# Functors between simplicial sets and cubical sets with connections

Let $sSet = Set^{\Delta^{op}}$, $cSet = Set^{\square^{op}}$, $ccSet = Set^{\square_c^{op}}$ be the categories of simplicial sets, cubical sets, and cubical sets with connections, respectively. Then there is a left adjoint functor $cSet \to sSet$ such that a representable cubical set $\square_n$ is mapped to $\Delta[1]^n$. But I'm looking for a left adjoint functor going in the other direction.

We could try to define a functor $\Delta \to cSet$ in such a way that $\Delta[n]$ is mapped to a quotient of $\square_n$. For example, $\Delta[2]$ is mapped to the square in which one of the faces is degenerated. This does not work since we cannot define a map of cubical sets corresponding to one of the degeneracy maps $\Delta[2] \to \Delta[1]$. But it seems that we can do this if we assume that our cubical sets have connections.

So, the question is: can we construct a (nontrivial) functor $\Delta \to ccSet$? Is there such a construction in the literature?

• One construction that comes to mind here is the homotopy coherent nerve which seems to involve turning simplicial structures into cubical ones (and then back to simplicial) in a sense that I'm not sure I can quite put my finger on. Let me also note that "change of shape" functors in higher category theory can be rather complicated to describe. The example I have in mind is the Street nerve which mediates between "globular" $n$-categories and "simplicial" ones. – Tim Campion Apr 21 '17 at 3:13
• The homotopy coherent nerve looks like it should factor through categories enriched in cubical sets with connections. But I don't know whether it is actually true. It's an interesting question on its own. – Valery Isaev Apr 21 '17 at 4:17
• @user62782 that's shown in a paper of Rivera and Zeinalian: arxiv.org/abs/1612.04801 – Tyler Lawson Apr 21 '17 at 13:18

I believe that the standard barycentric subdivision functor $\mathrm{Sd} \colon \mathsf{sSet} \to \mathsf{sSet}$ factors through cubical sets with connections. We define a functor $S \colon \Delta \to \mathsf{ccSet}$ by setting $S[m]$ to the cubical subset of $\square^{m+1}$ spanned by all vertices except the initial one. If we think of $\square^{m+1}$ as the poset of all subsets of $[m]$, then $S[m]$ corresponds to the poset of non-empty subsets. A simplicial operator $\varphi \colon [m] \to [n]$ induces a cubical map $S[m] \to S[n]$ by sending subsets of $[m]$ to their images under $\varphi$. This appears to be well-defined as a functor into $\mathsf{ccSet}$, e.g. a degeneracy operator $[2] \to [1]$ acts as a connection on one square of $S[2]$ and as a regular degeneracy on two other squares. It is directly verified that the composite of the resulting left Kan extension with the functor $\mathsf{ccSet} \to \mathsf{sSet}$ that you mention in the question is indeed $\mathrm{Sd} \colon \mathsf{sSet} \to \mathsf{sSet}$.

• You can also define a functor $S' : \Delta \to ccSet$ such that $S'[m]$ is the cubical subset of $\square^{m+1}$ consisting of singleton subsets. This functor is closer to the one I was thinking about. For example, $S'[m]$ has exactly $m+1$ vertices. It seems that this is not the functor that I tried to describe in the question, but I'm not sure. – Valery Isaev Apr 21 '17 at 20:01
• @user62782 If I understand correctly this functor sends a simplicial set to the discrete cubical set with the same vertices. I would assume that this is not what you wanted. – Karol Szumiło Apr 21 '17 at 22:25

This is only a partial answer but maybe it can lead you somewhere. We can define the join of two cubical sets as follows:

If $C$ and $D$ are cubical sets, then $(C \wedge D)_n$ is the set $C_n \sqcup \bigsqcup_{n = i+j+1} (C_i \times D_j) \sqcup D_n$. For $x \in C_i$ and $y \in D_j$, the faces of the cell $x \wedge y \in (C \wedge D)_{i+j+1}$ are given by:

• $\partial_k^\alpha (x \wedge y) = (\partial_k^\alpha x) \wedge y$ for $1 \leq k \leq i$
• $\partial_k^\alpha (x \wedge y) = \epsilon_{i+1}^j x$ for $k = i+1$ and $\alpha = -$ (where $\epsilon_{i+1}^j$ is $\epsilon_{i+1}$ applied $j$ times).
• $\partial_k^\alpha (x \wedge y) = \epsilon_1^i y$ for $k = i+1$ and $\alpha = +$
• $\partial_k^\alpha (x \wedge y) = x \wedge \partial^\alpha_{k-i-1} y$ for $i+2 \leq k \leq i+j+1$.

I haven't checked, but I expect that one should be able to find suitable formulas for the degeneracies (and connections if $C$ and $D$ come equipped with connections) so as to make the join into a bifunctor. However one has to be careful: this bifunctor does not define a monoidal structure on cubical sets (even with connections): it is not associative.

Anyway, if you start from the terminal cubical set $\top$, then $\top^{\wedge n}$ (warning: you have to choose a suitable bracketing for this to make sense: for example $(\top \wedge \top) \wedge \top$ and $\top \wedge (\top \wedge \top)$ are not isomorphic) is a good candidate for the image of $\Delta[n]$. I do not know however whether all the morphisms in $\Delta$ have an image between the $\top^{\wedge n}$.

A word on the join of cubical sets: AFAIK it appears nowhere in the literature. It doesn't have very good properties on cubical sets. On cubical categories however it should be a monoidal product, and should correspond to the join of $\infty$-categories defined by Ara and Maltsiniotis in "Joint et tranches pour les $\infty$-catégories strictes"

Alternatively there is a morphism from $\Delta$ to $\square_c$ that sends $[n]$ to $[n+1]$, degeneracies to connections $\Gamma^+$ and faces to faces $\partial^+$. This is just a way to say that in $\square_c$, $[1]$ is a monoid object. General considerations on Kahn extensions tell you that it induces a left-adjoin from simplicial sets to cubical sets with connections. I believe that is the same as teh construction given by @KarolSzumilo.

There is also a dual construction obtained by considering the other monoidal structure on $[1]$, using $\Gamma^-$ and $\partial^-$. Finally this construction can be adapted if you work between augmented simplicial sets and cubical sets with connections.

• Regarding your second suggestion, it definitely differs from the construction suggested by @KarolSzumiło and I think it is not well-behaved homotopically: the geometric realization of the image of $\partial \Delta[2]$ is contractible. – Valery Isaev Apr 22 '17 at 1:29

Such an adjunction between simplicial sets and cubical sets with connections is constructed in a paper by Krzysztof Kapulkin, Zachery Lindsey and Liang Ze Wong, A co-reflection of cubical sets into simplicial sets with applications to model structures. Moreover, it is shown there that the left adjoint $$\mathrm{sSet} \to \mathrm{ccSet}$$ is full and fathfull and induces a Quillen equivalent model structure on $$\mathrm{ccSet}$$ for any cofibrantly generated model structure on $$\mathrm{sSet}$$ in which all cofibrations are monomorphisms.