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Mathematics construct
In mathematics, a comma category is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects
Comma_category
General theory of mathematical structures
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the
Category_theory
Construction in category theory
we can define the category of cones to F as the comma category (Δ ↓ F). Morphisms of cones are then just morphisms in this category. This equivalence
Cone_(category_theory)
Mathematical object that generalizes the standard notions of sets and functions
that of an undercategory or coslice category, and both are special cases of a construction called comma category. Given a class W {\displaystyle W} of
Category_(mathematics)
Characterizing property of mathematical constructions
abstractly as initial or terminal objects of a comma category (see § Connection with comma categories, below). Universal properties occur almost everywhere
Universal_property
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
Generalization of category theory
In mathematics, higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows
Higher_category_theory
Mapping between categories
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic
Functor
Category admitting tensor products
In mathematics, a monoidal category (or tensor category) is a category C {\displaystyle \mathbf {C} } equipped with a bifunctor ⊗ : C × C → C {\displaystyle
Monoidal_category
Generalization of a category
representability amounts to saying the ∞-category of elements is equivalent to a comma category over C). More generally, a map between simplicial sets is called final
Quasi-category
In mathematics, invertible homomorphism
as the category of topological spaces or categories of algebraic objects (like the category of groups, the category of rings, and the category of modules)
Isomorphism
Punctuation mark (,)
The comma , is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or
Comma
Special objects used in (mathematical) category theory
object X to a functor U can be defined as an initial object in the comma category (X ↓ U). Dually, a universal morphism from U to X is a terminal object
Initial_and_terminal_objects
Most general completion of a commutative square given two morphisms with same domain
In category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the
Pushout_(category_theory)
Category theory concept
terms of the more general construction of a comma category. Let C {\displaystyle {\mathcal {C}}} be a category and X {\displaystyle X} a fixed object of
Overcategory
In mathematics, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become
Localization_of_a_category
Most general completion of a commutative square given two morphisms with same codomain
In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit
Pullback_(category_theory)
Relationship between two functors abstracting many common constructions
preserves the identity). (Note that this is precisely the definition of the comma category of R over the inclusion of unitary rings into rng.) The existence of
Adjoint_functors
Type of category in category theory
In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified
Cartesian_closed_category
Central object of study in category theory
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal
Natural_transformation
Category theory constructs
_{F}X)(b)=\varinjlim _{f:Fa\to b}X(a)} where the colimit is taken over the comma category ( F ↓ const b ) {\displaystyle (F\downarrow \operatorname {const} _{b})}
Kan_extension
Map (arrow) between two objects of a category
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures
Morphism
Generalization of category
category V, the category of (small) categories enriched over V is a 2-category. Also, if A {\displaystyle A} is a category, then the comma category C a t ↓ A
2-category
Overview of and topical guide to category theory
monoidal category Braided monoidal category Symmetric monoidal category Semigroupoid Comma category Localization of a category Enriched category Bicategory
Outline_of_category_theory
Embedding of categories into functor categories
The Yoneda lemma is a fundamental result in category theory, a branch of mathematics. It is an abstract result on functors of the type morphisms into a
Yoneda_lemma
Applications of category theory
Applied category theory is an academic discipline in which methods from category theory are used to study other fields including but not limited to computer
Applied_category_theory
Mathematical concept
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products
Limit_(category_theory)
Product of two categories, in category theory
the mathematical field of category theory, the product of two categories C and D, denoted C × D and called a product category, is an extension of the concept
Product_category
Comma before the conjunction in a list
The serial comma (also referred to as the Oxford comma or Harvard comma) is a comma placed after the penultimate term in a list (just before the conjunction)
Serial_comma
Category with direct sums and certain types of kernels and cokernels
prototypical example of an abelian category is the category of abelian groups, Ab. Abelian categories are very stable categories; for example they are regular
Abelian_category
Indexed collection of objects and morphisms in a category
In category theory, a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in
Diagram_(category_theory)
Set of arguments where two or more functions have the same value
proved that any equaliser in any category is a monomorphism. If the converse holds in a given category, then that category is said to be regular (in the
Equaliser_(mathematics)
operad with a single object. comma Given functors f : C → B , g : D → B {\displaystyle f:C\to B,g:D\to B} , the comma category ( f ↓ g ) {\displaystyle (f\downarrow
Glossary_of_category_theory
Mathematical category with weak equivalences, fibrations and cofibrations
In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences'
Model_category
Category of non-empty finite ordinals and order-preserving maps
In mathematics, the simplex category (or simplicial category or nonempty finite ordinal category) is the category of non-empty finite ordinals and order-preserving
Simplex_category
Category whose hom sets have algebraic structure
In category theory, a branch of mathematics, an enriched category generalizes the idea of a locally small category by replacing hom-sets with objects
Enriched_category
Mathematical category
category that behaves like the category of sheaves of sets on a topological space (or more generally, on a site). Topoi behave much like the category
Topos
factorizations correspond precisely to the objects and morphisms of the comma category ( Δ ↓ F ) {\displaystyle (\Delta \downarrow {\mathcal {F}})} , and a
Diagonal_functor
Mathematical category whose hom sets form Abelian groups
specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian
Preadditive_category
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)
Quotient space of a codomain of a linear map by the map's image
cokernel is called the corank of f. Cokernels are dual to the kernels of category theory, hence the name: the kernel is a subobject of the domain (it maps
Cokernel
Abstract mathematics relationship
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories
Equivalence_of_categories
Category theory
In category theory, a Kleisli category is a category naturally associated to any monad T. It is equivalent to the category of free T-algebras. The Kleisli
Kleisli_category
Construction in category theory
any category, although their existence depends on the category that is considered. They are a special case of the concept of a limit in category theory
Inverse_limit
Category-theoretic construction
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces
Coproduct
Concept in mathematical category theory
In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" ⊗ {\displaystyle
Symmetric_monoidal_category
Topics referred to by the same term
well-definition A mathematical symbol for "approaching from above" A comma category, in category theory Down (game theory), a mathematical game An ingressive
↓
Type of category in category theory
In mathematics, specifically in category theory, an additive category is a preadditive category admitting all finitary biproducts. There are two equivalent
Additive_category
Concept in category theory
Fibred categories (or fibered categories) are abstract entities in mathematics used to provide a general framework for descent theory. They formalise
Fibred_category
Mathematical construction used in homotopy theory
homotopy category of topological spaces. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets
Simplicial_set
Correspondence between properties of a category and its opposite
In category theory, a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the opposite
Dual_(category_theory)
Theorem in category theory
In mathematics, Lawvere's fixed-point theorem is an important result in category theory. It is a broad abstract generalization of many diagonal arguments
Lawvere's_fixed-point_theorem
Functor type
mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors
Representable_functor
Category in which all small limits exist
In mathematics, a complete category is a category in which all small limits exist. That is, a category C is complete if every diagram F : J → C (where
Complete_category
Mathematical concept
In category theory, an end of a functor S : C o p × C → X {\displaystyle S\colon \mathbf {C} ^{\mathrm {op} }\times \mathbf {C} \to \mathbf {X} } is a
End_(category_theory)
Category whose hom objects correspond (di-)naturally to objects in itself
In category theory, a branch of mathematics, a closed category is a special kind of category. In a locally small category, the external hom (x, y) maps
Closed_category
Relation of categories in category theory
In category theory, two categories C and D are isomorphic if there exist functors F : C → D and G : D → C that are mutually inverse to each other, i.e
Isomorphism_of_categories
Type of category in mathematics
In category theory, a branch of mathematics, the inserter category is a variation of the comma category where the two functors are required to have the
Inserter_category
Surjective homomorphism
In category theory, an epimorphism is a morphism f : X → Y that is right-cancellative in the sense that, for all objects Z and all morphisms g1, g2: Y
Epimorphism
Injective homomorphism
Y {\displaystyle X\hookrightarrow Y} . In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative
Monomorphism
Marks to indicate pacing of written text
comma" and the "exclamation comma". The question comma has a comma instead of the dot at the bottom of a question mark, while the exclamation comma has
Punctuation
Functors which are surjective and injective on hom-sets
In category theory, a faithful functor is a functor that is injective on hom-sets, and a full functor is surjective on hom-sets. A functor that has both
Full_and_faithful_functors
Special case of colimit in category theory
objects may be groups, rings, vector spaces or in general objects from any category. The way they are put together is specified by a system of homomorphisms
Direct_limit
Category whose objects and morphisms are inside a bigger category
In mathematics, specifically category theory, a subcategory of a category C {\displaystyle {\mathcal {C}}} is a category S {\displaystyle {\mathcal {S}}}
Subcategory
Aspect of category theory
In category theory, a coequalizer (or coequaliser) is a generalization of the quotient of a set by an equivalence relation to objects in an arbitrary category
Coequalizer
Concept in category theory
In mathematics, more specifically in the area of category theory, a forgetful functor (also known as a stripping functor) "forgets" or drops some or all
Forgetful_functor
Concept in mathematics
S are (possibly noncommutative) rings, and consider the right module categories (an analogous statement holds for left modules): C = M o d S and D = M
Tensor–hom_adjunction
Homological construction in category theory
In mathematics, specifically category theory, certain functors may be derived to obtain other functors closely related to the original ones. This operation
Derived_functor
Mathematical structures in category theory
In category theory, a branch of mathematics, a functor category D C {\displaystyle D^{C}} is a category where the objects are the functors F : C → D {\displaystyle
Functor_category
Species of butterfly
Polygonia comma, the eastern comma, is a North American butterfly in the family Nymphalidae, subfamily Nymphalinae. This butterfly is seasonally variable
Polygonia_comma
In mathematics, specifically in category theory, a functor F : C → D {\displaystyle F:C\to D} is essentially surjective if each object d {\displaystyle
Essentially surjective functor
Essentially_surjective_functor
Categorical generalization of a function space in set theory
specifically in category theory, an exponential object or map object is the categorical generalization of a function space in set theory. Categories with all
Exponential_object
Collection of maps which give the same result
In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and
Commutative_diagram
Bi-universal property in category theory
In category theory, a branch of mathematics, a zero morphism is a special kind of morphism exhibiting properties like the morphisms to and from a zero
Zero_morphism
Result in category theory
Grothendieck construction, h x {\displaystyle h_{x}} corresponds to the comma category C ↓ x {\displaystyle C\downarrow x} . So, the lemma is also frequently
2-Yoneda_lemma
Endofunctor on the category V of finite-dimensional vector spaces
In algebra, a polynomial functor is an endofunctor on the category V {\displaystyle {\mathcal {V}}} of finite-dimensional vector spaces that depends polynomially
Polynomial_functor
Functor that preserves short exact sequences
exact, but in ways that can still be controlled. Let P and Q be abelian categories, and let F: P→Q be a covariant additive functor (so that, in particular
Exact_functor
In mathematics, the free category or path category generated by a directed graph or quiver is the category that results from freely concatenating arrows
Free_category
Abstract homotopical model for topological spaces
objects in the category of simplicial sets (with the standard model structure). It is an ∞-category generalization of a groupoid, a category in which every
∞-groupoid
Graphical representation of a morphism
representing morphisms in monoidal categories, or more generally 2-cells in 2-categories. They are a prominent tool in applied category theory. When interpreted
String_diagram
Type of quotient object in mathematics
quotient category is a category obtained from another category by identifying sets of morphisms. Formally, it is a quotient object in the category of (locally
Quotient_category
Concept in mathematical category theory
(Ff)b=a} . An equivalent definition is that the category of elements of F {\displaystyle F} is the comma category ( ∗ ↓ F ) o p {\displaystyle (\ast \downarrow
Category_of_elements
Object in category theory
In category theory in mathematics, a natural numbers object (NNO) is an object endowed with a recursive structure similar to natural numbers. More precisely
Natural_numbers_object
Concept in category theory
In category theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor
Monoidal_functor
Two theorems needed for Quillen's Q-construction in algebraic K-theory
classifying space B ( d ↓ f ) {\displaystyle B(d\downarrow f)} of the comma category d ↓ f {\displaystyle d\downarrow f} is contractible for any object d
Quillen's_theorems_A_and_B
Monoidal category
Tannakian category is a particular kind of monoidal category C, equipped with some extra structure relative to a given field K. The role of such categories C
Tannakian_formalism
In mathematics, collection of classes
In mathematics, in the framework of a one-universe foundation for category theory, the term conglomerate is applied to arbitrary sets as a contraposition
Conglomerate_(mathematics)
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)
Category
In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels. Spelled out in more
Pre-abelian_category
Hypothesis in mathematical category theory
In category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states, homotopy-theoretically speaking, that the ∞-groupoids are spaces
Homotopy_hypothesis
In category theory and related fields of mathematics, a refinement is a construction that generalizes the operations of "interior enrichment", like bornologification
Refinement_(category_theory)
Variant of the notion of the center of a monoid, group, or ring to a category
In category theory, a branch of mathematics, the center (or Drinfeld center, after Soviet-American mathematician Vladimir Drinfeld) is a variant of the
Center_(category_theory)
Vertices connected in pairs by edges
distinguish different vertices or edges.) The category of directed multigraphs permitting loops is the comma category Set ↓ D where D: Set → Set is the functor
Graph_(discrete_mathematics)
In category theory, a branch of mathematics, a stable ∞-category is an ∞-category such that (i) It has a zero object. (ii) Every morphism in it admits
Stable_∞-category
Mathematical structure
{\displaystyle W=X} . Fix a topological space X {\displaystyle X} . Consider the comma category S p c / X {\displaystyle {\mathcal {S}}pc/X} of topological spaces with
Grothendieck_topology
Category whose objects are manifolds and whose morphisms are differentiable maps
q 0 . {\displaystyle F(p_{0})=q_{0}.} The category of pointed manifolds is an example of a comma category - Man•p is exactly ( { ∙ } ↓ M a n p ) , {\displaystyle
Category_of_manifolds
Topics referred to by the same term
Ice, Internet Communications Engine Slice category, in category theory, a special case of a comma category Slice genus, in knot theory Slice knot, in
Slice
space. In terms of category theory, the fundamental groupoid is a certain functor from the category of topological spaces to the category of groupoids. [
Fundamental_groupoid
Symmetric monoidal category with a special involution
In the mathematical field of category theory, a dagger symmetric monoidal category is a monoidal category ⟨ C , ⊗ , I ⟩ {\displaystyle \langle \mathbf
Dagger symmetric monoidal category
Dagger_symmetric_monoidal_category
Topological space with a distinguished point
maps as morphisms. Another way to think about this category is as the comma category, ( { ∙ } ↓ {\displaystyle \{\bullet \}\downarrow } Top) where { ∙ }
Pointed_space
COMMA CATEGORY
COMMA CATEGORY
Girl/Female
German, Greek
Order
Girl/Female
Greek
Of the universe.
Girl/Female
Australian, French, Irish
Dove
Boy/Male
Hindu, Indian
One of the Doll
Surname or Lastname
English
English : of uncertain origin. It may be a nickname for a beggar, from an agent derivative of maund ‘beg’ (probably from Old French mendier, Late Latin mendicare); this word is not attested before the 16th century, but may well have been in use earlier. Alternatively it may be an occupational name for a maker of baskets, from an agent derivative of Middle English maund ‘basket’ (Old French mande, of Germanic origin); or perhaps for someone in some position of authority, from a shortened form of Middle English coma(u)nder (from coma(u)nden ‘to command’).German : habitational name from places called Mandern, in Hesse and the Rhineland.Belgian (van der Mander) : habitational name from a place called Ter Mandere or Mandel, in West Flanders, derived from the river name Mandel.Indian (Panjab) : Sikh (Dogar, Jat) name of unknown meaning, based on the names of clans in these communities.
COMMA CATEGORY
COMMA CATEGORY
Boy/Male
American, Australian, British, Celtic, Christian, English, Irish, Welsh
White Haired; The Hollow; Flood; Gray-haired; Gray; Sacred; Gray Haired
Boy/Male
Tamil
Victory of beloved
Boy/Male
Hindu, Indian
Royal
Boy/Male
Sikh
Life which has been granted by God, God of heaven
Boy/Male
Hindu, Indian
Water; Life; Respective
Surname or Lastname
English
English : variant of Mansell.in some cases perhaps an Americanized spelling of German Munzel, a habitational name from a place so named near Hannover or from Monzel near Trier.
Boy/Male
Indian, Punjabi, Sikh
The Lord
Girl/Female
Indian
The generous
Boy/Male
Christian & English(British/American/Australian)
Wanderering Noble
Boy/Male
Hindu
Victory, One who always win
COMMA CATEGORY
COMMA CATEGORY
COMMA CATEGORY
COMMA CATEGORY
COMMA CATEGORY
a.
Encompassed with a coma, or bushy appearance, like hair; hairy.
n.
An interval equal to half a comma.
n.
A celestial body which revolves about the sun in an orbit of a moderate degree of eccentricity. It is distinguished from a comet by the absence of a coma, and by having a less eccentric orbit. See Solar system.
n.
The envelope of a comet; a nebulous covering, which surrounds the nucleus or body of a comet.
n.
A member of the scapolite, group, occuring in glassy crystals on Monte Somma, near Naples.
n.
A character or point [,] marking the smallest divisions of a sentence, written or printed.
n.
A tuft or bunch, -- as the assemblage of branches forming the head of a tree; or a cluster of bracts when empty and terminating the inflorescence of a plant; or a tuft of long hairs on certain seeds.
n.
The punctuation mark [;] indicating a separation between parts or members of a sentence more distinct than that marked by a comma.
a.
Relating to, or resembling, coma; drowsy; lethargic; as, comatose sleep; comatose fever.
n.
One of several similar sets of figures or terms usually marked by points or commas placed at regular intervals, as in numeration, in the extraction of roots, and in circulating decimals.
n.
A small interval (the difference between a major and minor half step), seldom used except by tuners.
n.
Coma with complete insensibility; deep lethargy.
v. i.
To raise from coma, languor, depression, or discouragement; to bring into action after a suspension.
n.
The nebulous covering of the head or nucleus of a comet; -- called also coma.
n.
An American butterfly (Polygonia, / Vanessa, Progne). It is orange and black above, grayish beneath, with an L-shaped silver mark on the hind wings. Called also gray comma.
n.
A comma.
v. t.
To liken; to compa/e.
n.
The aforesaid thing; the same (as before). Often contracted to do., or to two "turned commas" ("), or small marks. Used in bills, books of account, tables of names, etc., to save repetition.
n.
A state of profound insensibility from which it is difficult or impossible to rouse a person. See Carus.
n.
A mark of punctuation; a character used to mark the divisions of a composition, or the pauses to be observed in reading, or to point off groups of figures, etc.; a stop, as a comma, a semicolon, and esp. a period; hence, figuratively, an end, or conclusion.