Regular trisp

Clearly, if the vertices of this book, regular trisps will appear as nerves of acyclic categories. 

For example, all the cell spaces, except for the trisps that we have defined up to now are regular CW complexes. Regular trisps are also regular CW complexes. 

Many of the complexes that arise in combinatorics have the structure of regular trisps, allowing a completely combinatorial description. Let us describe one of the most prominent examples. As we have seen, one way to present an abstract simplicial complex by combinatorial means is to take the order complex of some poset consisting of combinatorial objects. Almost without exception, this poset allows an action of some finite group in fact, often the acting group is the finite symmetric group. 

Assume now that P is a poset and assume that a group G acts on P in an order-preserving way. Since the order complex construction is functorial, this in turn will induce a simplicial action on the complex A(P). In fact, it is easy to see that in this situation the topological quotient has a natural induced cell structure, with new cells being the orbits of the old cells, and that furthermore, one gets a regular trisp. More details on this will be provided in Chapter 14. 

It goes without saying that such quotient complexes serve as a bountiful source of combinatorially defined regular trisps. One example is derived... 

In Figure 9.10 we have pictured the regular trisp A Tlz)/S - This complex consists of five triangles four in the background, and the fifth one is in front filled with a somewhat darker color. The vertices of this quotient complex are... 

The construction of combinatorial regular trisps in Section 9.3 goes back to the work of Babson and the author on quotient constructions for partial... 

The Regular Trisp of Composable Morphism Chains in an Acyclic Category... 

The appearance of acyclic categories in Combinatorial Algebraic Topology is in large part motivated by the existence of a construction that associates to each acyclic category a geometric object, more precisely, a regular trisp. 

It is easy to see that this description yields a well-defined regular trisp, according to Definition 2.47. Indeed, a A -simplex indexed by a composable... 

Example 10.5. Figure 10.2 shows the regular trisps that realize the nerves of previously considered acyclic categories. Note that in these examples, the nerves are not (geometric realizations of) abstract simphcial complexes. 

For arbitrary acyclic categories C and D we have an isomorphism of regular trisps ... 

We can see how these formulas are satisfied for the example in Figure 10.6. To prove formula (10.2), let us describe a bijection between the sets of sim-plices on the left- and on the right-hand sides. By definition, the fc-simplices of the regular trisp f A c)X are indexed by those [k + l)-simplices of the regular trisp A C) that have x as a vertex. This is the same as the composable morphism chains... 

To check that the maps / and g are continuous, we just need to see that this is the case on closed simplices. This is straightforward, since all our manipulations with coefficients give continuous functions, including the case in which the coefficients are set to 0. Also, clearly these maps are inverses of each other. This shows that the spaces A(C) and A I C)) are homeomorphic. Finally, one can see that the image of any closed simplex of A I C)) under g is contained in some closed simplex of A C). Therefore A I C)) is actually a subdivision of the regular trisp A(C). ... 

If, furthermore, X is a finite regular trisp and the G-action satisfies Condition (S2), then the quotient trisp X/G is again regular. 

First, we prove in Proposition 14.11 that A is always surjective. Furthermore, G a) = [a] for a G 0(C), which means that, restricted to 0-skeletons, A is an isomorphism. If the two regular trisps were abstract simplicial complexes (only one face for any fixed vertex set), this would suffice to show isomorphism. Neither one is an abstract simplicial complex in general, but while the quotient of a complex A C)/G can have simplices with fairly arbitrary face sets in common, A C/G) has only one face for any fixed edge set. 

Having chosen the subcategory J and the function /, let us now first define an increasing filtration on the regular trisp A K). Let F = ai... 

Fact. Assume that X is a regular trisp, and that Y is a subtrisp of X such that the vertices in X Y have no edges in between them. Then we have... 

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