SAMS Subject Classification Number: 4

There are various incidences across different areas of mathematics where objects are defined as being connected. In graph theory these are connected graphs, and in topology they are connected spaces. Examining some properties of these connected objects allows us to generalise and come up with a collection of category-theoretic conditions describing a more general form of connectedness. When we apply these conditions to the category of graphs and the category of topological spaces, we get the same objects as when we used the original definitions. We will consider so-called lextensive categories, in which the notion of objects being connected is well-behaved. Such a category is said to be locally connected if every object in it is a coproduct. A locally connected category \(\mathbb{C}\) is equivalent to \(\text{Fam}(\mathbb{A})\) for some category \(\mathbb{A}\), where \(\text{Fam}(\mathbb{A})\) denotes the category of families of objects in \(\mathbb{A}\). We will consider some properties and examples of such categories.