Simplicial lusternik schnirelmann category

This master's thesis discusses the work on the notion of simplicial LS-category $(scatK)$ introduced by D. Fernàndez-Ternero, E. Macìas-Virgòs and J. A. Vilches (2015), along with some of its analogues. The thesis analyzes the basic properties of the simplicial LS-category in the context of simplexes, graphs, and homotopy theory. The thesis, composed of four chapters, examines the relationship between the notion of $scat(K)$ and certain related concepts such as $catX$ and $gscatK$. It also investigates how the simplicial LS-category is affected by the geometric realization and the barycentric subdivision of the complex $K$. The fat wedge is introduced and the Whitehead LS-category is defined using this notion. By adapting the definition of the Whitehead LS-category to simplicial complexes, the simplicial Whitehead LS-category is analyzed. Furthermore, a combinatorial analogue of the homotopy extension theorem for a simplicial pair $(K, L)$ is presented. In the final chapter, the graphs are considered, and their simplicial LS-category is examined. Calculations for $scatK$ are carried out using the concept of the arboricity.