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Explore the intersection of higher algebra and type theory in this colloquium talk that introduces dependetopes as a novel approach to understanding higher generalized algebraic theories. Learn how classical categorical algebra tools like Lawvere theories, finite limit theories, and operads can be generalized to higher algebra through the study of algebraic structures on homotopy types. Discover the challenges that arise when extending Generalized Algebraic Theories (GATs) from dependent type theory to higher settings, where the connection to concrete syntax becomes problematic despite the availability of sophisticated ∞-categorical frameworks. Examine the speaker's proposed solution through dependetopes, a dependently typed extension of Baez-Dolan's opetopes that captures the higher geometry of type-dependency while maintaining a concrete syntactic description rooted in type theory. Understand how this framework bridges the gap between the abstract world of (∞,1)-categories and the practical needs of computer scientists and type theorists who require syntactic presentations suitable for implementation. Gain insights into how dependetopes provide both categorical semantics for higher algebraic theories and computational tools for their manipulation, offering a more tractable approach to working with higher generalized algebraic theories than existing methods.
