Categories where every morphism is a monomorphism

Question

What is the name given to categories where every morphism is a monomorphism? Where have these been studied?

Answer 1

These categories are closely related to preorder categories (a.k.a. thin categories) and preordered groupoids:

Proposition. Let $\mathcal{C}$ be a category. The following are equivalent:

• Every morphism in $\mathcal{C}$ is a monomorphism.

• For every object $X$ in $\mathcal{C}$, the slice $\mathcal{C}_{/ X}$ is a preorder category.

• There exists a preorder category $\mathcal{D}$ and a surjective discrete fibration $P : \mathcal{D} \to \mathcal{C}$.

• (Assuming $\mathcal{C}$ is small.) The presheaf topos $[\mathcal{C}^\textrm{op}, \textbf{Set}]$ is an étendue, i.e. there exists a presheaf $D : \mathcal{C}^\textrm{op} \to \textbf{Set}$ such that the unique morphism $D \to 1$ is an epimorphism and the slice $[\mathcal{C}^\textrm{op}, \textbf{Set}]_{/ D}$ is a localic topos.

• $\mathcal{C}$ is the codescent category of a preordered groupoid whose domain and codomain maps are discrete fibrations, i.e. we have a codescent diagram in $\textbf{Cat}$ of the form below, $$\mathcal{G}_2 \mathrel{\begin{array}{c} \rightarrow \\ \rightarrow \\ \rightarrow \end{array}} \mathcal{G}_1 \mathrel{\begin{array}{c} \rightarrow \\ \leftarrow \\ \rightarrow \end{array}} \mathcal{G}_0 \to \mathcal{C}$$ where $\mathcal{G}_2$, $\mathcal{G}_1$, and $\mathcal{G}_0$ are preorder categories, $\mathcal{G}_1 \rightrightarrows \mathcal{G}_0$ are discrete fibrations, and the $\mathcal{G}_\bullet$ comprise a groupoid object.

Étendues were originally defined in [SGA4, Exposé IV, Exercice 9.8.2]. The connection with groupoids was already noted then, but was generalised (to non-étendues) by various authors, particularly Joyal and Tierney [An extension of the Galois theory of Grothendieck].

Answer 2

Such categories are called left-cancellative (see the link for references). A one-object left-cancellative category is a left-cancellative monoid.

Answer 3

One example I can think of is the category $FI$, the category of finite sets (objects) and injections (arrows). This category is very important in representation theory wherein people study $FI$-modules over a ring (or more generally, over a small preadditive category). The notion of $FI$-modules was first introduced by T.Church, J. Eilenberg and B. Farb to obtain a new approach to the theory of stability of representations of $S_n$.