The {\it Lusternik-Schnirelmann category} of a space $X$, denoted $cat\left( X\right) $, is defined to be the minimal integer $k$ such that there exists an open covering $\left\{ A_{0},\dots ,A_{k}\right\} $ of $X$ with each $A_{i}$ contractible to a point in $X$. The motivation for introducing this concept was that for a closed differentiable manifold $M$, $cat\left(M\right) +1$ gives a lower bound for the number of critical points of a differentiable real function $f$ on M.

In 1986, M. Clapp and D. Puppe proposed the following generalization: If $\mathcal{A}$ is any class of spaces they replace the condition that $A_{j}\subset X$ is nulhomotopic by requiring that it factors throught some $A\in \mathcal{A}$ up to homotopy and they obtain the notion of {\it $\mathcal{A}$-category}. Roughly, they show that the $\cal A$-category, under certain conditions, gives new information on the topological structure of the critical set.

In this talk, for a closed $3$-manifold $M$, we relate the $\mathcal{A}$-$cat\left( M\right) $ with classical $3$-manifold theory and give an overview of what is known about these invariants.