This monograph presents a rigorous topological and categorical analysis of the three fundamental careers available to the working mathematician: the Generalizer (creator of new mathematics), the Surveyor (synthesizer and translator of existing mathematics), and the Teacher (disseminator and perpetuator of mathematical knowledge).

Drawing on the #1457843 Schneider (2025) taxonomy of AI agent roles, we formalize each career as a distinct functor in a category of mathematical production.

The Generalizer constructs new objects and morphisms; the Surveyor composes and reduces complexity via natural transformations; the Teacher embeds mathematics into cognitive structures via adjunctions.

We prove theorems establishing the unique contributions of each role, their interdependence, and the conditions under which each career thrives.

The Y‑combinator Y = λf.(λx.f (x x)) (λx.f (x x)) emerges as the fixed‑point operator of mathematical self‑reference, while the Coulon squaring relation n² = m governs the scaling of mathematical communities. Complete lambda calculus implementations model the dynamics of each career.

The framework is intended both as a guide for mathematicians navigating their professional lives and as a contribution to the philosophy and sociology of mathematics.