Acyclic Categories

Two features of the cyclic model are particularly important synthetically. The first is that the selectivities can be significantly higher than for the acyclic category. Compare entries 2 and 6 of Table 4.3 the methoxy and trimethylsilyloxy groups chelate the magnesium (entry 2) whereas the triisopropylsilyloxy group does not (entry 6). This poorly selective example reacts by the acyclic pathway (also compare entries 1-5 with Tables 4.1 and 4.2). The second noteworthy point is that the product predicted by the cyclic and acyclic models are sometimes different. As shown in Scheme 4.1, the predictions of the acyclic and cyclic models are different for Table 4.3, entry 1 (see also entries 2 and 6). 

Section 2.3 is entirely devoted to triangulated spaces. These will be of crucial importance when we study nerves of acyclic categories in Chapter 10, and will also appear in various combinatorial quotient constructions in Chapter 14. Finally, the last section of this chapter considers the general CW complexes. 

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

Finally, let us remark that it is possible to take one of the functors F and G in Definition 4.36 contravariant. This will also yield a category, which we denote by 0(F°p G) if F is taken to be contravariant, and 0 F J. G°p) if G is taken to be contravariant. This construction will come in handy when we define the analogue of intervals for acyclic categories. 

Most concepts and constructions involving acyclic categories are introduced in analogy with those used in the context of posets. Often these translate into classical notions of category theory, while in some cases they 5deld notions that are not standard at all. 

Recall from Chapter 4 that any poset P can be viewed as a category in the following way the objects of this category are the elements of P, and for every pair of elements x,y P, the set of morphisms M x,y) has precisely one element if x > y, and is empty otherwise. Clearly, this determines the composition rule for the morphisms uniquely. When a poset is viewed as a category in this way, it is of course an acyclic category, and intuitively, if posets appear to be more comfortable gadgets, one may think of acyclic categories as... 

Another way to visualize acyclic categories is to think of them as those that can be drawn on a sheet of paper, with dots indicating the objects, and straight or slightly bent arrows, all pointing down, indicating the nonidentity morphisms see examples in Figure 10.1. 

One way the acyclic categories generalize posets can be formalized as follows. For any acyclic category C there exists a unique partial order > on the set of objects 0 C) such that Mc x,y) 0 implies x > y. We denote this poset by R C). Furthermore, it is immediate that an acyclic category C is a poset if and only if for every pair of objects x,y 0(C), the cardinahty of the set of morphisms Mc x,y) is at most 1. 

We call a morphism indecomposable if it cannot be represented as a composition of two nonidentity morphisms. An acyclic category C is called graded if there is a function r 0 C) —> Z such that whenever m a —> y is a nonidentity indecomposable morphism, we have r(x) = r(y) - -1. 

It is straightforward to generalize the classical notion of linear extension of posets, see Definition 2.20, to the context of acyclic categories. To underline this, we give here the definition using similar wording. 

It is easy to see, for example using induction, that an acyclic category has at least one linear extension. In a way, a converse of this statement is true as well if a category C has a linear extension and every morphism of an object into itself is an identity, then C must be acyclic. 

In the context of acyclic categories, the role of order-preserving maps is played by functors, see Chapter 4 for their definition. It is an important and very useful fact that all acyclic categories together with all possible functors between them form a full subcategory of Cat we call that category AC. 

Sometimes it is useful to further restrict our attention to the subcategories consisting only of finite acyclic categories, or posets. The subcategory of finite posets is the one in which most of Combinatorial Algebraic Topology has been... 

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. 

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. 

Proof. Given two acyclic categories C and D, and a functor P C D, each composable morphism chain oq... 

An example of a functor between acyclic categories and an associated trisp map between their nerves is shown in Figures 10.3 and 10.4. For posets, Proposition 10.6 translates into saying that an order-preserving map between posets P and Q (these are the morphisms in Posets) induces a simphcial map between the corresponding order complexes A P) and A(Q), which additionally preserves edge orientations. 

Fig. 10.4. The trisp map between nerves of acyclic categories induced by the functor in Figure 10.3.

Stacks of Acyclic Categories and Joins of Regular TAisps... 

Instead of putting two acyclic categories side by side, one can also put one on top of the other. 

The second acyclic category in Figure 10.1 can be seen as a stack of the acyclic category with two objects and two nonidentity morphisms and the acyclic category with one object. Another example of a stack is shown in Figure 10.5. 

We note that for an arbitrary acyclic category C, the acyclic category C 1 is obtained from C by adding a terminal object, while the category 1 C is obtained from C by adding an initial one. In the special case of posets we recover the following classical concept. 

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

One of the simplest operations one can do on an acyclic category is that of a deletion of a set of vertices. Once the vertex is deleted, all the simplices that contained this vertex in the nerve will be deleted as well hence for an arbitrary acyclic category C, and for arbitrary set of objects S C 0(C), we have A C S) = dU(c)S. 

Fig. 10.6. Acyclic categories of objects below x and strictly below x.

The natural generalization of the notion of a lattice would be to require that the acyclic category have all finite products and coproducts. Unfortunately, this does not bring an3d,hing new, as the next proposition shows. 

Proposition 10.9. Let C be an acyclic category that has all products of two elements. Then C is a poset. 

Also, by s3Tnmetry, any acyclic category C that has all finite coproducts of two elements must be a poset. 

The following definition allows us to turn an acyclic category into a poset, as far as the topology of the associated space is concerned. 

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