Enrollment

Formal definition

An Enrollment is a triple $\mathcal{E} = (p, V, T)$:

$$\mathcal{E} = (p : \mathrm{Person},\; V : \mathrm{Vessel},\; T : \mathrm{Interval})$$

together with:

• A role assignment $r = R(p) \in \mathrm{Roles}(V)$ — the specific position $p$ fills in $V$’s complement
• An enrollment act $\iota_p$ — the discrete event (signing of articles, assumption of duty, formal assignment) at which $p$ becomes a section of $V$’s role fiber for $T$
• A discharge predicate $\delta_p$ — the condition (end of voyage, relief, death) at which the section status terminates

The enrollment constitutes $p$ as a section of the role fibration over $V$ for period $T$: the assignment $R(p) = r$ is a right inverse of the role projection $\pi_r : \mathrm{Crew}(V, T) \to \mathrm{Roles}(V)$ at position $r$. The section condition is: $\pi_r(R(p)) = r$ — after enrollment, $p$ is provably in position $r$.

Four invariants. $\mathcal{E}$ is an enrollment iff:

1. Section condition: $R(p) = r$ and $r \in \mathrm{Roles}(V)$ — $p$ fills an actual role in $V$’s complement, not a phantom position. The section condition $\pi_r \circ R = \mathrm{id}_{\mathrm{Roles}(V)}$ ensures that every required role has a person assigned (Section: a right inverse of the projection).

2. Discreteness of enrollment act: enrollment is not continuous but instantaneous. There is a specific moment $\iota_p$ — the signing, the declaration, the formal assumption — at which $p$ transitions from non-crew to crew. Before $\iota_p$: $p \notin \mathrm{Crew}(V, T)$. After $\iota_p$: $p \in \mathrm{Crew}(V, T)$ with role $r$. This mirrors the Investiture structure: enrollment is the crew-level investiture.

3. Period-specificity: $p$’s section status is bounded by $T$. The enrollment constitutes a section for $T$ only — not an indefinite assignment. At $T$’s end (the discharge predicate $\delta_p$), the section status lapses. The bounded-period invariant is what distinguishes crew enrollment from permanent institutional membership.

4. Counit coherence: the enrollment section coheres with the presheaf structure of $V$ across $T$. Formally: for each restriction map $H(f) : H_{t'} \to H_t$ (for $t \leq t' \leq T$), $p$’s role-binding at $t'$ restricts correctly to $p$’s role-binding at $t$. This is the counit law of the directed comonad (Section): $\varepsilon_V \circ \gamma_p = \mathrm{id}_p$ — after the stepping map advances $p$’s state and the counit restricts it back, $p$’s assignment is unchanged. Perfect enrollment: the person coheres with the vessel across time.

The section reading

Section in the fibration sense: given the role fibration $\pi : \mathrm{Crew} \to \mathrm{Roles}(V)$ that projects each crew member to their assigned role, an enrollment of $p$ into role $r$ is a section $s_r : \{r\} \to \mathrm{Crew}$ with $\pi \circ s_r = \mathrm{id}_{\{r\}}$. The section assigns to the position $r$ the person $p = s_r(r)$ who fills it.

The section condition is the minimal structural occupancy condition. It says:

• $p$ is in the correct fiber (the role $r$ position in $V$)
• The assignment is right-invertible: knowing the role, you can recover the person

The section condition says nothing about dynamics (how $p$ came to be enrolled), obligation (whether $p$ is fulfilling their duties), or performance (how well $p$ is executing role $r$). Those dimensions are added by Duty (the deontic obligations of the role), Discharge (performance recognition), and Serving (the ongoing act of service).

Enrollment vs. investiture

Investiture installs an agent into a formal office with a Hohfeldian deontic profile — a position in the normative system. Enrollment installs a person into a role in a crew for a bounded operational period. The structures are related but distinct:

Dimension	Investiture	Enrollment
Scope	Formal office with deontic profile	Operational role in a vessel’s complement
Duration	Until relief act $I^{-1}$	Bounded by operational period $T$
Constituting act	Installation ceremony / orders	Articles of agreement / duty assignment
What persists	The office and its deontic profile	The vessel’s complement schema
Math grounding	Section of duty-fibration	Section of role-fibration

A commanding officer is both invested (holds a formal command office) and enrolled (serves as crew for the voyage). For ordinary crew, enrollment creates the role-section; investiture into the command office is the additional step that creates the authority section.

Discharge of enrollment

The section status terminates when the discharge predicate $\delta_p$ is satisfied:

• Voyage completion: the articles of agreement close at the end of $T$; all crew sections lapse simultaneously
• Individual relief: $p$ is relieved from their role; the section $s_r \mapsto p$ is replaced by $s_r \mapsto p'$ for the relief crew member $p'$
• Death or incapacity: the section lapses; the role position becomes vacant; complement coverage may fail

At discharge, $p$ leaves the crew fiber for $V$ and returns to the pool of persons. The vessel $V$ continues; the section $s_r$ is vacant until a new enrollment fills it.

Math grounding

The enrollment structure grounds in two math concepts:

Section: the abstract section condition $p \circ s = \mathrm{id}$ is what enrollment instantiates. The role fibration $\pi : \mathrm{Crew}(V) \to \mathrm{Roles}(V)$ is the fibration; enrollment is the section. The counit law ($\varepsilon_X \circ \gamma = \mathrm{id}_X$) is the period-coherence condition: the enrolled person’s assignment advances with the vessel’s history and restricts correctly.

Natural Transformation: a person’s enrollment across the operational period $T$ is a natural transformation $p_T : \mathbf{1} \Rightarrow \mathrm{Crew}(V, -)$ from the terminal presheaf to the crew presheaf — a coherent assignment of $p$ to the vessel’s crew at every history in $T$, compatible with the restriction maps.

Open questions

• Whether collective enrollment (all crew members signing articles simultaneously) is the categorical product of individual enrollments, or whether the articles constitute the crew as a unit (a limit of the individual sections) with additional coherence conditions not present in the product.
• Whether the minimum complement condition $|\mathrm{Crew}(V,T)| \geq k$ has a formal expression in terms of the section coverage: whether a vessel below minimum complement fails to have a jointly surjective family of sections — and whether joint surjectivity is a covering condition in the site topology $J$ of the vessel.
• Whether enrollment into command position (the first officer or captain’s section) has a special structure — a distinguished section among the crew sections — and whether the distinction corresponds to the section being in $H^*_{t^*}$ (doubly settled) rather than merely in $H_{t^*}$.