homotopy theory, (∞,1)-category theory, homotopy type theory
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The definition is to be understood from the point of view of space and quantity: a cubical set is a space characterized by the fact that, and the ways in which, it may be probed by mapping standard cellular cubes into it: the set $S_n$ assigned by a cubical set to the standard $n$-cube $[n]$ is the set of $n$-cubes in this space, hence the way of mapping a standard $n$-cube into this spaces.
Being a functor $S : \Box^{op} \to Set$, a cubical set $S$ also assigns maps between its sets $S_n$ of $n$-cubes which determine in which way smaller cubes sit inside larger cubes.
The face maps go from sets $S_{n+1}$ of $(n+1)$-dimensional cubes to the corresponding set $S_{n}$ of $n$-dimensional cubes and can be thought of as sending each cube in the cubical set to one of its faces, for instance for $n=1$ the set $S_2$ of 2-cubes would be sent in four different ways by four different face maps to the set of $1$-cubes, for instance one of the face maps would send
another one would send
On the other hand, the degeneracy maps go the other way round and send sets $S_n$ of $n$-cubes to sets $S_{n+1}$ of $(n+1)$-cubes by regarding an $n$-cube as a degenerate or “thin” $(n+1)$-cube in the various different ways that this is possible. For instance again for $n=1$ a degeneracy map may act by sending
Notice the $Id$-labels, which indicate that the edges and faces labeled by them are “thin” in much the same way as an identity morphism is thin (notice however that a cubical set by itself is not equipped with a notion of composition of cubes. If it were, we’d call it a cubical ∞-category).
In an ordinary cubical set all degeneracy maps act in the kind of way depicted above. One might also want to require a cubical set to contain “thin” cells between equal adjacent faces. These extra degeneracy maps act by sending 1-cells to degenerate 2-cells of the form
If the cubical set has this additional property, one calls it a cubical set with connection.
Daniel Kan‘s early work on homotopy theory used cubical sets instead of simplicial sets. But then a bit later it was found that plain cubical sets suffer from three disadvantages when it comes to modelling homotopy n-types:
normalization of chains is essential
cubical groups are not automatically fibrant
the geometric realization of cartesian products of cubical sets (see geometric realization below) tends to have the wrong homotopy type:
for instance the geometric realization of the cubical set $I \times I$ has non-trivial homotopy groups and hence does not model the topological space given by the standard square, which is contractible.
It was then realized that none of these problems is shared by simplicial sets:
Eilenberg and Mac Lane proved a normalisation theorem which may be found in Mac Lane’s book ‘Homology’.
every simplicial group is necessarily a Kan complex and therefore fibrant in the standard model structure on simplicial sets
geometric realization of simplicial sets is well behaved and in fact constitutes a cartesian monoidal Quillen equivalence
between SSet and Top. For more on this see homotopy hypothesis.
These observations led to a widespread use of simplicial methods, while cubical methods coexisted only with a kind of underground existence, to this date.
There is, however, a proof of parts of the homotopy hypothesis also for cubical sets: their homotopy category is equivalent, after all, to the standard homotopy catgeory Top. This is described at model structure on cubical sets.
Also, it turns out that the second and third of the above disadvantages of cubical sets over simplicial sets in homotopy theory can be dealt with to some extent.
The first problem can’t be avoided. See
Rosa Antolini and Bert Wiest, The singular cubical set of a topological space , Mathematical Proceedings of the Cambridge Philosophical Society, 126 (1), (1999)
The second problem is resolved, at least when restricting to strict omega-groupoids by using cubical sets with connection. See
A. P. Tonks, Cubical groups which are Kan, J. Pure Appl. Algebra, 81 (1), (1992)
the third problem is due to the fact that the cube category is a test category but not a strict test category. However, the category of cubes with connection is a strict test category, as shown by Georges Maltsiniotis, based on work by Denis-Charles Cisinski. See connection on a cubical set for details.
If $X$ is a cubical set, the geometric realization $|X|$ may be defined as the weighted colimit (a coend) in Top
where $I^{\bullet}: (\Box, \otimes, I) \to (\Top, \times, 1)$ is the unique (up to isomorphism) monoidal functor mapping the generating object $int$ to $[0, 1]$, and for $k = 0, 1$, mapping $i_k$ to the inclusion $\{k\} \hookrightarrow [0, 1]$. This is parallel to one way of defining the geometric realization of a simplicial set. Geometric realization is a functor left adjoint to the functor
which takes a space $S$ to the functor $\hom_{Top}(I^{\bullet}-, S)$.
A cubical subdivision functor $sd$ is discussed in Jardine 0, Section 5. It is an obvious subdivision of an $n$-cube, which is just a product of barycentric subdivisions of intervals. The (functorial) subdivision $sd X$ of a cubical set $X$ is constructed from this naive subdivision of the $n$-cube in the end. See Jardine’s lecture notes for details. There is a natural sequence of maps of cubical sets
defined similar to its simplicial counterparts. Let its right adjoint be denoted as usual by $Ex X$. So $n$-cubes of $Ex X$ are cubical maps from subdivision of the $n$-cube to $X$ (similar to the definition of simplicial Ex-functor). We therefore get maps
Let $Ex^\infty X$ be the union of the latter maps (similar to simplicial $Ex^\infty$).
question: Is $Ex^\infty X$ a fibrant cubical set for any cubical set $X$?.
Recall that a cubical set is fibrant if any cubical horn has a filler (similar to simplicial set: any Kan fibrant simplicial set has horn fillers). See also (Cisinski 2006) or Jardine’s lectures on cubical sets for definitions.
The first question is probably not true in general, but if we consider cubical sets with connections in the sense of Brown-Higgins (we add some degeneracy maps to cubical sets), see e.g. Maltsiniotis paper then the cubical subdivision remains the same and the $Ex^\infty X$ is defined similarly. The question is whether $Ex^\infty X$ with $X$ a cubical set with connections is fibrant. Is it true?
There is a model structure on cubical sets with the same homotopy theory as the standard model structure on simplicial sets (Jardine 02), which models homotopy types/infinity-groupoids.
In fact (Jardine 02, theorem 29, theorem 30) gives an adjunction between the model categories which, while not quite a Quillen adjunction, does have unit and counit being weak equivalences. Hence by the discussion at adjoint (∞,1)-functor it should indeed follow that the derived functors of the adjunction exhibit the simplicial localizations of cubical sets equivalent to that of simplicial sets, hence make their (∞,1)-categories equivalent (hence equivalent to ∞Grpd).
This generalizes to a local model structure on cubical presheaves over any site, which has at least the same homotopy category as the corresponding local model structure on simplicial presheaves.
The notion of cubical set and of homology theory and homotopy theory based on singular cubes goes way back in the literature. Serre’s work on spectral sequences and fibre spaces was based on cubes. Kan’s early work on combinatorial homotopy was based on cubes. However the category of cubical sets was found to have a major disadvantage compared with simplicial sets, in that the cartesian product in this category failed to have the correct homotopy type. This is in striking contrast to the cartesian product on simplicial sets.
Nonetheless cubical sets continued to have a kind of underground existence.
Brown and Higgins introduced the extra structure of connections $\Gamma^\pm_i$ on cubical sets, and included this structure into their cubical strict ∞-groupoids. All this structure was essential for the equivalence with crossed complexes and for the applications to homotopy theory. For example these $\omega$-groupoids have a canonical structure of thin elements, defined as any composition of elements of the form $\pm \epsilon_j x, \pm \Gamma^\pm_i$. Such elements have “commuting boundary”.
The geometric realisation of cubical sets with connections, and the relation with cartesian products, has been analysed by Maltsiniotis in the paper referred to below.
Nonetheless, the advantages of cubes are:
Easy notions of multiple compositions (compared with the globular pasting schemes); we refer to compositions in cubical sets.
Good notions of tensor product, because of the rule $I^m \times I^n \cong I^{m+n}$, and hence easy conceptual handling of homotopies. This is exploited in the paper by Brown and Higgins on tensor products, and also in the following paper
which gives a monoidal closed structure on cubical $\omega$-categories with connections, allowing the transfer of this to globular $\omega$-categories. The tensor product here generalises the Gray tensor product of 2-categories. This is also convenient in the homotopical structure on C*-algebras.
Cubical methods are a key feature in using higher homotopy groupoids to prove homotopy classification results.
Early references include
Jean-Pierre Serre, Homologie singulière des espaces fibrés , Ann.
Math. 54 no.3 (1951), pp.425-505. (pdf)
Dan Kan, Abstract homotopy I , Proc. Nat. Acad. Sci. U.S.A. 41 (1955) pp.1092–1096. (pdf)
General introductions of the cube category and of cubical sets are in
Rick Jardine, Cubical sets, Lecture 12 in Fields Lectures on simplicial presheaves (unpublished).
Sjoerd Crans, section 2 of Pasting schemes for the monoidal biclosed structure on $\omega$-$\mathbf{Cat}$ (web, ps, pdf)
The cubical identities satisfied by a cubical set are given there in proposition 2.8 on p. 9.
Cubical singular homology is discussed in
An axiomatization of cubical sets in constructive set theory/type theory (with the aim of building models of homotopy type theory) is in
Marc Bezem, Thierry Coquand, Simon Huber, A model of type theory in cubical sets, 2013 (web, pdf)
Ambrus Kaposi, Thorsten Altenkirch, A syntax for cubical type theory (pdf)
Simon Docherty, A model of type theory in cubical sets with connection, 2014 (pdf)
See also
For more on this see at relation between category theory and type theory.
The homotopy theory / model category structure on cubical sets is discussed in
The fact that the exponential object of two fibrant cubical sets is again fibrant follows from remark 8.4.33 in
in the context of Cisinski model structures.
Finally cubical sets as categorical semantics for homotopy type theory with univalence is discussed in
The strict test category nature of cubical sets with connection is discussed in
There is also the old work
in which “supercomplexes” are discussed, that combine simplicial sets and cubical sets (def 5). There are functors from simplicial sets to supercomplexes (after Defn 5) and, implicitly, from supercomplexes to cubical sets (in Appendix II). This was written in 1956, long before people were thinking as formally as nowadays and long before Quillen model theory, but a comparison of the homotopy categories might be in there.
A discussion of cubical sets and normal forms in several cases is in
Cubical sets as models for strict ∞-groupoids are discussed in
Their use for monoidal closed structures and homotopy classification is given in
and are essential in
Essential subtoposes of $Set^{\Box^{op}}$ are discussed in the context of Lawvere’s dialectical theory of dimension in
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