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There are grounds for suggesting that simplicial sets are convenient, but are not entirely adequate, since they cannot easily express multiple compositions. On the other hand, cubical sets with connections are adequate for this test, but not entirely convenient! Andy Tonks proved that cubical groups with connection are Kan complexes. One reason for inconvenience is that although they have been shown to form a strict test category in the sense of Grothendieck, in the paper given here, the geometric realisation of the categorical product is only of the homotopy type of the product of the realisations, not actually homeomorphic to the product as in the case of simplicial sets. The latter homeomorphism property implies that in the right convenient category, the geometric realisation of a simplicial group is a topological group.

There are grounds for suggesting that simplicial sets are convenient, but are not entirely adequate, since they cannot easily express multiple compositions. On the other hand, cubical sets with connections are adequate for this test, but not entirely convenient! Andy Tonks proved that cubical groups with connection are Kan complexes. One reason for inconvenience is that although they have been shown to form a strict test category in the sense of Grothendieck, in the paper given here, the geometric realisation of the categorical product is only of the homotopy type of the product of the realisations, not actually homeomorphic to the product as in the case of simplicial sets. The latter homeomorphism property implies that in the right convenient category, the geometric realisation of a simplicial group is a topological group.

But this question is about the homotopy theory of categories and the related question of "Why simplicial sets?", for which see also the good answers in March 2011 to Is there a high-concept explanation for why "simplicial" leads to "homotopy-theoretic"?.

By contrast, the singular cubical complex of a space, or filtered space, is ideally suited for the description of multiple compositions, using an array notation. I have already explained this in answer to this mathoverflow question.

But this question is about the homotopy theory of categories and the related question of "Why simplicial sets?", for which see also the good answers in March 2011 to Is there a high-concept explanation for why "simplicial" leads to "homotopy-theoretic"?.

By contrast, the singular cubical complex of a space, or filtered space, is ideally suited for the description of multiple compositions, using an array notation. I have already explained this in answer to this mathoverflow question.

http://math.stackexchange.com/questions/1112107/why-does-seifert-van-kampen-not-hold-with-n-th-homotopy-groups/

https://math.stackexchange.com/questions/1112107/why-does-seifert-van-kampen-not-hold-with-n-th-homotopy-groups/