# What is an object with finite length in a tensor category?

I read the definition of a tensor category: A tensor category is a rigid abelian monoidal category in which the object 1 is simple and all objects have finite length.

This definition is in "Lectures on tensor categories" by Calaque and Etingof.

I understand all terminology but "finite length".

What does it mean for an object has finite length?

• Where did you get your definition of a tensor category? In the literature, several distinct things are referred to as "tensor categories." – Ben Sheller Sep 10 '15 at 15:18
• @BenS. I edited the question. Definition is in "Lectures on tensor categories" by Calaque and Etingof. – Snow Sep 10 '15 at 15:22

An object $X$ of an abelian category has finite length if it has a (finite) composition series: i.e., a chain of subobjects $$0=X_0<X_1<\dots<X_{n-1}<X_n=X$$ such that $X_i/X_{i-1}$ is simple for $1\leq i\leq n$.