# Free category

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In mathematics, the free category or path category generated by a directed graph or quiver is the category that results from freely concatenating arrows together, whenever the target of one arrow is the source of the next.

Mathematics includes the study of such topics as quantity, structure, space, and change.

In mathematics, and more specifically in graph theory, a directed graph is a graph that is made up of a set of vertices connected by edges, where the edges have a direction associated with them.

In mathematics, a quiver is a directed graph where loops and multiple arrows between two vertices are allowed, i.e. a multidigraph. They are commonly used in representation theory: a representation V of a quiver assigns a vector space V(x) to each vertex x of the quiver and a linear map V(a) to each arrow a.

## Contents

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 as a finite sequence

${\displaystyle V_{0}{\xrightarrow {\;\;E_{0}\;\;}}V_{1}{\xrightarrow {\;\;E_{1}\;\;}}\cdots {\xrightarrow {E_{n-1}}}V_{n}}$

where ${\displaystyle V_{k}}$ is a vertex of the quiver, ${\displaystyle E_{k}}$ is an edge of the quiver, and n ranges over the non-negative integers. For every vertex ${\displaystyle V}$ of the quiver, there is an "empty path" which constitutes the identity morphisms of the category.

The composition operation is concatenation of paths. Given paths

${\displaystyle V_{0}{\xrightarrow {E_{0}}}\cdots {\xrightarrow {E_{n-1}}}V_{n},\quad V_{n}{\xrightarrow {F_{0}}}W_{0}{\xrightarrow {F_{1}}}\cdots {\xrightarrow {F_{n-1}}}W_{m},}$

their composition is

${\displaystyle \left(V_{n}{\xrightarrow {F_{0}}}W_{0}{\xrightarrow {F_{1}}}\cdots {\xrightarrow {F_{n-1}}}W_{m}\right)\circ \left(V_{0}{\xrightarrow {E_{0}}}\cdots {\xrightarrow {E_{n-1}}}V_{n}\right):=V_{0}{\xrightarrow {E_{0}}}\cdots {\xrightarrow {E_{n-1}}}V_{n}{\xrightarrow {F_{0}}}W_{0}{\xrightarrow {F_{1}}}\cdots {\xrightarrow {F_{n-1}}}W_{m}}$

Note that the result of the composition starts with the right operand of the composition, and ends with its left operand.

## Examples

• If Q is the quiver with one vertex and one edge f from that object to itself, then the free category on Q has as arrows 1, f, ff,fff, etc. [2]
• Let Q be the quiver with two vertices a, b and two edges e, f from a to b and b to a, respectively. Then the free category on Q has two identity arrows and an arrow for every finite sequence of alternating es and fs, including: e, f, ef, fe, fef, efe, etc. [1]
• If Q is the quiver ${\displaystyle a{\xrightarrow {f}}b{\xrightarrow {g}}c}$, then the free category on Q has (in addition to three identity arrows), arrows f, g, and gf.
• If a quiver Q has only one vertex, then the free category on Q has only one object, and corresponds to the free monoid on the edges of Q. [1]

In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero elements, often called the empty string and denoted by ε or λ, as the identity element. The free monoid on a set A is usually denoted A. The free semigroup on A is the subsemigroup of A containing all elements except the empty string. It is usually denoted A+.

## Properties

The category of small categories Cat has a forgetful functor U into the quiver category Quiv:

In mathematics, specifically in category theory, the category of small categories, denoted by Cat, is the category whose objects are all small categories and whose morphisms are functors between categories. Cat may actually be regarded as a 2-category with natural transformations serving as 2-morphisms.

In mathematics, in the area of category theory, a forgetful functor 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given signature, this may be expressed by curtailing the signature: the new signature is an edited form of the old one. If the signature is left as an empty list, the functor is simply to take the underlying set of a structure. Because many structures in mathematics consist of a set with an additional added structure, a forgetful functor that maps to the underlying set is the most common case.

U : CatQuiv

which takes objects to vertices and morphisms to arrows. Intuitively, U "[forgets] which arrows are composites and which are identities". [2] This forgetful functor is right adjoint to the functor sending a quiver to the corresponding free category.

### Universal property

The free category on a quiver can be described up to isomorphism by a universal property. Let C : QuivCat be the functor that takes a quiver to the free category on that quiver (as described above), let U be the forgetful functor defined above, and let G be any quiver. Then there is a graph homomorphism I : GU(C(G)) and given any category D and any graph homomorphism F : GU(B), there is a unique functor F' : C(G) → D such that U(F')∘I=F, i.e. the following diagram commutes:

In mathematics, the phrase up to appears in discussions about the elements of a set, and the conditions under which subsets of those elements may be considered equivalent. The statement "elements a and b of set S are equivalent up to X" means that a and b are equivalent if criterion X is ignored. That is, a and b can be transformed into one another if a transform corresponding to X is applied.

In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation.

In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.

The functor C is left adjoint to the forgetful functor U. [1] [2] [3]

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## References

1. Awodey, Steve (2010). Category theory (2nd ed.). Oxford: Oxford University Press. pp. 20–24. ISBN   0199237182. OCLC   740446073.
2. Mac Lane, Saunders (1978). Categories for the Working Mathematician (Second ed.). New York, NY: Springer New York. pp. 49–51. ISBN   1441931236. OCLC   851741862.
3. "free category in nLab". ncatlab.org. Retrieved 2017-09-12.