A groupoid is a category in which every morphism is an isomorphism. I'm looking at a dual case: categories in which none morphism (except identites) is an isomorphism.
- These kind of categories are well-studied?
- There are implications of the assumption in terms of existence of limits/colimits?
- There is some characterization of them?
For instance, notice that if $\mathbf{D}$ is without isomorphisms, then there is no functor $F:\mathbf{C}\rightarrow \mathbf{D}$ where $\mathbf{C}$ is a category which has at least one isomorphism $f:X\rightarrow Y$ with $F(Y) \neq F(X)$. This means that categories without isomorphisms are typically not codomain of functors.
On the other hand, one can always enlarge a category by adding isomorphisms (for instance, take a set of weak equivalences and consider some localization at them). This shows that categories without isomorphisms can be domain of functors.
Any help is welcome.