Functors between simplicial sets and cubical sets with connections

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 $\mathrm{Sd} \colon \mathsf{sSet} \to \mathsf{sSet}$.

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"

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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.

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.