We modify a previous result, which showed that certain diagrams of spaces are essentially simplicial monoids, to construct diagrams of spaces which model simplicial groups. Furthermore, we show that these diagrams can be generalized to models for Segal groupoids. We then modify Segalâs model for simplicial abelian monoids in such a way that it becomes a model for simplicial abelian groups. Much research has been done on various structures equivalent to topological or simplicial groups. Classical examples include Thomasonâs work with delooping machines [27] and Stashefiâs group-like A1-spaces [26], while more modern examples make use of the structure of algebraic theories [1], [2]. In [7], we focus instead on simplicial monoids. In particular, we consider diagrams of spaces which essentially give one of the spaces the structure of a monoid. While such a structure is given when the diagram is the theory of monoids, we obtain a simpler diagram which, from the viewpoint of homotopy theory, encodes the same structure. Here we return to the original situation and ask if this construction can be modifled to obtain a diagram encoding the structure of a simplicial group rather than a simplicial monoid. After all, we need only to flnd a way to include inverses. In this paper, we show that indeed we can represent simplicial groups in this manner. Furthermore, as the construction for simplicial monoids generalizes to the manyobject case of simplicial categories, we can obtain models for simplicial groupoids as well. In doing so, we extend a result relating simplicial categories to Segal categories (essentially categories with composition only given up to higher homotopy) to one relating simplicial groupoids to Segal groupoids. One might ask, then, if there are other algebraic structures which can be modelled in a similar way, in particular by some kind of diagram which is simpler than the one given by the corresponding algebraic theory. An answer for the case of simplicial abelian monoids is given by Segalâs category i [24], although it does not provide the