Definition 14.11.1. Let U be a simplicial set. We say x is an n-simplex of U to signify that x is an element of U_ n. We say that y is the jth face of x to signify that d^ n_ jx = y. We say that z is the jth degeneracy of x if z = s^ n_ jx. A simplex is called degenerate if it is the degeneracy of another simplex.
14.11 Simplicial sets
Let U be a simplicial set. It is a good idea to think of U_0 as the 0-simplices, the set U_1 as the 1-simplices, the set U_2 as the 2-simplices, and so on.
We think of the maps s^ n_ j : U_ n \to U_{n + 1} as the map that associates to an n-simplex A the degenerate (n + 1)-simplex B whose (j, j + 1)-edge is collapsed to the vertex j of A. We think of the map d^ n_ j : U_ n \to U_{n - 1} as the map that associates to an n-simplex A one of the faces, namely the face that omits the vertex j. In this way it become possible to visualize the relations among the maps s^ n_ j and d^ n_ j geometrically.
Here are a few fundamental examples.
Example 14.11.2. For every n \geq 0 we denote \Delta [n] the simplicial set
\Delta ^{opp} \longrightarrow \textit{Sets},\quad [k] \longmapsto \mathop{\mathrm{Mor}}\nolimits _{\Delta }([k], [n])
We leave it to the reader to verify the following statements. Every m-simplex of \Delta [n] with m > n is degenerate. There is a unique nondegenerate n-simplex of \Delta [n], namely \text{id}_{[n]}.
Lemma 14.11.3. Let U be a simplicial set. Let n \geq 0 be an integer. There is a canonical bijection
\mathop{\mathrm{Mor}}\nolimits (\Delta [n], U) \longrightarrow U_ n
which maps a morphism \varphi to the value of \varphi on the unique nondegenerate n-simplex of \Delta [n].
Proof. Omitted. \square
Example 14.11.4. Consider the category \Delta /[n] of objects over [n] in \Delta , see Categories, Example 4.2.13. There is a functor p : \Delta /[n] \to \Delta . The fibre category of p over [k], see Categories, Section 4.35, has as objects the set \Delta [n]_ k of k-simplices in \Delta [n], and as morphisms only identities. For every morphism \varphi : [k] \to [l] of \Delta , and every object \psi : [l] \to [n] in the fibre category over [l] there is a unique object over [k] with a morphism covering \varphi , namely \psi \circ \varphi : [k] \to [n]. Thus \Delta /[n] is fibred in sets over \Delta . In other words, we may think of \Delta /[n] as a presheaf of sets over \Delta . See also, Categories, Example 4.38.7. And this presheaf of sets agrees with the simplicial set \Delta [n]. In particular, from Equation (14.4.0.1) and Lemma 14.11.3 above we get the formula
\mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\Delta )}(\Delta /[n], U) = U_ n
for any simplicial set U.
Lemma 14.11.5. Let U, V be simplicial sets. Let a, b \geq 0 be integers. Assume every n-simplex of U is degenerate if n > a. Assume every n-simplex of V is degenerate if n > b. Then every n-simplex of U \times V is degenerate if n > a + b.
Proof. Suppose n > a + b. Let (u, v) \in (U \times V)_ n = U_ n \times V_ n. By assumption, there exists a \alpha : [n] \to [a] and a u' \in U_ a and a \beta : [n] \to [b] and a v' \in V_ b such that u = U(\alpha )(u') and v = V(\beta )(v'). Because n > a + b, there exists an 0 \leq i \leq a + b such that \alpha (i) = \alpha (i + 1) and \beta (i) = \beta (i + 1). It follows immediately that (u, v) is in the image of s^{n - 1}_ i. \square
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