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Construction (Joins of Augmented Simplicial Sets). For every pair of functors $X,Y: \operatorname{Lin}^{\operatorname{op}} \rightarrow \operatorname{Set}$, we let $(X \star Y): \operatorname{Lin}^{\operatorname{op}} \rightarrow \operatorname{Set}$ denote a new functor given on objects by the formula

\[ (X \star Y)(J) = \coprod _{I \sqsubseteq J} (X(I) \times Y(J \setminus I)). \]

Here the coproduct is indexed by the collection of all initial segments $I \sqsubseteq J$.

More formally, the functor $(X \star Y): \operatorname{Lin}^{\operatorname{op}} \rightarrow \operatorname{Set}$ can be described as follows:

  • For every finite linearly ordered set $J$, $(X \star Y)(J)$ is the collection of all triples $(I, x, y)$, where $I$ is an initial segment of $J$, $x$ is an element of $X(I)$, and $y$ is an element of $Y(J \setminus I)$.

  • If $\alpha : J' \rightarrow J$ is a nondecreasing function, then the induced map $(X \star Y)(\alpha ): (X \star Y)(J) \rightarrow (X \star Y)(J')$ is given by the construction

    \[ (I, x, y) \mapsto ( \alpha ^{-1}(I), X( \alpha |_{ \alpha ^{-1}(I) })(x), Y( \alpha |_{ \alpha ^{-1}(J \setminus I)} )(y) ). \]

We will refer to $X \star Y$ as the join of $X$ and $Y$.