### The poset of matrix properties

Below are the notes for the talk above, given at the Algebra, Geometry, Topology & Applications seminar. 1. Bird's-Eye View of Exactness Properties One of the active areas of research in Categorical Algebra is the study of various properties of categories expressed using limits and colimits. Such properties are usually referred to as exactness properties . This terminology comes from the fact that, historically, the first such properties emerged in the study of exact sequences in the sense of Homological Algebra. The matrix properties in the title of this post are particular types of exactness properties, which can be encoded using integer matrices. Before explaining what they are, let us first recall the notions of limit and colimit. Given a diagram of objects and arrows (objects are certain mathematical structures and arrows are morphisms between them), a limit (of the diagram) is a way to encode the information about the diagram in a single object; it is a terminal (commu