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I feel one should also be somewhat eclectic and consider not only advantages but also comparative disadvantages of any candidate for a central categorical role.

I admit to special pleading here since our 2001 EMS Tract on Nonabelian algebraic topology gives a major role for a homotopical foundation of algebraic topology to cubical sets. The point is that in the usual simplicial $\infty$-category theory the emphasis is on the Kan condition. In the cubical theory, the emphasis is on compositions. This allows for the replacement of the usual formal sums in basic homology theory by actual compositions of pieces for homotopically defined functors. See this mathoverflow discussion.

Dan Kan's thesis and first (1955) paper were cubical, as best for intuition and conjectures, but severe disadvantages were found in the category of cubical sets by workers at Princeton. One disadvantage was that cubical groups, unlike simplicial groups, were not Kan complexes. Another was the realisation of the cartesian product of cubical sets, which had the wrong homotopy type, again unlike simplicial sets. So it was assumed that the cubical theory was quite unfixable.

However work at Bangor in the 1970s with Chris Spencer and Philip Higgins, starting with the relation between double groupoids and crossed modules, found it necessary to introduce a new type of "degenerate" cube based on the monoid structures max and min on the unit interval $I=[0,1]$.The standard degeneracy in cubical sets yields cubes with opposite faces the same, where these new had some adjacent faces the same, and so made the theory a bit nearer to the simplicial theory. These new structures were called connections, because of a relation with path connections in differential geometry.

As Philippe Gaucher points out, Andy Tonks proved in 1992 that cubical groups with connections were Kan complexes. In 2005, Georges Maltsiniotis proved that the geometric realisation of of the cartesian product of cubical sets with connections has the correct homotopy type.

There are two main reasons for using cubical sets. One is the formula $I^m \times I^n \cong I^{m+n}$, which makes for a good tensor product of cubical sets, and also allows a convenient and direct definition of homotopies.

The second and for our purposed major reason for using cubical sets is expressed in the slogan "algebraic inverses to subdivision", which is an elaboration of the idea of composition in cubical sets. The standard singular cubical set $S^{\Box} X$ of a topological space ie easily equipped with $n$ partial compositions in dimension $n$ giving $S^{\Box} X$ the structure of weak $\infty$-groupoid. A simple matrix/array notation $[a_{(r)}]$ allows for the expression of multiple partial compositions of $n$-cubes which are compatible in all the directions. This gives meaning to "algebraic inverses to subdivision". This is one of the basic intuitions behind the first proofs of higher versions of the Seifert-van Kampen Theorem.

It is not at all clear to me how these ideas can be expressed in simplicial terms.

There is further discussion of the intuition of these uses of cubical sets in talks I gave in Paris on June 5, 2014, and Galway, December 2014, available on my preprint page.

March, 2015: I have since learned that cubical sets with connection are used in the theory of motives in this paper. It seems that the degeneracies in cubes work better here than in simplices. Another paper compares cubical and simplicial derived functors. Both these papers use cubical sets with connections, which were introduced in JPAA 1981 by Philip Higgins and me. These additional kind of degeneracies go some way to correcting some well known deficiences of the usual cubical sets, in terms of cubical groups and geometric realisations of cartesian products - see the talks referred to above. <------->

I feel one should also be somewhat eclectic and consider not only advantages but also comparative disadvantages of any candidate for a central categorical role.

I admit to special pleading here since our 2001 EMS Tract on Nonabelian algebraic topology gives a major role for a homotopical foundation of algebraic topology to cubical sets. The point is that in the usual simplicial $\infty$-category theory the emphasis is on the Kan condition. In the cubical theory, the emphasis is on compositions. This allows for the replacement of the usual formal sums in basic homology theory by actual compositions of pieces for homotopically defined functors. See this mathoverflow discussion.

Dan Kan's thesis and first (1955) paper were cubical, as best for intuition and conjectures, but severe disadvantages were found in the category of cubical sets by workers at Princeton. One disadvantage was that cubical groups, unlike simplicial groups, were not Kan complexes. Another was the realisation of the cartesian product of cubical sets, which had the wrong homotopy type, again unlike simplicial sets. So it was assumed that the cubical theory was quite unfixable.

However work at Bangor in the 1970s with Chris Spencer and Philip Higgins, starting with the relation between double groupoids and crossed modules, found it necessary to introduce a new type of "degenerate" cube based on the monoid structures max and min on the unit interval $I=[0,1]$.The standard degeneracy in cubical sets yields cubes with opposite faces the same, where these new had some adjacent faces the same, and so made the theory a bit nearer to the simplicial theory. These new structures were called connections, because of a relation with path connections in differential geometry.

As Philippe Gaucher points out, Andy Tonks proved in 1992 that cubical groups with connections were Kan complexes. In 2005, Georges Maltsiniotis proved that the geometric realisation of of the cartesian product of cubical sets with connections has the correct homotopy type.

There are two main reasons for using cubical sets. One is the formula $I^m \times I^n \cong I^{m+n}$, which makes for a good tensor product of cubical sets, and also allows a convenient and direct definition of homotopies.

The second and for our purposed major reason for using cubical sets is expressed in the slogan "algebraic inverses to subdivision", which is an elaboration of the idea of composition in cubical sets. The standard singular cubical set $S^{\Box} X$ of a topological space ie easily equipped with $n$ partial compositions in dimension $n$ giving $S^{\Box} X$ the structure of weak $\infty$-groupoid. A simple matrix/array notation $[a_{(r)}]$ allows for the expression of multiple partial compositions of $n$-cubes which are compatible in all the directions. This gives meaning to "algebraic inverses to subdivision". This is one of the basic intuitions behind the first proofs of higher versions of the Seifert-van Kampen Theorem.

It is not at all clear to me how these ideas can be expressed in simplicial terms.

There is further discussion of the intuition of these uses of cubical sets in talks I gave in Paris on June 5, 2014, and Galway, December 2014, available on my preprint page.

March, 2015: I have since learned that cubical sets with connection are used in the theory of motives in this paper. It seems that the degeneracies in cubes work better here than in simplices. Another paper compares cubical and simplicial derived functors. Both these papers use cubical sets with connections, which were introduced in JPAA 1981 by Philip Higgins and me. These additional kind of degeneracies go some way to correcting some well known deficiences of the usual cubical sets, in terms of cubical groups and geometric realisations of cartesian products - see the talks referred to above. <------->

I feel one should also be somewhat eclectic and consider not only advantages but also comparative disadvantages of any candidate for a central categorical role.

I admit to special pleading here since our 2001 EMS Tract on Nonabelian algebraic topology gives a major role for a homotopical foundation of algebraic topology to cubical sets. The point is that in the usual simplicial $\infty$-category theory the emphasis is on the Kan condition. In the cubical theory, the emphasis is on compositions. This allows for the replacement of the usual formal sums in basic homology theory by actual compositions of pieces for homotopically defined functors. See this mathoverflow discussion.

Dan Kan's thesis and first (1955) paper were cubical, as best for intuition and conjectures, but severe disadvantages were found in the category of cubical sets by workers at Princeton. One disadvantage was that cubical groups, unlike simplicial groups, were not Kan complexes. Another was the realisation of the cartesian product of cubical sets, which had the wrong homotopy type, again unlike simplicial sets. So it was assumed that the cubical theory was quite unfixable.

However work at Bangor in the 1970s with Chris Spencer and Philip Higgins, starting with the relation between double groupoids and crossed modules, found it necessary to introduce a new type of "degenerate" cube based o the monoid structures max and min on the unit interval $I=[0,1]$.The standard degeneracy in cubical sets yields cubes with opposite faces the same, where these new had some adjacent faces the same, and so made the theory a bit nearer to the simplicial theory. These new structures were called connections, because of a relation with path connections in differential geometry.

As Philippe Gaucher points out, Andy Tonks proved in 1992 that cubical groups with connections were Kan complexes. In 2005, Georges Maltsiniotis proved that the geometric realisation of of the cartesian product of cubical sets with connections has the correct homotopy type.

There are two main reasons for using cubical sets. One is the formula $I^m \times I^n \cong I^{m+n}$, which makes for a good tensor product of cubical sets, and also allows a convenient and direct definition of homotopies.

The second and for our purposed major reason for using cubical sets is expressed in the slogan "algebraic inverses to subdivision", which is an elaboration of the idea of composition in cubical sets. The standard singular cubical set $S^{\Box} X$ of a topological space ie easily equipped with $n$ partial compositions in dimension $n$ giving $S^{\Box} X$ the structure of weak $\infty$-groupoid. A simple matrix/array notation $[a_{(r)}]$ allows for the expression of multiple partial compositions of $n$-cubes which are compatible in all the directions. This gives meaning to "algebraic inverses to subdivision". This is one of the basic intuitions behind the first proofs of higher versions of the Seifert-van Kampen Theorem.

It is not at all clear to me how these ideas can be expressed in simplicial terms.

There is further discussion of the intuition of these uses of cubical sets in talks I gave in Paris on June 5, 2014, and Galway, December 2014, available on my preprint page.

March, 2015: I have since learned that cubical sets with connection are used in the theory of motives in this paper. It seems that the degeneracies in cubes work better here than in simplices. Another paper compares cubical and simplicial derived functors. Both these papers use cubical sets with connections, which were introduced in JPAA 1981 by Philip Higgins and me. These additional kind of degeneracies go some way to correcting some well known deficiences of the usual cubical sets, in terms of cubical groups and geometric realisations of cartesian products - see the talks referred to above.

I feel one should also be somewhat eclectic and consider not only advantages but also comparative disadvantages of any candidate for a central categorical role.

I admit to special pleading here since our 2001 EMS Tract on Nonabelian algebraic topology gives a major role for a homotopical foundation of algebraic topology to cubical sets. The point is that in the usual simplicial $\infty$-category theory the emphasis is on the Kan condition. In the cubical theory, the emphasis is on compositions. This allows for the replacement of the usual formal sums in basic homology theory by actual compositions of pieces for homotopically defined functors. See this mathoverflow discussion.

Dan Kan's thesis and first (1955) paper were cubical, as best for intuition and conjectures, but severe disadvantages were found in the category of cubical sets by workers at Princeton. One disadvantage was that cubical groups, unlike simplicial groups, were not Kan complexes. Another was the realisation of the cartesian product of cubical sets, which had the wrong homotopy type, again unlike simplicial sets. So it was assumed that the cubical theory was quite unfixable.

However work at Bangor in the 1970s with Chris Spencer and Philip Higgins, starting with the relation between double groupoids and crossed modules, found it necessary to introduce a new type of "degenerate" cube based on the monoid structures max and min on the unit interval $I=[0,1]$.The standard degeneracy in cubical sets yields cubes with opposite faces the same, where these new had some adjacent faces the same, and so made the theory a bit nearer to the simplicial theory. These new structures were called connections, because of a relation with path connections in differential geometry.

As Philippe Gaucher points out, Andy Tonks proved in 1992 that cubical groups with connections were Kan complexes. In 2005, Georges Maltsiniotis proved that the geometric realisation of of the cartesian product of cubical sets with connections has the correct homotopy type.

There are two main reasons for using cubical sets. One is the formula $I^m \times I^n \cong I^{m+n}$, which makes for a good tensor product of cubical sets, and also allows a convenient and direct definition of homotopies.

The second and for our purposed major reason for using cubical sets is expressed in the slogan "algebraic inverses to subdivision", which is an elaboration of the idea of composition in cubical sets. The standard singular cubical set $S^{\Box} X$ of a topological space ie easily equipped with $n$ partial compositions in dimension $n$ giving $S^{\Box} X$ the structure of weak $\infty$-groupoid. A simple matrix/array notation $[a_{(r)}]$ allows for the expression of multiple partial compositions of $n$-cubes which are compatible in all the directions. This gives meaning to "algebraic inverses to subdivision". This is one of the basic intuitions behind the first proofs of higher versions of the Seifert-van Kampen Theorem.

It is not at all clear to me how these ideas can be expressed in simplicial terms.

There is further discussion of the intuition of these uses of cubical sets in talks I gave in Paris on June 5, 2014, and Galway, December 2014, available on my preprint page.

March, 2015: I have since learned that cubical sets with connection are used in the theory of motives in this paper. It seems that the degeneracies in cubes work better here than in simplices. Another paper compares cubical and simplicial derived functors. Both these papers use cubical sets with connections, which were introduced in JPAA 1981 by Philip Higgins and me. These additional kind of degeneracies go some way to correcting some well known deficiences of the usual cubical sets, in terms of cubical groups and geometric realisations of cartesian products - see the talks referred to above. <------->

I feel one should also be somewhat eclectic and consider not only advantages but also comparative disadvantages of any candidate for a central categorical role.

I admit to special pleading here since our 2001 EMS Tract on Nonabelian algebraic topology gives a major role for a homotopical foundation of algebraic topology to cubical sets. The point is that in the usual simplicial $\infty$-category theory the emphasis is on the Kan condition. In the cubical theory, the emphasis is on compositions. This allows for the replacement of the usual formal sums in basic homology theory by actual compositions of pieces for homotopically defined functors. See this mathoverflow discussion.

Dan Kan's thesis and first (1955) paper were cubical, as best for intuition and conjectures, but severe disadvantages were found in the category of cubical sets by workers at Princeton. One disadvantage was that cubical groups, unlike simplicial groups, were not Kan complexes. Another was the realisation of the cartesian product of cubical sets, which had the wrong homotopy type, again unlike simplicial sets. So it was assumed that the cubical theory was quite unfixable.

However work at Bangor in the 1970s with Chris Spencer and Philip Higgins, starting with the relation between double groupoids and crossed modules, found it necessary to introduce a new type of "degenerate" cube based o the monoid structures max and min on the unit interval $I=[0,1]$.The standard degeneracy in cubical sets yields cubes with opposite faces the same, where these new had some adjacent faces the same, and so made the theory a bit nearer to the simplicial theory. These new structures were called connections, because of a relation with path connections in differential geometry.

As Philippe Gaucher points out, Andy Tonks proved in 1992 that cubical groups with connections were Kan complexes. In 2005, Georges Maltsiniotis proved that the geometric realisation of of the cartesian product of cubical sets with connections has the correct homotopy type.

There are two main reasons for using cubical sets. One is the formula $I^m \times I^n \cong I^{m+n}$, which makes for a good tensor product of cubical sets, and also allows a convenient and direct definition of homotopies.

The second and for our purposed major reason for using cubical sets is expressed in the slogan "algebraic inverses to subdivision", which is an elaboration of the idea of composition in cubical sets. The standard singular cubical set $S^{\Box} X$ of a topological space ie easily equipped with $n$ partial compositions in dimension $n$ giving $S^{\Box} X$ the structure of weak $\infty$-groupoid. A simple matrix/array notation $[a_{(r)}]$ allows for the expression of multiple partial compositions of $n$-cubes which are compatible in all the directions. This gives meaning to "algebraic inverses to subdivision". This is one of the basic intuitions behind the first proofs of higher versions of the Seifert-van Kampen Theorem.

It is not at all clear to me how these ideas can be expressed in simplicial terms.

There is further discussion of the intuition of these uses of cubical sets in talks I gave in Paris on June 5, 2014, and Galway, December 2014, available on my preprint page.

March, 2015: I have since learned that cubical sets with connection are used in the theory of motives in this paper. It seems that the degeneracies in cubes work better here than in simplices.

I feel one should also be somewhat eclectic and consider not only advantages but also comparative disadvantages of any candidate for a central categorical role.

I admit to special pleading here since our 2001 EMS Tract on Nonabelian algebraic topology gives a major role for a homotopical foundation of algebraic topology to cubical sets. The point is that in the usual simplicial $\infty$-category theory the emphasis is on the Kan condition. In the cubical theory, the emphasis is on compositions. This allows for the replacement of the usual formal sums in basic homology theory by actual compositions of pieces for homotopically defined functors. See this mathoverflow discussion.

Dan Kan's thesis and first (1955) paper were cubical, as best for intuition and conjectures, but severe disadvantages were found in the category of cubical sets by workers at Princeton. One disadvantage was that cubical groups, unlike simplicial groups, were not Kan complexes. Another was the realisation of the cartesian product of cubical sets, which had the wrong homotopy type, again unlike simplicial sets. So it was assumed that the cubical theory was quite unfixable.

However work at Bangor in the 1970s with Chris Spencer and Philip Higgins, starting with the relation between double groupoids and crossed modules, found it necessary to introduce a new type of "degenerate" cube based o the monoid structures max and min on the unit interval $I=[0,1]$.The standard degeneracy in cubical sets yields cubes with opposite faces the same, where these new had some adjacent faces the same, and so made the theory a bit nearer to the simplicial theory. These new structures were called connections, because of a relation with path connections in differential geometry.

As Philippe Gaucher points out, Andy Tonks proved in 1992 that cubical groups with connections were Kan complexes. In 2005, Georges Maltsiniotis proved that the geometric realisation of of the cartesian product of cubical sets with connections has the correct homotopy type.

There are two main reasons for using cubical sets. One is the formula $I^m \times I^n \cong I^{m+n}$, which makes for a good tensor product of cubical sets, and also allows a convenient and direct definition of homotopies.

The second and for our purposed major reason for using cubical sets is expressed in the slogan "algebraic inverses to subdivision", which is an elaboration of the idea of composition in cubical sets. The standard singular cubical set $S^{\Box} X$ of a topological space ie easily equipped with $n$ partial compositions in dimension $n$ giving $S^{\Box} X$ the structure of weak $\infty$-groupoid. A simple matrix/array notation $[a_{(r)}]$ allows for the expression of multiple partial compositions of $n$-cubes which are compatible in all the directions. This gives meaning to "algebraic inverses to subdivision". This is one of the basic intuitions behind the first proofs of higher versions of the Seifert-van Kampen Theorem.

It is not at all clear to me how these ideas can be expressed in simplicial terms.

There is further discussion of the intuition of these uses of cubical sets in talks I gave in Paris on June 5, 2014, and Galway, December 2014, available on my preprint page.

March, 2015: I have since learned that cubical sets with connection are used in the theory of motives in this paper. It seems that the degeneracies in cubes work better here than in simplices. Another paper compares cubical and simplicial derived functors. Both these papers use cubical sets with connections, which were introduced in JPAA 1981 by Philip Higgins and me. These additional kind of degeneracies go some way to correcting some well known deficiences of the usual cubical sets, in terms of cubical groups and geometric realisations of cartesian products - see the talks referred to above.