Know Time #138: Category Theory 101, Math, Higher Topos Theory in Physics & Philosophy ft. Urs Schreiber
Nov 8, 2025
Urs Schreiber, a theoretical physicist and mathematician at NYU Abu Dhabi, delves into the intriguing realms of category theory and topos theory. He explores the philosophical question of whether something can emerge from nothing, linking it to Hegel's logic. Urs simplifies category theory concepts like functors and adjunctions while addressing the applicability of higher topos theory to theoretical physics. He shares insights on the limitations of perturbation theory, his motivations for founding nLab, and his vision for a more interconnected understanding of math and physics.
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Oxford Eureka Moment
- Urs Schreiber discovered he needed a cohesive topos while seeking a global formulation for gauge fields.
- A talk in Oxford revealed William Lawvere's axiomatic cohesion, which matched and extended his insights.
Structure Over Elements
- Category theory formalizes structuralism by remembering objects and their maps instead of just sets of elements.
- Understanding an object by all maps into it (Yoneda idea) reveals its internal structure through relations.
Functors And Natural Transformations
- Functors are structure-preserving maps between categories and natural transformations compare functors pointwise.
- These three layers (categories, functors, natural transformations) let category theory reflect on its own structures.
