Homotopy Type Theory
algebraic formulation of dependent type theory

Idea

The goal of this line of work is to find an essentially algebraic formulation of dependent type theory. This automatically provides free models and quotients. The syntax will then be the initial model.

Rijke’s E-systems are a generalization of Voevodsky’s B-systems.

This line of research should be compared with Dybjer’s categories with families which is a generalized algebraic theory; see categorical model of dependent types for more information.

References

• Egbert Rijke, An algebraic formulation of dependent type theory, (mailing list discussion)

• Vladimir Voevodsky, B-systems, arXiv

• Vladimir Voevodsky, A C-system defined by a universe in a category, arXiv:1406.7413

• Vladimir Voevodsky, C-system of a module over a monad on sets, arXiv:1406.7413

• Vladimir Voevodsky, Subsystems and regular quotients of C-systems, arXiv:1406.7413