mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the free category or path category generated by a directed graph or quiver is the

category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

that results from freely concatenating arrows together, whenever the target of one arrow is the source of the next. 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 :

V_0\xrightarrow V_1\xrightarrow 
 \cdots \xrightarrow V_n

where

V_k

is a vertex of the quiver,

E_k

is an edge of the quiver, and ''n'' ranges over the non-negative integers. For every vertex

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 :

V_0\xrightarrow\cdots\xrightarrow V_n,\quad V_n\xrightarrowW_0\xrightarrow\cdots\xrightarrow W_m,

their composition is :

\left(V_n\xrightarrowW_0\xrightarrow\cdots\xrightarrow W_m \right) \circ \left(V_0\xrightarrow\cdots\xrightarrow V_n \right) := V_0\xrightarrow\cdots\xrightarrow V_n\xrightarrowW_0\xrightarrow\cdots\xrightarrow 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 is the quiver with one vertex and one edge from that object to itself, then the free category on has as arrows , , ∘,∘∘, etc. * Let be the quiver with two vertices , and two edges , from to and to , respectively. Then the free category on has two identity arrows and an arrow for every finite sequence of alternating s and s, including: , , ∘, ∘, ∘∘, ∘∘, etc. * If is the quiver

a\xrightarrowb\xrightarrowc

, then the free category on has (in addition to three identity arrows), arrows , , and ∘. * If a quiver has only one vertex, then the free category on has only one object, and corresponds to the

free monoid In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero ...

on the edges of .

Properties

The

category of small categories 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-c ...

Cat has a

forgetful functor 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 of the input's structure or properties mapping to the output. For an algebraic structure of ...

into the quiver category Quiv: : : Cat → Quiv which takes objects to vertices and morphisms to arrows. Intuitively, " orgetswhich arrows are composites and which are identities". 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 Two Mathematical object, mathematical objects and are called "equal up to an equivalence relation " * if and are related by , that is, * if holds, that is, * if the equivalence classes of and with respect to are equal. This figure of speech ...

isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

by a

universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...

. Let : Quiv → Cat be the functor that takes a quiver to the free category on that quiver (as described above), let be the forgetful functor defined above, and let be any quiver. Then there is a graph homomorphism : → (()) and given any category D and any graph homomorphism : → , there is a unique functor : () → D such that ()∘=, i.e. the following diagram commutes: The functor is left adjoint to the forgetful functor .

References

{{category theory Free algebraic structures

Examples

Properties

Universal property

See also

References