# Mathematics Colloquia and Seminars

The sl_N spider'' (a notion introduced by Greg Kuperberg) is a diagrammatically defined category whose idempotent completion is equivalent to the category of finite dimensional representations of sl_N (or rather, the associated quantum group). Primitive idempotents in the spider play the role of irreducuble representations. Of particular importance are the generalized Jones-Wenzl projectors,'' which are projections onto the symmetric powers of the fundamental representation. In this expository talk I will describe a categorification of sl_N spider, and I will discuss how the generalized Jones-Wenzl projectors are categorified by certain infinite chain complexes. The categorified projections posses a lot of interesting structure, and can be used to defined colored link homology theories.