topical guide to category theory
Category of magmas Initial object Terminal object Zero object Subobject Group object Magma object Natural number object Exponential object Epimorphism Monomorphism
Outline_of_category_theory
Generalizations of '"`UNIQ--math-00000000-QINU`"' in algebraic structures
under set union An empty sum or empty coproduct An initial object in a category (an empty coproduct, and so an identity under coproducts) An absorbing element
Zero_element
Abstract mathematics relationship
topos) if and only if D is cartesian closed (or a topos). Dualities "turn all concepts around": they turn initial objects into terminal objects, monomorphisms
Equivalence_of_categories
Mapping between categories
where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to
Functor
Map (arrow) between two objects of a category
composition. Morphisms and objects are constituents of a category. Morphisms, also called maps or arrows, relate two objects called the source and the target of
Morphism
Quotient space of a codomain of a linear map by the map's image
Hilbert spaces) is an object Q and a morphism q : Y → Q such that the composition q f is the zero morphism of the category, and furthermore q is universal
Cokernel
Category admitting tensor products
objects are lists (finite sequences) A1, ..., An of objects of C; there are arrows between two objects A1, ..., Am and B1, ..., Bn only if m = n, and
Monoidal_category
Functors which are surjective and injective on hom-sets
each X and Y in C. A faithful functor need not be injective on objects or morphisms. That is, two objects X and X′ may map to the same object in D (which
Full_and_faithful_functors
Category theory concept
this applies to limits and colimits as well. By construction, ( X , id ) {\displaystyle (X,\operatorname {id} )} is a terminal object of C / X {\displaystyle
Overcategory
Injective homomorphism
left-cancellative morphism. That is, an arrow f : X → Y such that for all objects Z and all morphisms g1, g2: Z → X, f ∘ g 1 = f ∘ g 2 ⟹ g 1 = g 2 . {\displaystyle
Monomorphism
Category with direct sums and certain types of kernels and cokernels
simple objects (meaning the only sub-objects of any X i {\displaystyle X_{i}} are the zero object 0 {\displaystyle 0} and itself) such that an object X ∈
Abelian_category
Generalization of category theory
features, such as the Eilenberg-MacLane space. An ordinary category has objects and morphisms, which are called 1-morphisms in the context of higher category
Higher_category_theory
Mathematical object that generalizes the standard notions of sets and functions
existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. Category
Category_(mathematics)
Construction in category theory
F is a universal morphism from Δ to F (thought of as an object in CJ), or a terminal object in (Δ ↓ F). Dually, a cone φ from F to L is a universal cone
Cone_(category_theory)
Category whose objects and morphisms are inside a bigger category
category S {\displaystyle {\mathcal {S}}} whose objects are objects in C {\displaystyle {\mathcal {C}}} and whose morphisms are morphisms in C {\displaystyle
Subcategory
In category theory, a branch of mathematics, given a morphism f: X → Y and a morphism g: Z → Y, a lift or lifting of f to Z is a morphism h: X → Z such
Lift_(mathematics)
Most general completion of a commutative square given two morphisms with same domain
to coproducts and coequalizers (if there is an initial object) in the sense that: Coproducts are a pushout from the initial object, and the coequalizer
Pushout_(category_theory)
Mathematical category
the étale topos, these form the foundational objects of study in anabelian geometry, which studies objects in algebraic geometry that are determined entirely
Topos
Mathematical construction used in homotopy theory
category. The objects of Δ are nonempty totally ordered finite sets, and the morphisms (non-strictly) order-preserving functions. Each object is uniquely
Simplicial_set
Central object of study in category theory
{\textbf {Cat}}} whose 0-cells (objects) are the small categories, 1-cells (arrows) between two objects C {\displaystyle C} and D {\displaystyle D} are the
Natural_transformation
Category whose hom sets have algebraic structure
must have a means of composing hom-objects in an associative manner: that is, there must be a binary operation on objects giving us at least the structure
Enriched_category
Applications of category theory
mechanics), natural language processing, control theory, probability theory and causality. The application of category theory in these domains can take different
Applied_category_theory
Generalization of a category
ordinary categories, they contain objects (the 0-simplices of the simplicial set) and morphisms between these objects (1-simplices). But unlike categories
Quasi-category
Theorem in category theory
B {\displaystyle b:1\rightarrow B} (where 1 {\displaystyle 1} is a terminal object in C {\displaystyle \mathbf {C} } ) such that g ∘ b = b {\displaystyle
Lawvere's_fixed-point_theorem
Concept in mathematical category theory
tensor product is the direct product of objects, and any terminal object (empty product) is the unit object. The category of bimodules over a ring R
Symmetric_monoidal_category
A category C consists of objects and morphisms between these objects. The morphisms reflect relations between the objects. In many situations, it is
Localization_of_a_category
Motion of a body subject only to gravity
the absence of other forces, objects and people will experience weightlessness in these situations. Examples of objects not in free-fall: Flying in an
Free_fall
Aspect of category theory
objects X and Y and two parallel morphisms f, g : X → Y. More explicitly, a coequalizer of the parallel morphisms f and g can be defined as an object
Coequalizer
Collection of maps which give the same result
defines a poset category, where: the objects are the nodes, there is a morphism between any two objects if and only if there is a (directed) path between
Commutative_diagram
Mathematical concept
category M, a fibrant object A of M is an object that has a fibration to the terminal object of the category. The fibrant objects of a closed model category
Fibrant_object
Mathematical category formed by reversing morphisms
G)^{\text{op}}\cong (G^{\text{op}}\downarrow F^{\text{op}})} (see comma category) Dual object Dual (category theory) Duality (mathematics) Adjoint functor Contravariant
Opposite_category
which maps objects as well as morphisms. This functor can be employed to give a succinct alternate description of the product of objects within the category
Diagonal_functor
Set of arguments where two or more functions have the same value
context, X and Y are objects, while f and g are morphisms from X to Y. These objects and morphisms form a diagram in the category in question, and the equaliser
Equaliser_(mathematics)
Concept in category theory
practice). For these objects, there are forgetful functors that forget the extra sets that are more general. Most common objects studied in mathematics
Forgetful_functor
Mathematics construct
looking at morphisms: instead of simply relating objects of a category to one another, morphisms become objects in their own right. This notion was introduced
Comma_category
Concept in category theory
{\displaystyle f^{*}} taking the considered objects defined on Y {\displaystyle Y} to the same type of objects on X {\displaystyle X} . This is indeed the
Fibred_category
Category of non-empty finite ordinals and order-preserving maps
finite ordinals as objects, thought of as totally ordered sets, and (non-strictly) order-preserving functions as morphisms. The objects are commonly denoted
Simplex_category
Concept in mathematics
is the statement that the tensor product − ⊗ X {\displaystyle -\otimes X} and hom-functor Hom ( X , − ) {\displaystyle \operatorname {Hom} (X,-)} form
Tensor–hom_adjunction
Mathematical category whose hom sets form Abelian groups
will be both terminal (a nullary product) and initial (a nullary coproduct), it will in fact be a zero object. Indeed, the term "zero object" originated
Preadditive_category
Correspondence between properties of a category and its opposite
language with objects and morphisms as distinct sorts, together with the relations of an object being the source or target of a morphism and a symbol for
Dual_(category_theory)
Categorical generalization of a function space in set theory
object or map object is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects
Exponential_object
Endofunctor on the category V of finite-dimensional vector spaces
powers V ↦ Sym n ( V ) {\displaystyle V\mapsto \operatorname {Sym} ^{n}(V)} and the exterior powers V ↦ ∧ n ( V ) {\displaystyle V\mapsto \wedge ^{n}(V)}
Polynomial_functor
Mathematical category with weak equivalences, fibrations and cofibrations
model category has a terminal object by completeness and an initial object by cocompleteness, since these objects are the limit and colimit, respectively
Model_category
Mathematical concept
{\displaystyle (e,\omega )} , where e {\displaystyle e} is an object of X {\displaystyle \mathbf {X} } and ω : e → ¨ S {\displaystyle \omega \colon e{\ddot {\to
End_(category_theory)
Airport in Sepang, Selangor, Malaysia
runways and two terminals each with two satellite terminals. Phase One involved the construction of the main terminal and one satellite terminal, giving
Kuala Lumpur International Airport
Kuala_Lumpur_International_Airport
Generalization of category
namely, it consists of the data a class of objects, for each pair of objects a , b {\displaystyle a,b} , a hom-object Hom ( a , b ) {\displaystyle \operatorname
2-category
Function type in category theory
construction is used to define group objects over an arbitrary category with finite products and a terminal object 1 {\displaystyle 1} . When the category
F-algebra
Unique ring consisting of one element
xy = 0 for all x and y. This article refers to the one-element ring.) In the category of rings, the zero ring is the terminal object, whereas the ring
Zero_ring
Functor type
or as an initial object in the category of elements of F. The natural transformation induced by an element u ∈ F(A) is an isomorphism if and only if (A
Representable_functor
Indexed collection of objects and morphisms in a category
and a diagram is then an object in this category. Given any object A in C, one has the constant diagram, which is the diagram that maps all objects in
Diagram_(category_theory)
Graphical representation of a morphism
by: a set Σ 0 {\displaystyle \Sigma _{0}} of generating objects, the lists of generating objects in Σ 0 ⋆ {\textstyle \Sigma _{0}^{\star }} are also called
String_diagram
Airport terminal at London Heathrow Airport
boundaries. Objects recovered from the dig site were immediately analysed and catalogued, allowing for the preparation works for Terminal 5 to occur simultaneously
Heathrow_Terminal_5
Surjective homomorphism
morphism f : X → Y that is right-cancellative in the sense that, for all objects Z and all morphisms g1, g2: Y → Z, g 1 ∘ f = g 2 ∘ f ⟹ g 1 = g 2 . {\displaystyle
Epimorphism
Homological construction in category theory
subcategory of “good” (fibrant or cofibrant) objects. By first taking a fibrant or cofibrant resolution of an object and then applying that functor, we have successfully
Derived_functor
Concept in category theory
every three objects A {\displaystyle A} , B {\displaystyle B} and C {\displaystyle C} of C {\displaystyle {\mathcal {C}}} the diagrams , and commute
Monoidal_functor
Certain generalizations of groups
finite products (i.e. C has a terminal object 1 and any two objects of C have a product). A group object in C is an object G of C together with morphisms
Group_object
Retarding force on a body moving in a fluid
≈9 m/s, and so on. Terminal velocity for very small objects (pollen, etc.) at low Reynolds numbers is determined by Stokes law. In short, terminal velocity
Drag_(physics)
For example, given a ring R, let C be a category whose objects are the elements of the set R and whose morphisms are only the identity morphisms. Then
2-ring
Abstract homotopical model for topological spaces
{G} } . This is defined as the category whose objects are finite ordinals [ n ] {\displaystyle [n]} and morphisms are given by σ n : [ n ] → [ n + 1 ]
∞-groupoid
finite-dimensional real vector spaces whose morphisms consist of all linear mappings, and let F be a covariant functor that maps Vect to itself. For vector spaces
Smooth_functor
next. More precisely, the objects of the category are the vertices of the quiver, and the morphisms are paths between objects. Here, a path is defined
Free_category
Type of category in category theory
it). The empty product, is a final object and the empty product in the case of an empty diagram, an initial object. Both being limits, they are not finite
Additive_category
Functor that preserves short exact sequences
calculations because they can be directly applied to presentations of objects. Much of the work in homological algebra is designed to cope with functors
Exact_functor
Product of two categories, in category theory
define bifunctors and multifunctors. The product category C × D has: as objects: pairs of objects (A, B), where A is an object of C and B of D; as arrows
Product_category
Category theory
over a category C. The Kleisli category of C is the category CT whose objects and morphisms are given by O b j ( C T ) = O b j ( C ) , H o m C T ( X ,
Kleisli_category
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
Glossary_of_category_theory
Bi-universal property in category theory
H. More generally, suppose C is any category with a zero object 0. Then for all objects X and Y there is a unique sequence of morphisms 0XY : X → 0 → Y
Zero_morphism
Symmetric monoidal category with a special involution
f:A\rightarrow B} , g : C → D {\displaystyle g:C\rightarrow D} and all A , B , C {\displaystyle A,B,C} and D {\displaystyle D} in O b ( C ) {\displaystyle Ob(\mathbf
Dagger symmetric monoidal category
Dagger_symmetric_monoidal_category
Mathematical structures in category theory
{\displaystyle D^{C}} is a category where the objects are the functors F : C → D {\displaystyle F:C\to D} and the morphisms are natural transformations η
Functor_category
Hypothesis in mathematical category theory
n-groupoid Π n ( X ) {\displaystyle \Pi _{n}(X)} of a space X where an object is a point in X, a 1-morphism f : x → y {\displaystyle f:x\to y} is a path
Homotopy_hypothesis
Monoidal category
algebraic geometry and number theory. The name is taken from Tadao Tannaka and Tannaka–Krein duality, a theory about compact groups G and their representation
Tannakian_formalism
Terminal at JFK Airport in Queens, New York
or TWA Terminal) is a building at John F. Kennedy International Airport (JFK) in Queens, New York, United States. Designed by Eero Saarinen and Associates
TWA_Flight_Center
surjective if each object d {\displaystyle d} of D {\displaystyle D} is isomorphic to an object of the form F c {\displaystyle Fc} for some object c {\displaystyle
Essentially surjective functor
Essentially_surjective_functor
INITIAL AND-TERMINAL-OBJECTS
INITIAL AND-TERMINAL-OBJECTS
Male
English
Unisex pet form of English Andrew and Andrea, ANDY means "man; warrior."
Female
Norwegian
Danish and Norwegian form of Greek Hanna, ANE means "favor; grace."
Female
Bulgarian
(Ðна), compassion, grace; and, prayers.
Boy/Male
American, Australian, British, English
Phonetic Name Based on Initials; Combination of Initials J and D
Boy/Male
German, Spanish
Famous Land
Girl/Female
Australian, Dutch
Loving and Musical
Girl/Female
Hindu, Indian, Tamil
Sweet
Surname or Lastname
English, Scottish, Danish, Norwegian, Swedish, German, and Jewish (Ashkenazic)
English, Scottish, Danish, Norwegian, Swedish, German, and Jewish (Ashkenazic) : topographic name for someone who lived on patch of sandy soil, from the vocabulary word sand. As a Swedish or Jewish name it was often purely ornamental.Dutch and Belgian : reduced form of Van den Sand(e), Van den Zande, a habitational name from places such as Zande in West Flanders or various minor places named with zand ‘sand’.English and Scottish : from a short form of Alexander.French : from a Germanic personal name, Sando.
Surname or Lastname
English and German
English and German : topographic name from Old English land, Middle High German lant, ‘land’, ‘territory’. This had more specialized senses in the Middle Ages, being used to denote the countryside as opposed to a town or an estate.English : topographic name for someone who lived in a forest glade, Middle English, Old French la(u)nde, or a habitational name from Launde in Leicestershire or Laund in West Yorkshire, which are named with this word.Norwegian : habitational name from any of three farmsteads so named, from Old Norse land ‘land’, ‘territory’ (see 1 above).
Female
Serbian
(Bulgarian and Serbian Ðна): Bulgarian and Serbian form of Greek Hanna, ANA means "favor; grace."
Boy/Male
Hindu, Indian
The Sprout; Initial
Female
Danish
, compassion, grace; and, prayers.
Surname or Lastname
English and German
English and German : nickname for someone with a deformed hand or who had lost one hand, from Middle English hand, Middle High German hant, found in such appellations as Liebhard mit der Hand (Augsburg 1383).Jewish (Ashkenazic) : nickname from German Hand ‘hand’ (see 1).Irish : Anglicized form of Gaelic Ó Flaithimh (see Guthrie), resulting from an erroneous association of the Gaelic name with the Gaelic word lámh ‘hand’. It is used as an English equivalent for several other names of Gaelic origin too, e.g. Claffey, Glavin, and McClave.Dutch : from a variant of hont ‘dog’, ‘hound’, either a derogatory nickname, or a habitational name for someone living at a house distinguished by the sign of a dog.
Girl/Female
Indian
The initial reality
Girl/Female
Tamil
The initial reality
Surname or Lastname
English, German, and Jewish (Ashkenazic)
English, German, and Jewish (Ashkenazic) : metonymic occupational name for a maker of hoops and bands, etc., from Middle English band, bond, Middle High German, Middle Low German bant, German Band denoting something used for tying or binding: ‘hoop’, ‘metal band’, ‘fetter’, ‘shackle’.Old spelling of the Dutch cognates Bant, Bande, from Middle Dutch bant ‘band’.
Female
Spanish
Portuguese and Spanish form of Latin Anna, ANA means "favor; grace."Â Compare with another form of Ana.
Girl/Female
Muslim
Clever
Girl/Female
Hebrew, Indian, Spanish
Ann
Female
Finnish
Estonian and Finnish pet form of Greek Hanna, ANU means "favor; grace."
INITIAL AND-TERMINAL-OBJECTS
INITIAL AND-TERMINAL-OBJECTS
Boy/Male
Indian, Sanskrit
Sparrow
Boy/Male
Muslim/Islamic
Ambassador handsome
Girl/Female
Muslim/Islamic
Morning Star
Girl/Female
Hungarian
meaning a peaceful ruler.
Male
Spanish
Spanish form of Latin Martinus, MARTÃN means "of/like Mars."
Boy/Male
Afghan, Arabic, Muslim
Servant of the Creator
Girl/Female
Indian
Fresh water, Green water
Boy/Male
Indian
Part of Sun
Girl/Female
Hindu, Indian, Tamil
Forest
Girl/Female
Hindu, Indian
Dimples
INITIAL AND-TERMINAL-OBJECTS
INITIAL AND-TERMINAL-OBJECTS
INITIAL AND-TERMINAL-OBJECTS
INITIAL AND-TERMINAL-OBJECTS
INITIAL AND-TERMINAL-OBJECTS
v. t.
To put an initial to; to mark with an initial of initials.
a.
Pertaining or belonging to a germ; as, the germinal vesicle.
n. pl.
A festival celebrated annually by the Romans on February 23 in honor of Terminus, the god of boundaries.
n.
A turbinal bone or cartilage.
a.
Pertaining to, containing, or consisting of, seed or semen; as, the seminal fluid.
a.
Of or pertaining to the beginning; marking the commencement; incipient; commencing; as, the initial symptoms of a disease.
p. pr. & vb. n.
of Initial
adv.
In an initial or incipient manner or degree; at the beginning.
a.
Contained in seed; holding the relation of seed, source, or first principle; holding the first place in a series of developed results or consequents; germinal; radical; primary; original; as, seminal principles of generation; seminal virtue.
v. t.
To terminate.
n.
A determining; as, in oyer and terminer. See Oyer.
n.
Growing at the end of a branch or stem; terminating; as, a terminal bud, flower, or spike.
imp. & p. p.
of Initial
a.
Terminal.
v. t.
To bring to an end or conclusion; to finish; to close; to terminate; as, to end a speech.
a.
Placed at the beginning; standing at the head, as of a list or series; as, the initial letters of a name.
n.
The terminal, and usually flexible, posterior appendage of an animal.
n.
Of or pertaining to the end or extremity; forming the extremity; as, a terminal edge.
v. t.
To put an end to; to make to cease; as, to terminate an effort, or a controversy.
pl.
of Terminus