# Kerodon

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Variant 3.3.2.10 ($\operatorname{Ex}$ for Semisimplicial Sets). Note that, for every nonnegative integer $n$, the simplicial set $\operatorname{N}_{\bullet }( \operatorname{Chain}[n] )$ is braced (Exercise 3.3.1.2). If $X$ is a semisimplicial set, we write $\operatorname{Ex}_{n}(X)$ for the collection of all morphisms of semisimplicial sets $\operatorname{N}_{\bullet }( \operatorname{Chain}[n] )^{\mathrm{nd}} \rightarrow X$; here $\operatorname{N}_{\bullet }( \operatorname{Chain}[n] )^{\mathrm{nd}}$ denotes the semisimplicial subset of $\operatorname{N}_{\bullet }(\operatorname{Chain}[n] )$ spanned by the nondegenerate simplices. The construction $[n] \mapsto \operatorname{Ex}_{n}(X)$ determines a semisimplicial set, which we denote by $\operatorname{Ex}(X)$.

Note that, if $X$ is the underlying semisimplicial set of a simplicial set $Y$, then $\operatorname{Ex}(X)$ is the underlying semisimplicial set of the simplicial set $\operatorname{Ex}(Y)$ given by Construction 3.3.2.5 (this is a special case of Proposition 3.3.1.5). In other words, the construction $X \mapsto \operatorname{Ex}(X)$ determines a functor from the category of semisimplicial sets to itself which fits into a commutative diagram

$\xymatrix@R =50pt@C=50pt{ \{ \text{Simplicial sets} \} \ar [r]^-{\operatorname{Ex}} \ar [d] & \{ \text{Simplicial sets} \} \ar [d] \\ \{ \text{Semisimplicial sets} \} \ar [r]^-{\operatorname{Ex}} & \{ \text{Semisimplicial sets} \} . }$