Let A be an algebra over a field k and V a vector space. A representation (Drozd, Kirichenko (1980)) of k-algebra A is a homomorphism h from A
h: A M(A), (6)
to the algebra *M(A)* of linear operators of V. If h is an isomorphism h(A) A (7)
then the representation is the exact.

Given a quiver Q its path category *PathQ* has as objects the vertices of Q and the morphisms i j (8)
are the paths from i to j which are by definition the formal compositions a1, ,an (9)
an starts in *i, a1* ends in j and the end point of *ai+1* coincides with the start point of ai for all i = 1, , *n-1.* (10)
 We extend *PathQ* to the linear category *PathLQ* with the same class of objects and with morphisms Mor (i, j) = {the set of all paths from i to j and all their linear combinations with coefficients from k}.

 An equivalent definition of the representation of a quiver is an additive functor from PathLQ to the category of k-*mod.*
A quiver Q is marked (Roiter, (2001)) if for each i, j V is attached a category Ci
, and to each arrow a: i j in Q is attached a functor Fa: (Ci
 )0 Cj Ens (11)
taking values in the category of sets.

A *categorification* is the category theory representation of the world that we want to investigate. If an ontology has a representation by a category then we may apply homological and homotopical algebra constructions to study the ontology (Glazunov, 2007, 2006). 

Below we give category theory concepts for our framework (an category theory ontology is given by Robert Kent (Kent, 2002)).

Category C (small or big), object (X, Y, ), morphism, class of morphisms HomC (X, Y)
(C(X, Y) or *MorC(X,* Y)) or set of morphisms homC (X, Y) (C(X, Y) or *morC(X,* Y)) of C, dual category, monoid (a category with a single object x and with \#hom(x, x) 1). Subcategory. Full subcategory. Category of binary relations. Let Ens be the category of sets, Annk be the category of commutative rings with unit over a commutative ring k. *Spec* R. The category Grp of all groups with group homomorphisms as morphisms. The category Ab of abelian groups and group homomorphisms. Bicategory. A monoidal category (or tensor category) is a bicategory with one object. n- categories. Groupoid. Picard groupoid. Stack. 