nLab synthetic homotopy theory

Idea

Homotopy type theory provides a synthetic formalization of homotopy theory in its modern and most powerful incarnation in the guise of (∞,1)-toposes.

Just like plain dependent type theory is the internal logic of locally cartesian closed categories, so homotopy type theory is the internal logic of locally cartesian closed (∞,1)-categories, and homotopy type theory with univalent type universes is the internal logic of elementary (∞,1)-toposes. This means that homotopy type theory provides a “structural” foundation of the kind that William Lawvere had found in topos theory, but refined to homotopy theory in the refined guise of (∞,1)-topos theory.

Homotopy type theory natively knows about the formalization of simplicial homotopy theory and higher topos theory without the need to first formalize several textbooks worth of material starting from bare set theory. For instance, the formal proof of the Blakers-Massey theorem (FFLL) does not need to begin by first formalizing what a simplicial set is, what a Kan fibrancy condition is, what the infinite tower of homotopy groups is, what weak homotopy equivalences are, what homotopy pushouts are, how they reflect in long exact sequences of homotopy groups; because all this is native to homotopy type theory. Accordingly, the proof, on top of being a formal proof, is elegantly transparent and of actual practical interest. Moreover, since the formal HoTT proof generalizes the traditional statement to more general (∞,1)-toposes, it is actually a genuine new mathematical result of genuine interest in modern homotopy theory, to practicing mathematicians not concerned about foundations.

Cubical synthetic homotopy theory

Mörtberg & Pujet (2020) make the case for the use of cubical versions of HoTT in synthetic homotopy theory since univalence and HITs are natively supported there, rather than axiomatically added as in HoTT. The path algebra in HoTT is made complicated by the fact that many equalities do not hold definitionally, even in the proof of simple results such as that the torus is equivalent to the product of two circles. The proof of this result is trivial in cubical type theory.

Related concepts

• synthetic topology

• synthetic differential topology

• synthetic $(\infty,1)$-category theory

References

The above text of the Idea section follows Schreiber 14.

The idea of regarding homotopy type theory as synthetic homotopy theory is essentially Awodey's proposal:

• Steve Awodey, §3 of: Type theory and homotopy, in: Dybjer P., Lindström S., Palmgren E., Sundholm G. (eds.), Epistemology versus Ontology, Springer (2012) 183-201 $[$arXiv:1010.1810, doi:10.1007/978-94-007-4435-6_9$]$

Exposition:

• Guillaume Brunerie, Dan Licata, Peter LeFanu Lumsdaine: Homotopy theory in type theory (2013) $[$pdf slides, pdf, blog entry 1, blog entry 2$]$

• Mike Shulman, slides 37 onwards in Homotopical trinitarianism: A perspective on homotopy type theory, 2018 (pdf)

Further discussion in homotopy type theory:

• Egbert Rijke, Classifying Types (arXiv:1906.09435)

• Kuen-Bang Hou (Favonia), Eric Finster, Dan Licata, Peter LeFanu Lumsdaine, A mechanization of the Blakers-Massey connectivity theorem in Homotopy Type Theory, (arXiv:1605.03227)

Therefore, most of the articles listed at

• mathematics presented in homotopy type theory

count as synthetic homotopy theory.

The use of cubical type theory for synthetic homotopy theory is discussed in:

• Anders Mörtberg, Loïc Pujet, Cubical synthetic homotopy theory, CPP 2020: Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs (Jan 2020) 158–171 [doi:10.1145/3372885.3373825, pdf]