12.1 Algebraic theories in topoi

Algebraic theories in topoi provide a powerful framework for studying mathematical structures. They allow us to describe and analyze groups, rings, and other algebraic objects within a generalized universe of sets, offering new insights and perspectives.

Free models play a crucial role in this context, embodying universal properties and facilitating the construction of algebraic structures. The interplay between algebraic and Lawvere theories, along with the unique features of topoi, opens up exciting avenues for mathematical exploration.

Foundations of Algebraic Theories in Topoi

Algebraic theories in topoi

• Algebraic theories serve as formal systems describing algebraic structures encompassing sorts, operations, and equations (groups, rings)
• Models in a topos manifest as objects satisfying theory axioms with morphisms respecting operations
• Operations interpreted as arrows in topos representing functions (addition, multiplication)
• Equations satisfied through commutative diagrams in topos (associativity, distributivity)

Free models in topoi

• Free model concept embodies universal property in category theory acting as initial object in model category
• Construction process utilizes coproducts and quotients in topos, iteratively applying operations
• Adjoint functor pair consists of forgetful functor and free functor
• Free functor as left adjoint preserves limits (products, equalizers)

Connections and Frameworks

Algebraic vs Lawvere theories

• Lawvere theories offer category-theoretic formulation of algebraic theories with objects representing arities and morphisms representing terms
• Equivalence exists between algebraic and Lawvere theories, with models of algebraic theories corresponding to product-preserving functors
• Lawvere theories provide more categorical approach facilitating work in certain contexts (abstract algebra, universal algebra)

Topoi for algebraic structures

• Topos functions as generalized universe of sets with internal logic, subobject classifier, and power objects
• Algebraic structures interpreted as objects in topos (groups, rings, modules)
• Geometric morphisms preserve algebraic structures between topoi
• Sheaf models allow algebraic structures to vary over a space (vector bundles, local rings)
• Synthetic approach enables axiomatization of mathematics within a topos (synthetic differential geometry, smooth infinitesimal analysis)