**$FinInj$** (or **$FinSet_inj$**) is the wide subcategory of FinSet with its morphisms restricted to monomorphisms (injective functions).

It may be characterised as:

- the free (symmetric) semicocartesian category on a single object
- the free symmetric monoidal category on a pointed object $I \to X$ (where $I$ is the monoidal unit)
- the free monoidal category on an object $X$ equipped with an involution $s : X \otimes X \to X \otimes X$ satisfying the braid relations, and a morphism $i : I \to X$ satisfying $s \circ i \otimes X = s \circ X \otimes i$

Created on September 25, 2022 at 12:13:02. See the history of this page for a list of all contributions to it.