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Special objects used in (mathematical) category theory
one with a zero object. A strict initial object I is one for which every morphism into I is an isomorphism (strict terminal objects are defined analogously)
Initial_and_terminal_objects
Object in category theory
In the mathematical discipline of category theory, a strict initial object is an initial object 0 of a category C with the property that every morphism
Strict_initial_object
Mathematical set containing no elements
space is the unique initial object in the category of topological spaces with continuous maps. In fact, it is a strict initial object: only the empty set
Empty_set
Category admitting tensor products
coproducts is monoidal with the coproduct as the monoidal product and the initial object as the unit. Such a monoidal category is called cocartesian monoidal
Monoidal_category
Concurrency control method
serializability. A transaction is holding a lock on an object if that transaction has acquired a lock on that object which has not yet been released. For 2PL, the
Two-phase_locking
focuses on strict 2-groups. A strict 2-group is a strict monoidal category in which every morphism is invertible and every object has a strict inverse (so
2-group
Characterizing property of mathematical constructions
Universal morphisms can also be thought more abstractly as initial or terminal objects of a comma category (see § Connection with comma categories,
Universal_property
Generalization of category theory
concept is too strict for some purposes in for example, homotopy theory, where "weak" structures arise in the form of higher categories, strict cubical higher
Higher_category_theory
Class of mathematical orderings
a non-strict well ordering, then < is a strict well ordering. A relation is a strict well ordering if and only if it is a well-founded strict total order
Well-order
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
Glossary_of_category_theory
Relationship between two functors abstracting many common constructions
isomorphism Φ : homC(F−,−) → homD(−,G−). For each object X in C, each object Y in D, as (F(Y), ηY) is an initial morphism, then ΦY, X is a bijection, where ΦY
Adjoint_functors
In mathematics, invertible homomorphism
examples, the "equal" objects contain elements that are not set-theoretically identical, so they are not equal in this strict sense. However, because
Isomorphism
General theory of mathematical structures
category is 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
Category_theory
Generalization of category
2-morphism is a natural transformation between functors. The concept of a strict 2-category was first introduced by Charles Ehresmann in his work on enriched
2-category
Object whose state cannot be modified after it is created
In object-oriented (OO) and functional programming, an immutable object (unchangeable object) is an object whose state cannot be modified after it is
Immutable_object
Order of execution of transactions in transaction processing
satisfied: If the transaction T i {\displaystyle T_{i}} in S1 reads an initial value for object X, so does the same transaction T i {\displaystyle T_{i}} in S2
Database_transaction_schedule
Embedding of categories into functor categories
fixed object. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a miniature category with just one object and only
Yoneda_lemma
Proposed parameter in linguistics
and non-rigid) and head-initial types. The identification of headedness is based on the following: the order of subject, object, and verb the relationship
Head-directionality_parameter
Linguistic classification
In syntax, verb-initial (V1) word order is a word order in which the verb appears before the subject and the object. In the more narrow sense, this term
Verb-initial_word_order
Most general completion of a commutative square given two morphisms with same domain
and coequalizers (if there is an initial object) in the sense that: Coproducts are a pushout from the initial object, and the coequalizer of f, g : X
Pushout_(category_theory)
Abstract mathematics relationship
object c of C is an initial object (or terminal object, or zero object), if and only if Fc is an initial object (or terminal object, or zero object)
Equivalence_of_categories
Mathematical set with an ordering
also called strict partial orders. Strict and non-strict partial orders can be put into a one-to-one correspondence, so for every strict partial order
Partially_ordered_set
Order whose elements are all comparable
ISSN 0031-952X. JSTOR 24340068. This definition resembles that of an initial object of a category, but is weaker. Roland Fraïssé (December 2000). Theory
Total_order
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
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
Mathematical concept
also referred to as universal co-cones. They can be characterized as initial objects in the category of co-cones from F {\displaystyle F} . As with limits
Limit_(category_theory)
Central object of study in category theory
\eta _{X}:F(X)\to G(X)} is natural in X {\displaystyle X} . If, for every object X {\displaystyle X} in C {\displaystyle {\mathcal {C}}} , the morphism η
Natural_transformation
Programming language and superset of JavaScript
libraries, much like C++ header files can describe the structure of existing object files. This enables other programs to use the values defined in the files
TypeScript
Object in category theory
numbers object (NNO) is an object endowed with a recursive structure similar to natural numbers. More precisely, in a category E with a terminal object 1,
Natural_numbers_object
Mathematical category
of X {\displaystyle X} and Y {\displaystyle Y} over their sum is the initial object in C {\displaystyle C} . All equivalence relations in C {\displaystyle
Topos
Map (arrow) between two objects of a category
composition when it is defined, and existence of an identity morphism for every object), and the outcome of the composition is a morphism. Morphisms and categories
Morphism
WWII Soviet heavy tank
on the basis of the Object 220, in the form of the Object 221 (with an 85 mm gun), Object 222 (with the F-32 76.2 mm gun) and Object 223 (built to develop
KV_tank_family
Generalized object in category theory
In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas
Product_(category_theory)
Collection of objects and morphisms
the 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)
Type of category in category theory
x^{y+z}=x^{y}\times x^{z}} the initial object is the coproduct identity: 0 + x = x {\displaystyle 0+x=x} the initial object is the product zero: x × 0 =
Cartesian_closed_category
Markup language which places HTML in XML form
1.0 Strict document.<br /> <img id="validation-icon" src="http://www.w3.org/Icons/valid-xhtml10" alt="Valid XHTML 1.0 Strict"/><br /> <object id="pdf-object"
XHTML
Basic word order type
In linguistic typology, a verb–object–subject or verb–object–agent language, commonly abbreviated VOS or VOA, is one in which most sentences arrange their
Verb–object–subject word order
Verb–object–subject_word_order
Equivalently, it has an initial object, binary coproducts and binary coequalizers. E {\displaystyle {\mathcal {E}}} has exponential objects (in addition to binary
Elementary_topos
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
Category theory constructs
( a ) {\displaystyle \delta _{F}(a)=\delta (Fa):MF(a)\to RF(a)} for any object a {\displaystyle a} of A . {\displaystyle \mathbf {A} .} The functor R is
Kan_extension
High-level programming language
GitHub., however, TypeScript, which is a version of JavaScript that is strict about types and is statically typed, overtook JavaScript on Github by August
JavaScript
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
Functor type
universal morphism from the one-point set {•} to the functor F or as an initial object in the category of elements of F. The natural transformation induced
Representable_functor
Certain generalizations of groups
The strict 2-group is the group object in the category of small categories. Given a category C with finite coproducts, a cogroup object is an object G of
Group_object
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-theoretic construction
vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is
Coproduct
Mathematical category whose hom sets form Abelian groups
terminal (a nullary product) and initial (a nullary coproduct), it will in fact be a zero object. Indeed, the term "zero object" originated in the study of
Preadditive_category
Quotient space of a codomain of a linear map by the map's image
of the domain (it maps to the domain), while the cokernel is a quotient object of the codomain (it maps from the codomain). Intuitively, given an equation
Cokernel
Category theory concept
(X,\operatorname {id} )} is a terminal object of C / X {\displaystyle {\mathcal {C}}/X} and an initial object of X / C {\displaystyle X/{\mathcal {C}}}
Overcategory
terminal object 1, binary coproducts (denoted by +), and binary products (denoted by ×), a list object over A can be defined as the initial algebra of
List_object
Object that represents a simple entity whose equality is not based on identity
"VALJO" (VALue Java Object) has been coined to refer to the stricter set of rules necessary for a correctly defined immutable value object. public class StreetAddress
Value_object
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
Category with direct sums and certain types of kernels and cokernels
mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable
Abelian_category
Theoretical paradox resulting from time travel
ontological paradox, occurs when any event, such as an action, information, an object, or a person, ultimately causes itself, as a consequence of either retrocausality
Temporal_paradox
Most general completion of a commutative square given two morphisms with same codomain
the morphisms f {\displaystyle f} and g {\displaystyle g} consists of an object P {\displaystyle P} and two morphisms p 1 : P → X {\displaystyle p_{1}:P\rightarrow
Pullback_(category_theory)
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
adjoint to the forgetful functor U. Mathematics portal Free strict monoidal category Free object Adjoint functors Awodey, Steve (2010). Category theory (2nd ed
Free_category
Category whose hom objects correspond (di-)naturally to objects in itself
the external hom (x, y) maps a pair of objects to a set of morphisms. So in the category of sets, this is an object of the category itself. In the same vein
Closed_category
Variant of the notion of the center of a monoid, group, or ring to a category
{\mathcal {Z(C)}}} , is the category whose objects are pairs ( A , u ) {\displaystyle (A,u)} consisting of an object A {\displaystyle A} of C {\displaystyle
Center_(category_theory)
Software component technology from Microsoft
Component Object Model (COM) is a binary-interface technology for software components from Microsoft that enables using objects in a language-neutral
Component_Object_Model
Type of quotient object in mathematics
another category by identifying sets of morphisms. Formally, it is a quotient object in the category of (locally small) categories, analogous to a quotient group
Quotient_category
Concept in mathematical category theory
{\displaystyle A\otimes B} is, in a certain strict sense, naturally isomorphic to B ⊗ A {\displaystyle B\otimes A} for all objects A {\displaystyle A} and B {\displaystyle
Symmetric_monoidal_category
Concept in category theory
maps of strong monoidal functors are invertible. The coherence maps of strict monoidal functors are identity maps. Although we distinguish between these
Monoidal_functor
Mathematical concept
\mathbf {C} \to \mathbf {X} } is a universal dinatural transformation from an object e {\displaystyle e} of X {\displaystyle \mathbf {X} } to S {\displaystyle
End_(category_theory)
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
Abstract homotopical model for topological spaces
for topological spaces. One model uses Kan complexes which are fibrant objects in the category of simplicial sets (with the standard model structure)
∞-groupoid
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
Aspect of category theory in mathematics
involved into equalities. A rig category is semi-strict if the two monoidal structures involved are strict, both of its annihilators are equalities and one
Rig_category
Mathematical category with weak equivalences, fibrations and cofibrations
closed 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
Applications of category theory
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Applied_category_theory
Mathematics construct
free abelian group having that set as its basis. In particular, the initial object of ( s ↓ T ) {\displaystyle (s\downarrow T)} is the canonical injection
Comma_category
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
Theorem in category theory
} and given an object B {\displaystyle B} in it, if there is a weakly point-surjective morphism f {\displaystyle f} from some object A {\displaystyle
Lawvere's_fixed-point_theorem
Bi-universal property in category theory
morphisms. The category of sets does not have a zero object, but it does have an initial object, the empty set ∅. The only right zero morphisms in Set
Zero_morphism
Indexed collection of objects and morphisms in a category
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 J to A, and
Diagram_(category_theory)
Construction in category theory
a universal cone from F is a universal morphism from F to Δ, or an initial object in (F ↓ Δ). The limit of F is a universal cone to F, and the colimit
Cone_(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
Concept in software engineering
often seen in terms of services, but objects could be an equally powerful approach. The DSP's initial 'Naked Object Architecture' was developed by an external
Naked_objects
Category theory
{\displaystyle C} as above, we associate with each object X {\displaystyle X} in C {\displaystyle C} a new object X T {\displaystyle X_{T}} , and for each morphism
Kleisli_category
Category in which all small limits exist
not products, coproducts, equalizers, coequalizers, terminal objects, or initial objects. Abstract and Concrete Categories, Jiří Adámek, Horst Herrlich
Complete_category
Concept in mathematics
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Tensor–hom_adjunction
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Lift_(mathematics)
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
Soviet main battle tank
coming back from Nizhniy Tagil, with Morozov at its head. A project named object 430 led to three prototypes which were tested in Kubinka in 1958. Those
T-64
Software optimization technique
which delays the evaluation of an expression until its value is needed (non-strict evaluation) and which avoids repeated evaluations (by the use of sharing)
Lazy_evaluation
Set whose elements all belong to another set
A is a proper (or strict) subset of B, denoted by A ⊊ B {\displaystyle A\subsetneq B} , or equivalently, B is a proper (or strict) superset of A, denoted
Subset
Set of arguments where two or more functions have the same value
consists of an object E and a morphism eq : E → X satisfying f ∘ e q = g ∘ e q {\displaystyle f\circ eq=g\circ eq} , and such that, given any object O and morphism
Equaliser_(mathematics)
Collection of maps which give the same result
commutes (the notion of diagram strictly generalizes commutative diagram). As a simple example, the diagram of a single object with an endomorphism ( f : X
Commutative_diagram
Family of higher-order functions
reason, such languages often provide a stricter variant of left folding which forces the evaluation of the initial parameter before making the recursive
Fold_(higher-order_function)
Library of modules (software)
The Perl Object Environment (POE) is a library of Perl modules written in the Perl programming language by Rocco Caputo et al. From CPAN: "POE originally
Perl_Object_Environment
Concept in category theory
the inverse image functors are not strictly compatible with composed maps: if z {\displaystyle z} is an object over Z {\displaystyle Z} (a vector bundle
Fibred_category
Mathematical structures in category theory
a functor category D C {\displaystyle D^{C}} is a category where the objects are the functors F : C → D {\displaystyle F:C\to D} and the morphisms are
Functor_category
Higher categorical generalization of a topos
an ∞-topos (infinity-topos) is, roughly, an ∞-category such that its objects behave like sheaves of spaces with some choice of Grothendieck topology;
∞-topos
Category whose objects and morphisms are inside a bigger category
strictly full. A subcategory of C {\displaystyle {\mathcal {C}}} is wide or lluf (a term first posed by Peter Freyd) if it contains all the objects of
Subcategory
Functors which are surjective and injective on hom-sets
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 is why the range of a full
Full_and_faithful_functors
Philosophical study of beauty and art
appreciation. Aesthetic properties are features that influence the appeal of objects. They include aesthetic values, which express positive or negative qualities
Aesthetics
Product of two categories, in category theory
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 from (A1, B1) to (A2
Product_category
Set of rules defining correctly structured programs
adding a property to the global object (usually window) will also create a new global variable. Note that JavaScript's strict mode forbids the assignment
JavaScript_syntax
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
Small Solar System body with an orbit that can bring it close to Earth
Atens: 2,952 (7.90%) Comets: 123 (0.33%) Atiras: 34 (0.09%) A near-Earth object (NEO) is by definition any small Solar System body orbiting the Sun whose
Near-Earth_object
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