What is $\text{Col}(\kappa,\lambda)$ in set theory

Question

I saw somewhere thing like this: $\text{Col}(\kappa,\lambda)$. What is this?

Answer 1

This is a notation from forcing. This is the Levy collapse of $\lambda$ to $\kappa$.

In actual details this is a partial order whose elements are functions from bounded subsets of $\kappa$ to $\lambda$. And we say that $p\leq q$ if $q\subseteq p$ (or if you work with Shelah, the other way around).

When we add a generic filter to $\text{Col}(\kappa,\lambda)$ the result is a new set is introduced to the model and this set is a surjection from $\kappa$ onto $\lambda$. So we collapse $\lambda$ to be the same cardinality as $\kappa$.