Truncations of topological spaces have been a key part of algebraic topology. They can help us define homotopy groups and simplify complicated topological structure, as they give us a natural filtration on spaces. Despite those benefits constructing a functorial truncation is always a challenge and involves some set theoretical assumptions, such as the small object argument or local presentability.
In this talk we give an elementary way to construct localizations and truncations using the notion of reflective subuniverses. We will then illustrate the strength of this result by constructing truncations in a specific category without infinite colimits and show how it results in new internal truncation levels. | 
