For example, let A A, B B and C C be categories, T:A → B T: A → B be a functor. Let U: Cring !Monoid be the forgetful functor that forgets ring addition. Consider a category C of algebraic structures; the objects can be thought of as sets plus operations, obeying some laws. \mathcal {B}G is precisely a representing object for this functor; the universal element is the (isomorphism class of the) classifying [\pi: \mathcal. There is a functor π1: Top → Group π 1: T o p → G r o u p that associates to every topological space* X X a group π1(X) π 1 ( X), called the fundamental group of X X, and which sends every continuous function X f Y X f Y to a group homomorphism π1(X) π1(f) π1(Y) π 1 ( X) π 1 ( f) π 1 ( Y) . It has a single method, called fmap. It is a high level concept of implementing polymorphism. In computer science a functor is an object that can be called as though it were a. e. The integral monoid ring construction gives a functor from monoids to rings. The following diagram depicts how an Applicative Functor acts as an endofunctor in the Hask category. mathcal {B}G is precisely a representing object for this functor; the universal element is the (isomorphism class of the) classifying [pi: mathcal. It is well-known that the pullback construction is invariant with respect to homotopic deformations; that is, this presheaf descends to a functor on the. Example #2: the fundamental group. The functor Hom (–, B) is also called the functor of points of the object B . A representable functor F is any functor naturally isomorphic to Mor C(X; ). Nonton dan Download Indo Viral Funcrot. In addition, certain conditions are satisfied by a functor. Functor is a concept from category theory and represents the mapping between two categories. In addition to defining a wrapping monadic type, monads define two operators: one to wrap a value in the monad type, and another to compose together. Thus, inverse limits can be defined in any category although their existence depends on the category that is considered. e a mapping of the category to category. confused about function as instance of Functor in haskell. Since it overloads the function-call operator, code can call its major method using the same syntax they would for a function call. Roughly, it is a general mathematical theory of structures and of systems of structures. In this asymmetrc interpretation, the theorem (if you define adjoints via universal morphisms. Naperian functors are closed under constant unit (Phantom), product, exponentiation (a ->) aka Reader, identity. They all motivate functor of points this way : In general, for any object Z of a category X, the association X ↦ Hom X ( Z, X) defines a functor ϕ from the category X to the category of sets. The intuitive description of this construction as "most efficient" means "satisfies a universal property" (in this case an initial property), and that it is intuitively "formulaic" corresponds to it being functorial, making it an "adjoint" "functor". There are three non-trivial well-known functors. A lambda expression creates an nameless functor, it's syntactic sugar. fmap takes a function and a structure, then returns the same. Pacar Toci Cakep Ngewe Meki Sempit | Mukacrot merupakan salah satu situs bokep terlengkap yang menyajikan konten-konten dewasa vulgar syur dan penuh gairah sex yang ada diseluruh dunia yang di bagi dalam beberapa genre dengan persentase bokep lokal mendominasi 80% khusus bagi anda pecinta bokep maupun pecandu bokep atau. I mentioned proper and smooth base change, but there are many more : projection formula, Verdier duality, gluing. What is less well known is that the second actually follows from the first and parametricity, so you only need to sit down and prove one Functor law when you go. Functor Type Syntax and Semantics# The simplest syntax for functor types is actually the same as for functions:In mathematics higher-order functions are also termed operators or functionals. , it is a regular epimorphism , in fact an absolute ? coequalizer , being the coequalizer of a pair ( e , 1 B ) (e, 1_B) where e = i ∘ r : B → B e = i circ r: B o B is idempotent). An example of a functor generating list combinators for various types of lists is given below, but this example has a problem: The various types of lists all have advantages -- for example, lazy lists can be infinitely long, and concantenation lists have a O(1) concat operator. Functors are objects that can be treated as though they are a function or function pointer--you could write code that looks like this: 1. It is common for the same conceptual function or operation to be implemented quite differently for different types of arguments: adding two integers is very different from adding two. map (function) (promise) = fmap (function) (promise) promise <- async (return 11) wait (map (sub2) (promise)) -- 9. In your particular example, the functor-based approach has the advantage of separating the iteration logic from the average-calculation logic. Definition of a Function. There are two example of this notion that come to my mind: if you have a graph-map between the graphs underlying two categories, such map is functorial if it verifies the condition for being a functor: namely it preserves composites and identities morphisms. 00:00. #include <iostream> #include <algorithm> #include. e. Indeed, by definition, a functor is composed by two "functions": one that assigns objects to objects, and one that assigns maps to maps. For C++, a functor is simply a class supporting operator (); what one might refer to as a callable in Python. Funcrot Website Dewasa Terlengkap, Nonton "Goyangan Nikmat Dari Pacar Mesum" Di Funcrot, Nonton Dan Baca Cerita Dewasa Hanya Di Funcrot. Functions are not something on their own anymore, but they are always connected to objects in a modular fashion. functor: [noun] something that performs a function or an operation. Roughly, it is a general mathematical theory of structures and of systems of structures. "Kalo lagi jenuh doang sih biasanya" ujarnya. x stackrel {f} { o} y,. " Let F:C → D F: C → D be a functor. Stack Exchange Network. 96580 views 100%. opposite The opposite category of a category is obtained by reversing the arrows. In other words, a contravariant functor acts as a covariant functor from the opposite category C op to D. In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. Bokepfull Avtub Terbaru. g. every one of them can be assigned a well-defined morphism-mapping through Haskell's typeclass mechanism. From monoids and groups to rings. In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Here are a few other examples. Like other languages, Haskell does have its own functional definition and declaration. See also at idempotent monad – Properties – Algebras for an idempotent monad and localization. Hence, a natural transformation can be considered to be a "morphism of functors". "Iya ibu gak kaku soalnya". 0 seconds of 2 minutes, 16 secondsVolume 90%. Prelude. When covering the vital Functor and Monad type classes, we glossed over a third type class: Applicative, the class for applicative functors. It can be proven that in this case, both maps are equal. Functor. In mathematics, specifically in category theory, an exponential object or map object is the categorical generalization of a function space in set theory. Google "Naperian Functor": Hancock gives them that name, because the domain of the representing function is log_x (f x). In programming languages like Scala, we can find a lot of uses for Functors. How to use scrot- in a sentence. Examples of such type constructors are List, Option, and Future. It is a minimal function which maps an algebra to another algebra. Postingan Terbaru Ngintip Abg Di Kamar Mandi Kolam Renang. (all of the information of a category is in its arrows so we can reduce all various-shaped elements to arrows and equations between them, but we don't have to) A functor is a design pattern used in functional programming inspired by the definition used in category theory. Nowadays. Ome Tv Ngaku Abg Tapi Body Udah Jadi. a function may be applied to the values held within the structure/container without changing the (uh!) structure of the structure/container. The ZipList is an applicative functor on lists, where liftA2 is implemented by zipWith. By the way, [] is not Naperian, because the length of the list may vary. JUL-756 Orang Yang Membuliku Meniduri Ibuku - Asahi Mizuno. To derive from this the definition of natural transformations above, it is sufficient to consider the interval category A := I := {a o b}. They are a. T {displaystyle T} , which assigns to each object. Funcrot Website Dewasa Terlengkap, Nonton "Putri Lestari Hijab Binal Pamer Body" Di Funcrot, Nonton Dan Baca Cerita Dewasa Hanya Di Funcrot. function object implementing x - y. Another interesting reason why categories cannot be identified always with categories having functions for morphisms is given in this paper, by Peter Freyd in which is proven that there are some categories which aren't concrete: i. The functor F induces a function. A functor takes a pure function (and a functorial value) whereas a monad takes a Kleisli arrow, i. More specifically, every morphism f : x → y in C must be assigned to a morphism F(f) : F(y) → F(x) in D. object. What's a typeclass? One of Haskell's selling points is its strong type system. Movie. fox, dog , and cat (nouns) sly, brown, and lazy (adjectives) gracefully (adverb) jumped (main verb) Function words include: the (determiner) over (preposition) and (conjunction) Even though the function words don't have concrete meanings, sentences would make a lot less sense without them. Applicative functors allow for functorial computations to be sequenced (unlike plain functors), but don't allow using results from prior computations in the definition. The functor Hom (–, B) is also called the functor of points of the object B . 1 Answer. Covers many abstractions and constructions starting from basics: category, functor up to kan extensions, topos, enriched categories, F-algebras. 6. The reason this helps is that type constructors are unique, i. A functor is a special case of a functionoid: it is a functionoid whose method is the “function-call operator,” operator()(). The notion of morphism recurs in much of contemporary mathematics. For every value of the index and for every value of the Representable, we can call the. Many books (eg Kashiwara, Schapira) give an exhaustive list of these properties. Then there's an induced functor F ∗: [B, Set] → [A, Set] defined by composition with F. The functor implementation for a JavaScript array is Array. A functor, in the mathematical sense, is a special kind of function on an algebra. Funcrot Website Dewasa Terlengkap, Nonton "Ome Tv Abg SMP Temenin Pascol" Di Funcrot, Nonton Dan Baca Cerita Dewasa Hanya Di Funcrot. A functor is the mapping of one category to another category. Nonton dan Download. Now let’s see what a functor is. Functions are blocks of code that can be called by their name. Some type constructors with two parameters or more have a Bifunctor instance that. In this example, we will look at a predefined C++ functor greater<T>(), where T is the type of the functor parameter with the STL algorithm sort. fmap g = fmap (f . As category theory is still evolving, its functions are correspondingly developing, expanding. Functor is a term that refers to an entity that supports operator in expressions (with zero or more parameters), i. Indeed, we already saw in Remark 3. Home Indo Indo Hijab Indo Live Full Pack Jav Sub Jav Uncensored Cerita. Like monads, applicative functors are functors with extra laws and operations; in fact, Applicative is an intermediate class between Functor and Monad. So one could say a functor is composed of two "parts", one that maps Objects to Objects, and one that maps Morphisms to Morphisms. Mukacrot tempat streaming maupun download video bokep termudah tanpa iklan jangan ragu kehabisan stock video karena koleksi video kami tak terbatas mulai dari Bokep Indo, Bokep Jepang, Bokep Asia, Bokep Barat, dengan aktor dari berbagai belahan dunia. Haskell's combination of purity, higher order functions, parameterized algebraic data types, and typeclasses allows us to implement polymorphism on a much higher level than possible in other languages. 00:03:20. Jiří Adámek, Jiri Rosicky, , Cambridge UP, 1994. Bokep Indo Skandal Abdi Negara Yuk Viralin Sangelink. Functor is not necessarily an object of some class with overloaded operator (). 2-2. Hence by the fully faithfulness of the Yoneda embedding, there is an. Ab Ab -enriched categories with finite. 105114 views 100%. Download Image. gửi email cho tác giả. In functional programming, an applicative functor, or an applicative for short, is an intermediate structure between functors and monads. such that each. 01:02:26 Indo Keseringan Diewe Titit Sampai Kendor. Recall from Functions that a function is an object that maps a tuple of arguments to a return value, or throws an exception if no appropriate value can be returned. Proof. Public access must be granted to the overloading of the operator in order to be used as intended. Tên của bạn Địa chỉ email Nội dung. The free theorem for fmap. The default definition is fmap . thus you always start with something like. A functor that has both properties is called a fully faithful functor. Simontok– Nonton Video Bokep Indo Viral Funcrot Abg Mesum Di Gudang Sekolah terbaru durasi panjang full HD disini. A diagram is a collection of objects and morphisms, indexed by a fixed category; equivalently, a functor from a fixed index category to some category . A functor is a typed data structure that encapsulates some value (s). A post in Functional JavaScript Blog states that a functor is a function that, “given a value and a function, unwraps the values to get to its inner value (s), calls the given function with the. C++ defines several function objects that represent common arithmetic and logical operations: Arithmetic operations. Tante Keenakan Ngewe Sampai Crot Dalam. Note that fixing the first argument of Hom naturally gives rise to a covariant functor and fixing the second argument naturally gives a contravariant functor. 1. g. Advertisements. In Haskell, the term functor is also used for a concept related to the meaning of functor in category theory. A functor containing values of type a; The output it produces is a new functor containing values of type b. 4. You could add static variables to your function, but those would be used for any invocation of the function. Functions are not something on their own anymore, but they are always connected to objects in a modular fashion. Experts point out that a functor is created by overloading the operator and passing one argument the way that one would to a conventional function, albeit with different results. site for free in terms of their online performance: traffic sources, organic keywords, search rankings, authority, and much. Dual (category theory) 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 category Cop. "Minimality" is expressed by the functor laws. Functors. Functors exist in both covariant and contravariant types. BOKEP INDO | BOKEP ASIA | BOKEP JEPANG | BOKEP BARAT | FILM SEMI. fmap is used to apply a function of type (a -> b) to a value of type f a, where f is a functor, to produce a value of type f b. Yet more generally, an exponential. In Haskell this idea can be captured in a type class : classFunctorfwherefmap::(a->b)->fa->fb. , every arrow is mapped to an arrow . If f is some function then, in terms of your diagrams' categorical language, F (f) is . The functor will be producing a new module N as part of that application. What's a Functor? At the highest level of abstraction, a functor is a concept in Category Theory, a branch of mathematics that formalizes relationships between abstract objects via formal rules in any given collection of objects, referred to as Categories. Functor categories are of interest for two main reasons: $\begingroup$ This is slightly more intuitive for a less mathematically knowledgeable crowd. You can look at such a function as a mapping of a product (a pair, in Haskell) to another type (here, c ). This map function lifts an a -> b function to being a “list of a to list of b ” function. We introduce three concepts of CT which include (i) category; (ii) inclusion functor and expansion functor; and, most importantly, (iii) natural transformation. The fibres of the the two functors are the hom-sets, and the fact that $phi$ is a functor corresponds to naturality of the bijection. Various examples of non-representable functors. Second, the compiler can inline calls to the functor; it cannot do the same for a function pointer. The name is perhaps a bit itimidating, but **a functor is simply a "function" from structures to structures. The functor G : Ab → R-Mod, defined by G(A) = hom Z (M,A) for every abelian group A, is a right adjoint to F. The pullback is written. Nonton Video Porno HD BOKEP INDONESIA, Download Jav HD Terbaru Gratis Tanpa Iklan dan masih banyak video bokep yang kami sediakan seperti BOKEP BARAT, FILM SEMI. This follows from the results of chap II sections 2. Related concepts. In homotopy type theory. Functor categories serve as the hom-categories in the strict 2-category Cat. The fundamental group functor $pi_1$, maps (sends) topological spaces to groups and continuous maps to group homomorphisms:Website Video Live Record Aplikasi Streaming Mango, Bling2, Papaya, Dream Live, Bigo, Ome TV, Rekaman VCS dan Skandal LainnyaSang mudir ini sangat disegani, begitu pula istrinya Nyi Laila. Let’s see if we can figure out just what it means. So we have two cases: So we have two cases: [ pure x = (\_ -> x) ]: For pure we need to wrap a given -> r x into some functor but we are defining a function that just ignores input data type and returns data type x . When covering the vital Functor and Monad type classes, we glossed over a third type class: Applicative, the class for applicative functors. Covariant Functor, Functor , Hom. Nonton / streaming bokep Crot di Dalam Memek Sampai Tumpeh Tumpeh. So, you can think about a functor as a "function" (which indeed is not) between both objects and morphisms. Kalau anda suka video bokep Crot di Dalam Memek Sampai Tumpeh Tumpeh mungkin tertarik untuk menelusuri bokep sejenis lainnya yang berada dalam kegori Bokep Indo. Remark A split epimorphism r ; B → A r; B \to A is the strongest of various notions of epimorphism (e. 85795 views 100%. In particular, we’ve used modules to organize code into units with specified interfaces. Bokep Indo Viral Funcrot Abg Mesum Di Gudang Sekolah | Video Viral Thursday, 09/11/2023 Video yang Sedang viral saat ini di twitter Tiktok. 19:40 Mantan Bahenol Memek Terempuk. The closest thing to typeclasses in Elixir is protocols. A forgetful functor (also called underlying functor) is defined from a category of algebraic gadgets (groups, Abelian groups, modules, rings, vector spaces, etc. The maps. But what the hell does this mean. , nouns, verbs, adjectives, or adverbs, new words may be added readily, such as slang words, technical terms, and adoptions and adaptations of foreign words. Function Objects (Functors) - C++ allows the function call operator () to be overloaded, such that an object instantiated from a class can be "called" like a function. example pure (*2) should return. In this example I am taking an Array of Numbers and morphing it into an Array of Strings. Essentially, the product of a family. It is good for similar things that the const. Definition. They can be created by defining the function call operator (operator ()) They are created by declaring a function with a specific name and syntax. ) Wikipedia contains no definition. In other words, a contravariant functor acts as a covariant functor from the opposite category C op to D. Ia Melihat Royhan yg berjalan ke gedung Ri'ayah berdasarkan perintah kyainya tadi. This might seem a bit artificial at first but becomes useful for example in the study of topos theory: if we have a category C with pullbacks and a morphism f ∈ HomC(X, Y) where X, Y ∈ Ob(C), then the pullback construction induces a functor between slice categories C / Y → C / X. For instance, lists are this kind of container, such that fmap (+1) [1,2,3,4] yields [2,3,4,5]. Trnková, How large are left exact functors?, Theory and Applications of Categories 8 (2001), pp. Suppose given functors L: C → D L \,\colon\, C \to D, R: D → C R: D \to C and the structure of a pair of adjoint functors in the form of a. For Haskell, a functor is a structure/container that can be mapped over, i. Miss V Prank Ojol 156 3 Mb) — Jilbabviral Com. The meaning of SCROT- is scrotum. toString() const array = [1, 2, 3]. (Here C / X has as objects a pair of an object Z in C and a. [1] It is said that commutative diagrams play the role in category theory. A foldable container is a container with the added property that its items can be 'folded'. 3. 3 of Grothendieck. 4. Functors are objects that behave as functions. 00:20:56. Pesantren itu awalnya hanyalah Kobong Biasa yg terbuat dari Bale. For example, we could define a functor for std::vector like this:A contravariant functor F: C → D is like a covariant functor, except that it "turns morphisms around" ("reverses all the arrows"). You can define this functor for every four categories and two functors between them. But before detailing the latter, let's refresh our definition of a Haskell typeclass. A Functor is an inbuilt class with a function definition like −. Nonton dan Download Indo Viral Funcrot Abg Mesum Di Gudang Sekolah Skandal abg mesum tiktok Video Bokep Viral Tiktok, Instagram, Twitter, Telagram VIP Terbaru GratisIn mathematics, specifically category theory, a functor is a mapping between categories. 3. 121-152. We would like to show you a description here but the site won’t allow us. Proposition 0. It generalises the notion of function set, which is an exponential object in Set. See also weak equivalence of internal categories. 03:17. Note that the (<$) operator is provided for convenience, with a default implementation in terms of fmap; it is included in the class just to give Functor instances the opportunity to provide a more efficient implementation than the default. What Are Functor Laws? Every Functor implementation has to satisfy two laws: Identity, and Associativity. Functor. A Foldable type is also a container. If a type constructor takes two parameters, like. In the context of enriched category theory the functor category is generalized to the enriched functor category. A pragmatic new design for high-level abstractions. 7). Yes, all Functor instances are endofunctors on Hask --in fact, endofunctors from all of Hask to a proper subcategory whose objects are the types obtained by applying a particular type constructor. So, we can see that Array is a functor, because it respects the same type (results in other Array instance) and the connections too (have the same number of items). Function; interface. The differential operator in calculus is a common example, since it maps a function to its derivative, also a function. Thus, universal properties can be used for defining some objects independently from the method. An array is a good example of a functor, but many other kinds of objects can be mapped over as well, including promises, streams, trees, objects, etc. But there is an equivalent definition that splits the multi-argument function along a different boundary. For one, the functor can contain internal state; a state that is valid for this invocation of the function object only. Funcrot Website Dewasa Terlengkap, Nonton "Putri Lestari Hijab Binal. You cannot do this with functors. Postingan Terbarufunction word: [noun] a word (such as a preposition, auxiliary verb, or conjunction) that expresses primarily a grammatical relationship. ) to the category of sets. g. which are natural in C ∈ 𝒞 C in mathcal{C}, where we used that the ordinary hom-functor respects (co)limits as shown (see at hom-functor preserves limits), and that the left adjoint C ⊗ (−) C otimes (-) preserves colimits (see at adjoints preserve (co-)limits). sets and functions) allowing one to utilize, as much as possible, knowledge about. What does functor mean? Information and translations of functor in the most comprehensive dictionary definitions resource on the web. the “most optimized solution” to the. HD. So we can think of Id as taking a set and sending it to the set of all its elements. 2. Wolfram MathWorld defines it in terms of functors from algebraic categories to the category of sets, but then says, "Other forgetful functors. Any exact sequence can be broken down into short exact sequences (the Ci C i are kernels/images): So, since your functor F F preserves short exact sequences, you can apply F F and the diagonal sequences will remain exact. Functor. Methods. [1] They may be defined formally using enrichment by saying that a 2-category is exactly a Cat -enriched category and a 2-functor is a Cat -functor. Applicative functors allow for functorial computations to be sequenced (unlike plain functors), but don't allow using results from prior computations in the definition of subsequent ones (unlike monads). Then Id ≅ Set(1, −). An enriched functor is the appropriate generalization of the notion of a functor to enriched categories. fmap g = fmap (f . 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 endpoints lead to the same result. Note that for any type constructor with more than one parameter (e. 0 then 0 else 2 would then represent a value which switches at time 2. First there is a functor, denoted H 08:21 Gadis Mulus Kena Crot 2 Kali. HD 3881 View 00:05:13. Local Kan extension. It is a generalization of the map higher-order function. Retracts are clearly preserved by any functor. The reason this helps is that type constructors are unique, i. A sequence of one or more arguments, enclosed in brackets and. 21. , b in `Either a b`). g. According to Wikipedia: Let C and D be categories. a function that returns a monad (and a monadic value). Aku pun melihat sekitar rupanya banyak yang memperhatikanku. The class is defined as follows: class Functor f where fmap :: (a -> b) -> f a -> f b. Functor. It is a typical example of an applicative functor that is. If you tell this functor some location a a, it will spit out all the different routes you could take from a a to the library L L. In Haskell terms, fmap is a method in the typeclass Functor, not the functor itself. Okay, that is a mouth full. Hence you can chain two monads and the second monad can depend on the result of the previous one. But OCaml’s module system can do much more than that, serving as a powerful tool for building generic code and structuring large-scale systems. The concept of derived functors explains and clarifies many of these observations. 0 seconds of 5 minutes, 0Volume 90%. This means that it's both a "function" that takes a set and gives you back a set, and also a "function" that takes a morphism of sets (a function) and gives you back a morphism of sets. You can parameterise a functor more easily. Suppose that if Fi is a -functor such that for any i>0 Fi is e↵acable. HD 0 View 00:00:12. 31:11 Bokep Jepang Konoha Threesome Crot Didalam. Bokep Hot Crot Berkali-Kali Sampai Lemes | Foto Memek, Nonton film bokep,bokep barat,film bokep barat,video bokep,video. In functional programming, an applicative functor, or an applicative for short, is an intermediate structure between functors and monads. Funcrot Website Dewasa Terlengkap, Nonton "Ngintip Abg Di Kamar Mandi. – user6445533. com for free in terms of their online performance: traffic sources, organic keywords, search rankings, authority, and much. An adjunction is a pair of functors that interact in a particularly nice way. e. Meaning of functor. 6. The diagonal functor ΔJ C: C → CJ Δ C J: C → C J and the constant functors ΔJ C(c): J → C Δ C J ( c): J → C definitions are a bit too generous and lead to contradictions when applied to J = 0 J = 0 (the initial category). Then Fi = RiF0. 9. Such functors give representations of an abstract category in terms of known structures (i. Such left adjoints to a precomposition are known as left Kan extensions. The promise functor. In programming languages like Scala, we can find a lot of uses for Functors. There are video recordings with those content: part 1, part II and part III. map with type (A => B) => F [B]. is called a forgetful functor and there are many such functors. 00:00. HD. Episodes Abg SMP Cantik Mulus Colok Meki Bokep Indo Viral 4This includes the infamous Monad, the unknown Applicative, and the subject of this post: Functor. The book "Manifolds, Sheaves, and Cohomology" (written by Torsten Wedhorn) gives the following definition of adjoint functors: Definition: Let C, D be two categories and let F: [C] → [D] and G: [D] → [C] be functors. Given a statement regarding the category C, by interchanging the source and target of each morphism as well as interchanging the order of composing two. 9. Instances of std::function can store, copy, and invoke any CopyConstructible Callable target-- functions (via pointers thereto), lambda expressions, bind expressions, or other function objects, as well as pointers to member functions and pointers to data members. The notion appears in representation theory as well. Relationship with well-pointedness. A book that I states that functions take numbers and return numbers, while functionals take functions and return numbers - it seems here that you are saying functors can take both 1) functions and return functions, and 2) take numbers and return functions. In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". 2. Even though the indexed family isn't literally the same as the corresponding functor - the latter has the additional data of which morphisms go where, even though that data is trivial in the sense of being completely determined by the functor's action on objects alone - we can draw a conclusion about the latter by analyzing the former.