1 A Categorical and Homotopical Toolkit

The language required for categorical identification and decision

ORACLE and Universal Online Decision Learning combine several mathematical languages. Ordinary category theory describes typed composition and universal extension. Algebraic theories and sketches separate compact generators and relations from their functorial models. Simplicial sets encode histories of every finite compositional depth. Categorical homotopy theory distinguishes presentation from semantic content and retains coherent spaces of possible repairs. Tangent categories describe infinitesimal variation. This chapter introduces exactly the portion of each language required by the book.

It is a tutorial, not a replacement for standard references. Mac Lane and Riehl provide systematic introductions to ordinary category theory [ Mac Lane , 1971 , Riehl , 2016 ] ; Riehl develops the homotopical and enriched material used here [ Riehl , 2014 ] . The purpose of the chapter is to make later declarations readable and to expose every assumption used by the first identification and derived UODL theorems.

Guiding question.

What is the smallest mathematical language in which an online learner can construct a composite, preserve it across revelation, discover that its presentation was nonessential, and repair it coherently when new evidence arrives?