Free category

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.

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

$V_0\stackrel{E_0}{\to }{V}_1\stackrel{E_1}{\to }\cdots \stackrel{E_{n-1}}{\to }{V}_n$

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

$V_0\stackrel{E_0}{\to }\cdots \stackrel{E_{n-1}}{\to }{V}_n,V_n\stackrel{F_0}{\to }{W}_0\stackrel{F_1}{\to }\cdots \stackrel{F_m}{\to }{W}_m,$

their composition is

$(V_n\stackrel{F_0}{\to }{W}_0\stackrel{F_1}{\to }\cdots \stackrel{F_m}{\to }{W}_m)\circ (V_0\stackrel{E_0}{\to }\cdots \stackrel{E_{n-1}}{\to }{V}_n):=V_0\stackrel{E_0}{\to }\cdots \stackrel{E_{n-1}}{\to }{V}_n\stackrel{F_0}{\to }{W}_0\stackrel{F_1}{\to }\cdots \stackrel{F_m}{\to }{W}_m$.

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

Examples

• If Template:Var is the quiver with one vertex and one edge Template:Var from that object to itself, then the free category on Template:Var has as arrows Template:Var, Template:Var, Template:Var∘Template:Var,Template:Var∘Template:Var∘Template:Var, etc.
• Let Template:Var be the quiver with two vertices Template:Var, Template:Var and two edges Template:Var, Template:Var from Template:Var to Template:Var and Template:Var to Template:Var, respectively. Then the free category on Template:Var has two identity arrows and an arrow for every finite sequence of alternating Template:Vars and Template:Vars, including: Template:Var, Template:Var, Template:Var∘Template:Var, Template:Var∘Template:Var, Template:Var∘Template:Var∘Template:Var, Template:Var∘Template:Var∘Template:Var, etc.
• If Template:Var is the quiver $a\stackrel{f}{\to }b\stackrel{g}{\to }c$, then the free category on Template:Var has (in addition to three identity arrows), arrows Template:Var, Template:Var, and Template:Var∘Template:Var.
• If a quiver Template:Var has only one vertex, then the free category on Template:Var has only one object, and corresponds to the free monoid on the edges of Template:Var.

Properties

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

Template:Var : Cat → Quiv

which takes objects to vertices and morphisms to arrows. Intuitively, Template:Var "[forgets] which 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 isomorphism by a universal property. Let Template:Var : Quiv → Cat be the functor that takes a quiver to the free category on that quiver (as described above), let Template:Var be the forgetful functor defined above, and let Template:Var be any quiver. Then there is a graph homomorphism Template:Var : Template:Var → Template:Var(Template:Var(Template:Var)) and given any category D and any graph homomorphism Template:Var : Template:Var → Template:Var, there is a unique functor Template:Var : Template:Var(Template:Var) → D such that Template:Var(Template:Var)∘Template:Var=Template:Var, i.e. the following diagram commutes:

The functor Template:Var is left adjoint to the forgetful functor Template:Var.

See also

• Free strict monoidal category
• Free object
• Adjoint functors