Consider the category of pointed involutive sets $\pm\mathsf{Fin}_\ast^t$. An object of this category is a set $X$, an involution $x\mapsto -x$ of $X$, and a unique zero element $0_X \in X$ satisfying $-0_X = 0_X$. (This is the set of transitive pointed $\{\pm\}$-sets, see the appendix.)
For $X, Y \in \pm\mathsf{Fin}_\ast^t$, an $n$-alternating map is a map
$$
f \colon X^{n} \to Y
$$
such that
- $f(..., x, x', ...) = -f(..., x', x, ...)$ for all $x, x' \in X$ and
- $f(..., -x, ...) = -f(..., x, ...)$ for all $x \in X$
There is a functor
$$
\Lambda^n \colon \pm\mathsf{Fin}_\ast^t \to \pm\mathsf{Fin}_\ast^t
$$
such that $\mathrm{Hom}_{\pm\mathsf{Fin}_\ast^t}(\Lambda^n X, Y)$ is naturally equivalent to $n$-alternating maps from $X$ to $Y$.
Lemma $\Lambda^{\|X\|} X $ is a pointed $\{\pm 1\}$-torsor. (i.e., equivalent to $\{-1, 0, 1\}$).
Definition An orientation on a finite set $K$ is a trivialization of $\Lambda^{|K|} (\pm K_+)$, where $\pm K_+ = +K \sqcup -K \sqcup \{0\}$.
Unfortunately, just as in the case of vector spaces, the set of orientations is not preserved by arbitrary maps of finite sets, only equivalences. Indeed, the formula $\Lambda^{|K_1|} (\pm (K_2)_+)$ is contravariant in $K_1$, and covariant in $K_2$. So there is no "obvious" way to create a category $\mathsf{OrFin}$ of oriented finite sets concrete over $\mathsf{Fin}$.
That being said, given a partition $K = \bigsqcup_{i: I} K_i$ indexed by $I$, I think that giving an orientation on $K$ is equivalent to giving an orientation on $I$ and each $K_i$. This could be used to give a notion of "oriented map" (which would be extra structure on top of a usual function) $K_1 \to K_2$ which might not be bijective, since a map of sets is equivalent to a partition of the source indexed by the target.
Appendix
here I set up notation/definitions for $G$-sets (where $G$ is abelian), which we will use for the case that $G = \pm := \{\pm 1\}$.
- A pointed $G$-set will mean a $G$-set $X$ equipped with a distinguished $0_X \in X^G$ - an element fixed by $G$.
- Denote the category of finite pointed $G$-sets by $G\text{-}\mathsf{Fin}_\ast$.
- Given $X, Y, Z \in G\text{-}\mathsf{Fin}_\ast$, a map of underlying sets $X\times Y \to Z$ is G-bilinear if
- $f(gx, y) = f(x, gy) = g \cdot f(x, y)$ for all $x \in X$, $y \in Y$, $g \in G$
- $f(0_X, y) = f(x, 0_Y) = 0_Z$ for all $x \in X$, $y \in Y$.
- For $X,Y \in G\text{-}\mathsf{Fin}_\ast$, we may define $X \times_G Y \in G\text{-}\mathsf{Fin}_\ast$ co-representing $G$-bilinear maps:
$$
\mathrm{Hom}_{G\text{-}\mathsf{Fin}_\ast}(X \times_G Y, Z) =
\mathrm{Hom}_{G\text{-}\mathrm{bilin}}(X \times Y, Z)
$$
(here is where we used that $G$ is abelian)
- We will denote the coproduct in $G\text{-}\mathsf{Fin}_\ast$ by $X \vee Y$, since it identifies $0_X$ and $0_Y$.
- For $K \in \mathsf{Fin}$ a finite set, denote $GK_+ = G \times K \sqcup {0}$ the free pointed $G$-set on $K$. It is monoidal for the cartesian monoidal structure on $\mathsf{Fin}$ and $\times_G$ on $G\text{-}\mathsf{Fin}_\ast$
- The category $G\text{-}\mathsf{Fin}_\ast^t$ of
transitive finite pointed $G$-sets consists of those $X$ such that $X^G = \{0_X\}$. It is closed under $\times_G$ and is the full essential image of the free functor $K \mapsto GK_+$.
- For $X \in G\text{-}\mathsf{Fin}_\ast^t$, let $\|X\|$ denote the number of $G$-orbits of $X \setminus \{0_X\}$, so that $\|GK_+\| = |K|$.
- For $X \in G\text{-}\mathsf{Fin}_\ast$, let $X^{n, G}$ denote $X \times_G \dots \times_G X$, where $X$ is repeated $n$ times.
- A pointed $G$-torsor is a pointed $G$-set equivalent to $G\{\ast\}_+$ , the free pointed $G$-set on one element.
