Simplicial Subdivision of the Direct Product

The product of two arbitrary objects, see Definition 4.20, exists in the category Cat. The next definition describes an explicit construction. 

Definition 10.20. LetC and D he arbitrary categories. The direct product 

The next theorem is a classical statement about acyclic categories. 

Theorem 10.21. For arbitrary acyclic categories C and D, the topological space A C x D) is homeomorphic to A(C) x A D). In fact, a stronger statement is true  

On the right-hand side (RHS), the points can be described as pairs of linear combinations (souo + - + Sfcafc,to oH- -tmbm), where oq i 

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