With the initial motivation of looking for a categorical characterization of groups among monoids, we introduced some conditions on objects in general categories, describing good algebraic properties holding locally for these objects although not globally in the category. These conditions, arising from different sources (like universal algebra or homological algebra), are not equivalent in general, but in the category of monoids they all characterize groups. We will describe and compare them, pointing out implications between them and examples showing their differences. Joint work with Xabier Garcia Martinez, Diana Rodelo and Tim Van der Linden.