nLab constructive model structure on simplicial sets

Idea

A model structure on simplicial sets valid in constructive mathematics that reduced to the classical model structure on simplicial sets in presence of the law of excluded middle and axiom of choice.

The original proofs of the existence of the classical model structure on simplicial sets are based in classical mathematics as they use the principle of excluded middle and the axiom of choice, and are hence not valid in constructive mathematics. This becomes more than a philosophical issue with the relevance of this model category-structure in homotopy type theory, where internalization into the type theory requires constructive methods for interpreting proofs as programs.

A constructively valid model structure on simplicial sets and coinciding with the classical model structure if excluded middle and axiom of choice are assumed was found in Henry 19. Alternative simpler proofs were found in Gambino-Sattler-Szumiło 19

Properties

Fibrations and lifting properties

Fibrations and acyclic fibrations

are defined using the right lifting property with respect to the usual sets of horn inclusions and boundary inclusions?. However, the meaning of this definition is more refined: due to the constructive nature of the existential quantifier, being an (acyclic) fibration means we have a section of the map from the set of solutions to lifting problems to the set of lifting problems.

Weak equivalences

Simplicial weak equivalences? must also be defined more carefully.

Suppose $f\colon X\to Y$ is a simplicial map.

We follow GambinoSattlerSzumilo19. They give two (equivalent) definitions.

In the first proof:

• If $X$ and $Y$ are cofibrant Kan complexes, then $f$ is a weak equivalence if it is a simplicial homotopy equivalence;
• If $X$ and $Y$ are Kan complexes, then $f$ is a weak equivalence if there is a simplicial map $f'\colon X'\to Y'$ and acyclic fibrations with cofibrant domains $X'\to X$ and $Y'\to Y$ such that $f'$ is a weak equivalence in the previous sense and the square with $f$ and $f'$ commutes;
• If $X$ and $Y$ are cofibrant, then $f$ is a weak equivalence if for any Kan complex $K$ the map $Hom(f,K)$ is a weak equivalence in the previous sense (notice that its domain and codomain are Kan complexes);
• If $X$ and $Y$ are arbitrary, then $f$ is a weak equivalence if it has a replacement $f'\colon X'\to Y'$ like above, with $f'$ being a weak equivalence in the previous sense.

In the second proof:

• Within cofibrant simplicial sets, weak equivalences are those maps writing as an acyclic cofibration followed by an acyclic fibration.
• In general, weak equivalences are defined via (strong) cofibrant replacement.

Cofibrant objects

While in the classical model structure on simplicial sets all objects (simplicial sets) are cofibrant, a simplicial set is cofibrant in the constructive model structure if and only if the inclusion of degenerate $n$-simplices into all $n$-simplices is a decidable inclusion of sets.

Likewise, cofibrations $A\to B$ are simplicial maps such that $A_n\to B_n$ is a decidable inclusion of sets and the inclusion of degenerate $n$-simplices as a subset into $B_n\setminus A_n$ is also a decidable inclusion.

Since not all objects are cofibrant, left properness is no longer automatic, but can still be established through nontrivial arguments.

This model structure is also right proper.

See Henry 19 and Gambino-Sattler-Szumiło 19 for a proof of these claims.

References

• Simon Henry, Weak model categories in classical and constructive mathematics (arXiv:1807.02650)

• Simon Henry, A constructive account of the Kan-Quillen model structure and of Kan’s $Ex^\infty$ functor (arXiv:1905.06160)

• Nicola Gambino, Simon Henry, Towards a constructive simplicial model of Univalent Foundations (arXiv:1905.06281)

• Nicola Gambino, Christian Sattler, Karol Szumiło, The constructive Kan-Quillen model structure: two new proofs (arXiv:1907.05394)