A category C consists of the following three mathematical entities: A class ob(C), whose elements are called objects; A class hom(C), whose elements are called morphisms or maps or arrows. Each morphism f has a source object a and target object b. The expression f : a → b, would be verbally stated as "f is a morphism from a to b".The expression hom(a, b) – alternatively expressed as homC(a, b), mor(a, b), or C(a, b) – denotes the hom-class of all morphisms from a to b. A binary operation ∘, called composition of morphisms, such that for any three objects a, b, and c, we have ∘ : hom(b, c) × hom(a, b) → hom(a, c). The composition of f : a → b and g : b → c is written as g ∘ f or gf,[a] governed by two axioms: 1. Associativity: If f : a → b, g : b → c, and h : c → d then h ∘ (g ∘ f) = (h ∘ g) ∘ f 2. Identity: For every object x, there exists a morphism 1x : x → x called the identity morphism for x, such that for every morphism f : a → b, we have 1b ∘ f = f = f ∘ ida.[b] From the axioms, it can be proved that there is exactly one identity morphism for every object. Some authors[who?] deviate from the definition just given, by identifying each object with its identity morphism.

A category is formed by two sorts of objects, the objects of the category, and the morphisms, which relate two objects called the source and the target of the morphism

Category theory is a general theory of mathematical structures and their relations

In probability theory, Kolmogorov equations, including Kolmogorov forward equations and Kolmogorov backward equations, characterize continuous-time Markov processes. In particular, they describe how the probability that a continuous-time Markov process is in a certain state changes over time.

The Lotka–Volterra system of equations is an example of a Kolmogorov mode

The Lotka–Volterra equations, also known as the predator–prey equations