nLab product of simplices

Idea

This page describes aspects of the combinatorics of Cartesian products of simplicial sets, mostly by describing (Prop. ) the non-degenerate simplicices inside a Cartesian product $\Delta[p] \times \Delta[q]$ of basic n-simplices for $n = p,q$. These are enumerated by $(p,q)$-(un-)shuffles in a manner that for simplicial abelian groups is originally known from the Eilenberg-Zilber map.

Products of simplicial sets

Proposition

(cartesian product of simplicial sets)
For $K, L \,\in\,$ SimplicialSets, their Cartesian product,

$$K \times L \;\in\; SimplicialSets$$

is the simplicial set whose $k$th component set is the Cartesian product of Sets of the components of the two factors

(1)$$(K\times L)_k \;=\; K_k \times L_k \,,$$

and whose face- and degeneracy maps are, similarly, the image under the Cartesian product-functor of the face- and degeneracy maps of the factors:

(2)$$d^{K \times L}_i(x,y) \;=\; \big( d^K_i(x), \, d^L_i(y) \big) \,, \;\;\;\;\; s^{K \times L}_i(x,y) \;=\; \big( s^K_i(x), \, s^L_i(y) \big)$$

Proof

Since SimplicialSets is a category of presheaves, namely over the simplex category, this is a special case of the general fact that limits of presheaves are computed objectwise.

But it is also immediate to check that (2) with (1) satisfies the defining universal property of the Cartesian product.

Remark

(degenerate simplices in product of simplicial sets) Prop. means in particular that a simplex of the form $\big(s_\alpha(x),\, s_\beta(y)\big) \,\in\, K \times L$ may be non-degenerate even though its two components are each degenerate (see the archetypical Example below).

Indeed, the proposition says that the degenerate simplices in $K \times L$ are precisely those such that their two simplex-components in $K$ and $L$, respectively, are in the image of the same degeneracy map $\alpha = \beta$.

Non-degenerate cells in a product of simplices

Using the above basic facts about products of simplicial sets, we have the following explicit characterization of the non-degenerated cells in products $\Delta[p] \times \Delta[q]$ of the basic simplices (the simplicial sets represented by the objects in the simplex category).

Characterization

Proposition

(non-degenerate $(p+q)$-simplices in $\Delta[p] \times \Delta[q]$)
For $p,q \,\in\, \mathbb{N}$, the non-degenerate simplices in the Cartesian product (Prop. )

$$\Delta[p] \times \Delta[q]$$

of standard simplices in sSet correspond, under the Yoneda lemma, to precisely those morphisms of simplicial sets

(3)$$\Delta[p+q] \xrightarrow{\;\; \sigma \;\;} \Delta[p] \times \Delta[q]$$

which satisfy the following equivalent conditions:

• as morphisms of posets they are strictly monotone;

• as permutations of $(p+q)$ elements they are $(p,q)$-shuffles;

• as morphisms of finitely generated categories they take generating morphisms to generating morphisms;

(e.g. Kerodon 2.5.7.2: 00RH)

Here the first statement says that such morphisms may hence be represented by paths

• on a $(p+1)\times(q+1)$-lattice,

• from one corner to its opposite corner,

• consisting of $p+q$ unit steps,

• each either horizontally or vertically

and the second statement says that such paths are characterized by the $(p,q)$-shuffle $(\mu_1 \lt \cdots \lt \mu_p, \, \nu_1 \lt \cdots \lt \nu_q)$ of lists of step numbers $\mu_i$ where the path proceeds horizontally and step numbers $\nu_i$ where it proceeds vertically:

Proof

From Prop. it is clear (Rem. ) that a simplex $\sigma$ (3) is degenerate precisely if, when regarded as a path as above, it contains a constant step, i.e. one which moves neither horizontally nor vertically. But then – by degree reasons, since we are looking at paths of $p + q$ steps in a lattice of side length $p$ and $q$ – it must be that the path proceeds by $p + q$ unit steps.

Remark

(sequence- and shuffle-notation for simplices in a product of simplices)
Written as a pair of $(p+q)$-simplices, one in $\Delta[p]$ and one in $\Delta[q]$, the non-degenerate simplex (3) is hence a pair of monotone lists of natural numbers

(4)$$\left( \begin{array}{ccccccccc} 0& \ldots & 0 & 1 & \ldots & 1 & 2 & \ldots & p \\ 0& \ldots & i & i & \ldots & j & j & \ldots & q \end{array} \right) \,,$$

such that one row has a constant step precisely where the other has not.

Recording the step numbers where either of these lists is non-constant yields a $(p,q)$-shuffle:

(5)$$\begin{array}{cc} \text{positions of horizontal steps} & \text{positions of vertical steps} \\ \mu_1 \lt \mu_2 \lt \cdots \lt \mu_p & \nu_1 \lt \nu_2 \lt \cdots \lt \nu_p \end{array}$$

namely a permutation of $(p+q)$ elements where each element is larger than its left neighbour, except possibly when going from the $p$th to the $p+1$st element.

The shuffle-data yields a conveniently explicit re-construction of the product of simplices, as follows:

Definition

For $\mu = (\mu_1 \lt \mu_2 \lt \cdots \lt \mu_p)$ a sequence of natural numbers in $\{1, \cdots, p+q\}$, write

$$\sigma_\mu \;\colon\; X_q \to X_{p+q}$$

for the composite

$$[p+q] \overset{\;\; \sigma_{\mu_q - 1 } \;\;}{\longrightarrow} [p+q-1] \longrightarrow \cdots \longrightarrow [q + 2] \overset{\;\; \sigma_{ \mu_{2} - 1 } \;\;}{\longrightarrow} [q + 1] \overset{\;\; \sigma_{ \mu_1 - 1 } \;\;}{\longrightarrow} [q] \,,$$

where $\sigma_i \,\colon\, [n+1] \to [n]$ denotes the surjective monotone map (the codegeneracy map) that repeats the index $i$.

Proposition

The non-degenerate simplices in the Cartesian product (Prop. )

$$\Delta[p] \times \Delta[q]$$

of simplices in sSet are precisely those of the form

$$(\sigma_\nu, \sigma_\mu) \;\in\; Hom\big( \Delta[p + q] ,\, \Delta[p] \times \Delta[q] \big) \;=\; \big( \Delta[p] \times \Delta[q] \big)_{p+q}$$

for $(\mu,\nu)$ a $(p,q)$-shuffle.

Examples

Example

(cylinders of simplices)

The non-degenerate (n+1)-simplices in the cylinder over the n-simplex

$$\begin{aligned} & \Delta[1] \\ \times & \Delta[n] \end{aligned} \;\in\; sSet$$

are, according to Prop. and in the notation of Remark , as follows:

$$\begin{array}{cc} \text{path} &\phantom{A}& (1,n)\text{-shuffle} \\ \phantom{A} \\ \left( \;\, \array{ \mathllap{[} 0 & 1 & 1 & 1 & 1 & \cdots & 1 & 1 \mathrlap] \\ \mathllap[ 0 & 0 & 1 & 2 & 3 & \cdots & n-1 & n \mathrlap] } \;\, \right) && \Big( 1 , \; 2, 3, 4, 5, 6, \cdots, n \Big) \,, \\ \phantom{A} \\ \left( \;\, \array{ \mathllap{[} 0 & 0 & 1 & 1 & 1 & \cdots & 1 & 1 \mathrlap] \\ \mathllap[ 0 & 1 & 1 & 2 & 3 & \cdots & n-1 & n \mathrlap] } \;\, \right) && \Big( 2 ,\; 1, 3, 4, 5, 6, \cdots, n \Big) \\ \phantom{A} \\ \left( \;\, \array{ \mathllap{[} 0 & 0 & 0 & 1 & 1 & \cdots & 1 & 1 \mathrlap] \\ \mathllap[ 0 & 1 & 2 & 2 & 3 & \cdots & n-1 & n \mathrlap] } \;\, \right) && \Big( 3 ,\; 1, 2, 4, 5, 6, \cdots, n \Big) \\ \phantom{A} \\ \left( \;\, \array{ \mathllap{[} 0 & 0 & 0 & 0 & 1 & \cdots & 1 & 1 \mathrlap] \\ \mathllap[ 0 & 1 & 2 & 3 & 3 & \cdots & n-1 & n \mathrlap] } \;\, \right) && \Big( 4 ,\; 1, 2, 3, 5, 6 \cdots, n \Big) \end{array}$$

and so on.

Example

(non-degenerate simplices in simplicial square)
The complete set of non-degenerate simplices in $\Delta[1] \times \Delta[1]$ is, in specialization of Example , according to Prop. and in the notation of Remark , the following:

See also Friedman 2008, Fig. 20.

Example

The non-degenerate simplices in

$$\begin{aligned} & \Delta[2] \\ \times & \Delta[1] \end{aligned}$$

are, in specialization of Example , according to Prop. and in the notation of Remark , the following three:

$$\begin{array}{cc} \text{ path } && (2,1)\text{-shuffle} \\ \phantom{A} \\ \left( \;\, \array{ \mathllap{[} 0 & 1 & 2 & 2 \mathrlap] \\ \mathllap[ 0 & 0 & 0 & 1 \mathrlap] } \;\, \right) &\phantom{AA}& \Big( 1, 2 , \; 3 \Big) \\ \phantom{A} \\ \left( \;\, \array{ \mathllap{[} 0 & 1 & 1 & 2 \mathrlap] \\ \mathllap[ 0 & 0 & 1 & 1 \mathrlap] } \;\, \right) && \Big( 1, 3 , \; 2 \Big) \\ \phantom{A} \\ \left( \;\, \array{ \mathllap{[} 0 & 0 & 1 & 2 \mathrlap] \\ \mathllap[ 0 & 1 & 1 & 1 \mathrlap] } \;\, \right) && \Big( 2, 3 , \; 1 \Big) \end{array}$$

Example

Some non-degenerate simplices in $\Delta[2] \times \Delta[2]$:

$$\begin{array}{cc} \text{path} &\phantom{AA}& (2,2)\text{-shuffle} \\ \phantom{A} \\ \left( \;\, \array{ \mathllap{[} 0 & 1 & 2 & 2 & 2 \mathrlap] \\ \mathllap[ 0 & 0 & 0 & 1 & 2 \mathrlap] } \;\, \right) && \Big( 1, 2, \; 3, 4 \Big) \\ \phantom{AA} \\ \left( \;\, \array{ \mathllap{[} 0 & 1 & 1 & 2 & 2 \mathrlap] \\ \mathllap[ 0 & 0 & 1 & 1 & 2 \mathrlap] } \;\, \right) && \Big( 1, 3, \; 2, 4 \Big) \\ \phantom{AA} \\ \left( \;\, \array{ \mathllap{[} 0 & 1 & 1 & 1 & 2 \mathrlap] \\ \mathllap[ 0 & 0 & 1 & 2 & 2 \mathrlap] } \;\, \right) && \Big( 1, 4, \; 2, 3 \Big) \\ \phantom{AA} \\ \left( \;\, \array{ \mathllap{[} 0 & 0 & 0 & 1 & 2 \mathrlap] \\ \mathllap[ 0 & 1 & 2 & 2 & 2 \mathrlap] } \;\, \right) && \Big( 3, 4, \; 1, 2 \Big) \end{array}$$

Related concepts

• Eilenberg-Zilber map

References

The shuffle-formula is due (in terms of simplicial abelian groups of chains on a simplicial set, see at Eilenberg-Zilber map) to:

• Samuel Eilenberg, Saunders MacLane, (5.3) in: On the groups $H(\Pi,n)$, I, Ann. of Math. (2) 58, (1953), 55–106. (jstor:1969820)

following:

• Samuel Eilenberg, Joseph A. Zilber, American Journal of Mathematics Vol. 75, No. 1 (Jan., 1953), pp. 200-204 (jstor:2372629, doi:10.2307/2372629)

The streamlined perspective of strictly monotonic morphisms $\Delta[p+q] \to \Delta[p] \times \Delta[q]$ is highlighted in

• Kerodon, 2.5.7.2 $(p,q)$-Shuffles (00RH)

Exposition:

• Greg Friedman, §5 of: An elementary illustrated introduction to simplicial sets, Rocky Mountain J. Math. 42(2): 353-423 (2012) (arXiv:0809.4221, doi:10.1216/RMJ-2012-42-2-353)

Textbook accounts:

• Greg Friedman, §2.5.1 & §B.6 in: Singular Intersection Homology, Cambridge University Press 2020 (doi:10.1017/9781316584446, pdf)