Pullback abstract simplicial complex

Definition 2.25. Assume that Cl is an abstract simplicial complex on B. We define the pullback abstract simplicial complex f Cl) by setting... 

The pullback abstract simplicial complex satisfies the following universal property any abstract simplicial complex on A such that / is a simplicial map must be an abstract simplicial subcomplex of f Cl). Again, one could say that f Q) is obtained by pulling the simplicial structure Cl back, and it is the maximal one that makes the set map / simplicial. 

The reader is invited to check that the following properties of pullbacks and pushforwards hold for any abstract simplicial complex A ... 

Intuitively, one can think of the map g as the pullback map. It is important to remark that if tp is not injective, it may happen that dim g r]) > dim 77. Since g is an order-preserving map, the induced map between abstract simphcial complexes A g) Bd (Horn (G, K)) — Bd (Horn (T, K)) is simplicial and gives the corresponding map of topological spaces, which we denote by V k- It is important to notice that the map g does not always come from a cellular map. 

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See also in sourсe #XX -- [ ]

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