In the mathematical field of algebraic topology , the fundamental group is a mathematical group associated to any given pointed topological space that provides a way to determine when two paths , starting and ending at a fixed base point , can be continuously deformed into each other. It records information about the basic shape, or holes, of the topological space.
The fundamental group is the first and simplest homotopy group. The fundamental group is a topological invariant : homeomorphic topological spaces have the same fundamental group. Fundamental groups can be studied using the theory of covering spaces , since a fundamental group coincides with the group of deck transformations of the associated universal covering space. The abelianization of the fundamental group can be identified with the first homology group of the space.
When the topological space is homeomorphic to a simplicial complex , its fundamental group can be described explicitly in terms of generators and relations. It describes the monodromy properties of complex-valued functions , as well as providing a complete topological classification of closed surfaces. Start with a space for example, a surface , and some point in it, and all the loops both starting and ending at this point — paths that start at this point, wander around and eventually return to the starting point.
Two loops can be combined together in an obvious way: travel along the first loop, then along the second. Two loops are considered equivalent if one can be deformed into the other without breaking. The set of all such loops with this method of combining and this equivalence between them is the fundamental group for that particular space. Throughout this article, X is a topological space. A typical example is a surface such as the one depicted at the right.
As is explained below, its role is rather auxiliary. The idea of the definition of the homotopy group is to measure how many broadly speaking curves on X can be deformed into each other. The precise definition depends on the notion of the homotopy of loops, which is explained first. More precisely, h is required to satisfy the following conditions:. The purpose of considering the equivalence classes of loops up to homotopy, as opposed to the set of all loops the so-called loop space of X is that the latter, while being useful for various purposes, is a rather big and unwieldy object.
By contrast the above quotient has, in many cases, a manageable and computable size. It becomes a group and therefore deserves the name fundamental group using the concatenation of loops. The inverse of a homotopy class of a loop is the same loop, but traversed in the opposite direction. More formally,.
By comparison,. Thus, because of the differing speeds, the two paths are not literally the same. The associativity axiom.
Although the fundamental group in general depends on the choice of base point, it turns out that, up to isomorphism actually, even up to inner isomorphism , this choice makes no difference as long as the space X is path-connected. More generally, any star domain and, yet more generally any contractible space has a trivial fundamental group.
Thus, the fundamental group does not distinguish between such spaces. A path-connected space whose fundamental group is trivial is called simply connected. Instead, each homotopy class consists of all loops which wind around the circle a given number of times which can be positive or negative, depending on the direction of winding.
This fact can be used to give proofs of the Brouwer fixed point theorem and the Borsuk—Ulam theorem in dimension 2. Since the fundamental group is a homotopy invariant, the theory of the winding number for the complex plane minus one point is the same as for the circle.
The fundamental group of the figure eight is the free group on two letters. More generally, the fundamental group of a bouquet of r circles is the free group on r letters. The fundamental group of a wedge sum of two locally contractible spaces X and Y can be computed as the free product of the individual fundamental groups:. The fundamental group of the plane punctured at n points is also the free group with n generators. The i -th generator is the class of the loop that goes around the i -th puncture without going around any other punctures. The fundamental group of any graph is a free group.
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F gives rise to For example Java's Optional type violates the functor laws e. This index includes documentation for many Haskell modules. There is no color for Traversing as it's almost the combination of Traversing1 and Choice. This note explores the algebraic perspective on the notion of generalized Reedy category introduced by Berger and Moerdijk [BM08]. For example, if you use Church encoding for your numbers, how do you do static analysis of bounds checking? If you implement if-statements by passing two functions to a function which calls one or the other for truth and falsity, how do you determine dead code at compile time?
The limit L of such a diagram is called an equalizer of those morphisms. Submitted for publication. It is just the Enumerable function above. Just what is a Functor, really? This page is no longer maintained — Please continue to the home page at www. Haskell Hierarchical Libraries.
The tests only showcase five examples, but I hope it gives you an intuition how any rose tree is a bifunctor. Bifunctor base The applicative functor definition is actually equivalent to above monoidal functor definition. Bifunctor module provides the Bifunctor class, of which pairs and Either are instances. Binary binary We have seen a technique for approximating a hypothetical Generic2 representation with only using Generic.
Moving on the section 3. Support for deriving Functor, Foldable, and Traversable instances. A Bifunctor is very much like a Functor, except that there are two independent ways of mapping functions over it, corresponding to the first and second methods. For example, the set of functions from any set X into a vector space have a natural vector space structure given by pointwise addition and GitHub Gist: star and fork jnape's gists by creating an account on GitHub.
The Fantasy Land Specification "specifies interoperability of common algebraic structures" by defining a number of type classes. GHC 6. We give a simpler proof of this theorem which generalizes the proof mentioned above for the case of modules. Functor recap.
A bifunctor also known as a binary functor is a functor whose domain is a product category. A multifunctor is fine too. One that performs an operation or a function. After all, the SelectNode, SelectLeaf, and SelectBoth methods are all generic, and they behave the same for all generic type arguments. The product does not necessarily exist. Functional programming, however, has a relatively high number of programmers, that care about correctness and mathematical formalism, that leads to it.
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The limit of this diagram is called the J th power of X and denoted X J. Let Rbe a commutative ring. The Data. We capture this constraint as a type class.
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A lot of effort went into this, I also took a look at the languages I'm familiar with. If J is a category with two objects and two parallel morphisms from one object to the other, then a diagram of shape J is a pair of parallel morphisms in C. Figure out which is a Bifunctor and which is a Profunctor. Bifunctor, which is available in base, Data. Earlier bifunctor was part of bifunctors package.
They even have screenshots of the results and provide an excellent way of showing exactly what you would input and see. Instancias comunes de Bifunctor. You can define a Bifunctor by either defining bimap or by defining both first and second. Now we can derive even more interesting Bifunctor instances.
For example, Moors et al. It can be read as Functor, or Wrap, which wraps a value into an applicative functor monoidal functor.