The category 3Cob has closed oriented surfaces as objects and 3-dimensional cobordisms, i.e. 3-dimensional compact oriented manifolds (possibly with boundary) with canonical orientation preserving (reversing) identification of the incoming (outgoing) boundary. The composition is defined in terms of gluing. We present this category using a diagrammatic language similar to the language of standard surgery presentation of closed, orientable, connected 3-manifolds, save that besides framed links we use wedges of circles in our diagrams.

We will explain how to interpret such a diagram as an arrow of 3Cob and give an outline of the composition calculus for diagrams. This is a joint work with Jovana Nikolic and Mladen Zekic.