American Mathematical Society

The homotopy theory of fusion systems

By Carles Broto, Ran Levi, Bob Oliver

Abstract

We define and characterize a class of p -complete spaces upper X which have many of the same properties as the p -completions of classifying spaces of finite groups. For example, each such upper X has a Sylow subgroup upper B upper S long right-arrow upper X , maps upper B upper Q long right-arrow upper X for a p -group upper Q are described via homomorphisms upper Q long right-arrow upper S , and upper H Superscript asterisk Baseline left-parenthesis upper X semicolon double-struck upper F Subscript p Baseline right-parenthesis is isomorphic to a certain ring of “stable elements” in upper H Superscript asterisk Baseline left-parenthesis upper B upper S semicolon double-struck upper F Subscript p Baseline right-parenthesis . These spaces arise as the “classifying spaces” of certain algebraic objects which we call “ p -local finite groups”. Such an object consists of a system of fusion data in upper S , as formalized by L. Puig, extended by some extra information carried in a category which allows rigidification of the fusion data.

The main goal of this paper is to identify and study a certain class of spaces which in many ways behave like p -completed classifying spaces of finite groups. These spaces occur as the “classifying spaces” of certain algebraic objects, which we call p -local finite groups. A p -local finite group consists, roughly speaking, of a finite p -group upper S and fusion data on subgroups of upper S , encoded in a way explained below. Our starting point is our earlier paper ReferenceBLO on p -completed classifying spaces of finite groups, together with the axiomatic treatment by Lluís Puig ReferencePu, ReferencePu2 of systems of fusion among subgroups of a given p -group.

The p -completion of a space upper X is a space upper X Subscript p Superscript logical-and which isolates the properties of upper X at the prime p , and more precisely the properties which determine its mod p cohomology. For example, a map of spaces upper X right-arrow Overscript f Endscripts upper Y induces a homotopy equivalence upper X Subscript p Superscript logical-and Baseline ModifyingAbove right-arrow With asymptotically-equals upper Y Subscript p Superscript logical-and if and only if f induces an isomorphism in mod p cohomology; and upper H Superscript asterisk Baseline left-parenthesis upper X Subscript p Superscript logical-and Baseline semicolon double-struck upper F Subscript p Baseline right-parenthesis approximately-equals upper H Superscript asterisk Baseline left-parenthesis upper X semicolon double-struck upper F Subscript p Baseline right-parenthesis in favorable cases (if upper X is “ p -good”). When upper G is a finite group, the p -completion upper B upper G Subscript p Superscript logical-and of its classifying space encodes many of the properties of upper G at p . For example, not only the mod p cohomology of upper B upper G , but also the Sylow p -subgroup of upper G together with all fusion among its subgroups, are determined up to isomorphism by the homotopy type of upper B upper G Subscript p Superscript logical-and .

Our goal here is to give a direct link between p -local structures and homotopy types which arise from them. This theory tries to make explicit the essence of what it means to be the p -completed classifying space of a finite group, and at the same time yields new spaces which are not of this type, but which still enjoy most of the properties a space of the form upper B upper G Subscript p Superscript logical-and would have. We hope that the ideas presented here will have further applications and generalizations in algebraic topology. But this theory also fits well with certain aspects of modular representation theory. In particular, it may give a way of constructing classifying spaces for blocks in the group ring of a finite group over an algebraically closed field of characteristic p .

A saturated fusion system script upper F over a p -group upper S consists of a set upper H o m Subscript script upper F Baseline left-parenthesis upper P comma upper Q right-parenthesis of monomorphisms, for each pair of subgroups upper P comma upper Q less-than-or-equal-to upper S , which form a category under composition, include all monomorphisms induced by conjugation in upper S , and satisfy certain other axioms formulated by Puig (Definitions 1.1 and 1.2 below). In particular, these axioms are satisfied by the conjugacy homomorphisms in a finite group. We refer to ReferencePu and ReferencePu2 for more details of Puig’s work on saturated fusion systems (which he calls “full Frobenius systems” in ReferencePu2). The definitions and results given here, in Section 1 and in Appendix A are only a very brief account of those results of Puig used in our paper.

If script upper F is a saturated fusion system over upper S , then two subgroups upper P comma upper P prime less-than-or-equal-to upper S are called script upper F -conjugate if upper I s o Subscript script upper F Baseline left-parenthesis upper P comma upper P Superscript prime Baseline right-parenthesis not-equals normal empty-set . A subgroup upper P is called script upper F -centric if upper C Subscript upper S Baseline left-parenthesis upper P prime right-parenthesis less-than-or-equal-to upper P prime for all upper P prime that are script upper F -conjugate to upper P ; this is equivalent to what Puig calls “ script upper F -selfcentralizing”. Let script upper F Superscript c be the full subcategory of script upper F whose objects are the script upper F -centric subgroups of upper S . A centric linking system associated to script upper F is a category script upper L whose objects are the script upper F -centric subgroups of upper S , together with a functor script upper L right-arrow Overscript pi Endscripts script upper F Superscript c which is the identity on objects and surjective on morphisms, and which satisfies other axioms listed below in Definition 1.7. For example, for each object upper P , the kernel of the induced map upper A u t Subscript script upper L Baseline left-parenthesis upper P right-parenthesis right-arrow Overscript Endscripts upper A u t Subscript script upper F Baseline left-parenthesis upper P right-parenthesis is isomorphic to upper Z left-parenthesis upper P right-parenthesis , and upper A u t Subscript script upper L Baseline left-parenthesis upper P right-parenthesis contains a distinguished subgroup isomorphic to upper P .

The motivating examples for these definitions come from finite groups. If upper G is a finite group and p is a prime, then script upper F equals script upper F Subscript upper S Baseline left-parenthesis upper G right-parenthesis is the fusion system over upper S element-of upper S y l Subscript p Baseline left-parenthesis upper G right-parenthesis such that for each upper P comma upper Q less-than-or-equal-to upper S , upper H o m Subscript script upper F Baseline left-parenthesis upper P comma upper Q right-parenthesis is the set of homomorphisms induced by conjugation in upper G (and inclusion). The script upper F -centric subgroups of upper S are the p -centric subgroups: those upper P less-than-or-equal-to upper S such that upper C Subscript upper G Baseline left-parenthesis upper P right-parenthesis approximately-equals upper Z left-parenthesis upper P right-parenthesis times upper C prime Subscript upper G Baseline left-parenthesis upper P right-parenthesis for some upper C prime Subscript upper G Baseline left-parenthesis upper P right-parenthesis of order prime to p (see ReferenceBLO, Lemma A.5). In ReferenceBLO, we defined a category script upper L Subscript upper S Superscript c Baseline left-parenthesis upper G right-parenthesis whose objects are the p -centric subgroups of upper G which are contained in upper S , and where upper M o r Subscript script upper L Sub Subscript upper S Sub Superscript c Subscript left-parenthesis upper G right-parenthesis Baseline left-parenthesis upper P comma upper Q right-parenthesis equals upper N Subscript upper G Baseline left-parenthesis upper P comma upper Q right-parenthesis slash upper C prime Subscript upper G Baseline left-parenthesis upper P right-parenthesis . Here, upper N Subscript upper G Baseline left-parenthesis upper P comma upper Q right-parenthesis is the set of elements of upper G which conjugate upper P into upper Q . The category script upper L Subscript upper S Superscript c Baseline left-parenthesis upper G right-parenthesis , together with its projection to script upper F Subscript upper S Baseline left-parenthesis upper G right-parenthesis which sends the morphism corresponding to an element g element-of upper N Subscript upper G Baseline left-parenthesis upper P comma upper Q right-parenthesis to conjugation by g , is the example which motivated our definition of an associated centric linking system.

We define a p -local finite group to be a triple left-parenthesis upper S comma script upper F comma script upper L right-parenthesis , where script upper L is a centric linking system associated to a saturated fusion system script upper F over a p -group upper S . The classifying space of such a triple is the space StartAbsoluteValue script upper L EndAbsoluteValue Subscript p Superscript logical-and , where for any small category script upper C , the space StartAbsoluteValue script upper C EndAbsoluteValue denotes the geometric realization of the nerve of script upper C . This is partly motivated by the result that StartAbsoluteValue script upper L Subscript p Superscript c Baseline left-parenthesis upper G right-parenthesis EndAbsoluteValue Subscript p Superscript logical-and Baseline asymptotically-equals upper B upper G Subscript p Superscript logical-and for any finite upper G ReferenceBLO, Proposition 1.1. But additional motivation comes from Proposition 2.2 below, which says that if script upper L is a centric linking system associated to script upper F , then StartAbsoluteValue script upper L EndAbsoluteValue asymptotically-equals ModifyingBelow normal h times normal o times normal c times normal o times normal l times normal i times normal m With long right-arrow Subscript script upper O Sub Superscript c Subscript left-parenthesis script upper F right-parenthesis Baseline left-parenthesis upper B overTilde right-parenthesis , where script upper O Superscript c Baseline left-parenthesis script upper F right-parenthesis is a certain quotient “orbit” category of script upper F Superscript c , and upper B overTilde is a lifting of the homotopy functor which sends upper P to upper B upper P . The classifying space of a p -local finite group thus comes equipped with a decomposition as the homotopy colimit of a finite diagram of classifying spaces of p -groups.

We now state our main results. Our first result is that a p -local finite group is determined up to isomorphism by its classifying space. What is meant by an isomorphism of p -local finite groups will be explained later.

Theorem A (Theorem 7.4).

A p -local finite group left-parenthesis upper S comma script upper F comma script upper L right-parenthesis is determined by the homotopy type of StartAbsoluteValue script upper L EndAbsoluteValue Subscript p Superscript logical-and . In particular, if left-parenthesis upper S comma script upper F comma script upper L right-parenthesis and left-parenthesis upper S prime comma script upper F prime comma script upper L prime right-parenthesis are two p -local finite groups and StartAbsoluteValue script upper L EndAbsoluteValue Subscript p Superscript logical-and Baseline asymptotically-equals StartAbsoluteValue script upper L prime EndAbsoluteValue Subscript p Superscript logical-and , then left-parenthesis upper S comma script upper F comma script upper L right-parenthesis and left-parenthesis upper S prime comma script upper F prime comma script upper L prime right-parenthesis are isomorphic.

Next we study the cohomology of p -local finite groups. As one might hope, we have the following result, which appears as Theorem 5.8.

Theorem B

For any p -local finite group left-parenthesis upper S comma script upper F comma script upper L right-parenthesis , upper H Superscript asterisk Baseline left-parenthesis StartAbsoluteValue script upper L EndAbsoluteValue Subscript p Superscript logical-and Baseline semicolon double-struck upper F Subscript p Baseline right-parenthesis is isomorphic to the ring of “stable elements” in upper H Superscript asterisk Baseline left-parenthesis upper B upper S semicolon double-struck upper F Subscript p Baseline right-parenthesis ; i.e., the inverse limit of the rings upper H Superscript asterisk Baseline left-parenthesis upper B upper P semicolon double-struck upper F Subscript p Baseline right-parenthesis as a functor on the category script upper F . Furthermore, this ring is noetherian.

The next theorem gives an explicit description of the mapping space from the classifying space of a finite p -group into the classifying space of a p -local finite group. It is stated precisely as Corollary 4.5 and Theorem 6.3.

Theorem C

For any p -local finite group left-parenthesis upper S comma script upper F comma script upper L right-parenthesis , and any p -group upper Q ,

left-bracket upper B upper Q comma StartAbsoluteValue script upper L EndAbsoluteValue Subscript p Superscript logical-and Baseline right-bracket approximately-equals upper R e p left-parenthesis upper Q comma script upper L right-parenthesis equals Overscript def Endscripts upper H o m left-parenthesis upper Q comma upper S right-parenthesis slash left-parenthesis script upper F hyphen conjugacy right-parenthesis period

Furthermore, each component of the mapping space has the homotopy type of the classifying space of a p -local finite group which can be thought of as the “centralizer” of the image of the corresponding homomorphism upper Q right-arrow Overscript Endscripts upper S .

The next result describes the space of self equivalences of the classifying space of a p -local finite group. It is a generalization of ReferenceBLO, Theorem C. For a small category script upper C , let script upper A u t left-parenthesis script upper C right-parenthesis denote the groupoid whose objects are self equivalences of script upper C , and whose morphisms are natural isomorphisms of functors. Let script upper L be a centric linking system associated to a saturated fusion system script upper F . Self equivalences of script upper L which are structure preserving, in a sense to be made precise in section 7 below, are said to be isotypical. We let script upper A u t Subscript typ Baseline left-parenthesis script upper L right-parenthesis denote the subgroupoid of script upper A u t left-parenthesis script upper L right-parenthesis whose objects are the isotypical self equivalences of script upper L . For a space upper X , let upper A u t left-parenthesis upper X right-parenthesis denote the topological monoid of all self homotopy equivalences of upper X . The following theorem is restated below as Theorem 8.1.

Theorem D

Fix a p -local finite group left-parenthesis upper S comma script upper F comma script upper L right-parenthesis . Then upper A u t left-parenthesis StartAbsoluteValue script upper L EndAbsoluteValue Subscript p Superscript logical-and Baseline right-parenthesis and StartAbsoluteValue script upper A u t Subscript typ Baseline left-parenthesis script upper L right-parenthesis EndAbsoluteValue are equivalent as topological monoids in the sense that their classifying spaces are homotopy equivalent. In particular, their groups of components are isomorphic, and each component of upper A u t left-parenthesis StartAbsoluteValue script upper L EndAbsoluteValue Subscript p Superscript logical-and Baseline right-parenthesis is aspherical.

The statement of Theorem 8.1 also includes a description of the homotopy groups of upper A u t left-parenthesis StartAbsoluteValue script upper L EndAbsoluteValue Subscript p Superscript logical-and Baseline right-parenthesis .

So far, we have not mentioned the question of the existence and uniqueness of centric linking systems associated to a given saturated fusion system. Of course, as pointed out above, any finite group upper G gives rise to an associated p -local finite group. However there are saturated fusion systems which do not occur as the fusion system of any finite group. Thus a tool for deciding existence and uniqueness would be useful. The general obstructions to the existence and uniqueness of associated centric linking systems, which lie in certain higher limits taken over the orbit category script upper O Superscript c Baseline left-parenthesis script upper F right-parenthesis of the fusion system, are described in Proposition 3.1; and a means of computing these groups is provided by Proposition 3.2. The following result is just one special consequence of this, which settles the question for p -groups of small rank. Here, for any finite group upper G , we write r k Subscript p Baseline left-parenthesis upper G right-parenthesis for the largest rank of any elementary abelian p -subgroup of upper G .

Theorem E (Corollary 3.5).

Fix a saturated fusion system script upper F over a p -group upper S . If r k Subscript p Baseline left-parenthesis upper S right-parenthesis less-than p cubed , then there exists a centric linking system associated to script upper F , and if r k Subscript p Baseline left-parenthesis upper S right-parenthesis less-than p squared , then the associated centric linking system is unique.

In the last section, we present some direct constructions of saturated fusion systems and associated p -local finite groups (see Examples 9.3 and 9.4). The idea is to look at the fusion system over a p -group upper S (for p odd only) generated by groups of automorphisms of upper S and certain of its subgroups, and show that under certain hypotheses the resulting system is saturated. In all of these cases, the p -group upper S is nonabelian, and has an index p subgroup which is abelian and homocyclic (a product of cyclic groups of the same order). We then give a list of all finite simple groups which have Sylow subgroups of this form, based on the classification theorem, and use that to show that certain of the fusion systems which were constructed are not the fusion systems of any finite groups. In all cases, Theorem E applies to show the existence and uniqueness of centric linking systems, and hence p -local finite groups, associated to these fusion systems.

The basic definitions of saturated fusion systems and their associated centric linking systems are given in Section 1. Homotopy decompositions of classifying spaces of p -local finite groups are constructed in Section 2. The obstruction theory for the existence and uniqueness of associated centric linking systems, as well as some results about those obstruction groups, are shown in Section 3. Maps from the classifying space of a p -group to the classifying space of a p -local finite group are studied in Sections 4 and 6, while the cohomology rings of classifying spaces of p -local finite groups are dealt with in Section 5. A characterization of classifying spaces of p -local finite groups is given in Section 7, and their spaces of self equivalences are described in Section 8. The “exotic” examples of p -local finite groups are constructed in Section 9. Finally, some additional results on saturated fusion systems are collected in an appendix.

We would like to thank Dave Benson and Jesper Grodal for their many suggestions throughout the course of this work. In particular, Dave had earlier written and distributed notes which contained some of the ideas of our centric linking systems. We would also like to thank Lluís Puig for giving us a copy of his unpublished notes on saturated fusion systems. Markus Linckelmann, Haynes Miller, Bill Dwyer, and Jon Alperin have all shown interest and made helpful comments and suggestions. Kasper Andersen and Kari Ragnarsson both read earlier versions of this paper in detail, and sent us many suggestions for improvements. Two of the authors would also like to thank Slain’s Castle, a pub in Aberdeen, for their hospitality on New Year’s Day while we worked out the proof that the nerve of a centric linking system is p -good.

We would especially like to thank the universities of Aberdeen and Paris-Nord, the CRM and the UAB in Barcelona, and the Max-Planck Institut in Bonn for their hospitality in helping the three authors get together in various combinations; and also the European Homotopy Theory Network for helping to finance these visits.

1. Fusion systems and associated centric linking systems

We begin with the precise definitions of saturated fusion systems and their associated centric linking systems. Additional results about fusion systems due to Puig ReferencePu, ReferencePu2 are in Appendix A.

Given two finite groups upper P , upper Q , let upper H o m left-parenthesis upper P comma upper Q right-parenthesis denote the set of group homomorphisms from upper P to upper Q , and let upper I n j left-parenthesis upper P comma upper Q right-parenthesis denote the set of monomorphisms. If upper P and upper Q are subgroups of a larger group upper G , then upper H o m Subscript upper G Baseline left-parenthesis upper P comma upper Q right-parenthesis subset-of-or-equal-to upper I n j left-parenthesis upper P comma upper Q right-parenthesis denotes the subset of homomorphisms induced by conjugation by elements of upper G , and upper A u t Subscript upper G Baseline left-parenthesis upper P right-parenthesis the group of automorphisms induced by conjugation in upper G .

Definition 1.1

A fusion system script upper F over a finite p -group upper S is a category whose objects are the subgroups of upper S , and whose morphism sets upper H o m Subscript script upper F Baseline left-parenthesis upper P comma upper Q right-parenthesis satisfy the following conditions:

(a)

upper H o m Subscript upper S Baseline left-parenthesis upper P comma upper Q right-parenthesis subset-of-or-equal-to upper H o m Subscript script upper F Baseline left-parenthesis upper P comma upper Q right-parenthesis subset-of-or-equal-to upper I n j left-parenthesis upper P comma upper Q right-parenthesis for all upper P comma upper Q less-than-or-equal-to upper S .

(b)

Every morphism in script upper F factors as an isomorphism in script upper F followed by an inclusion.

Note that what we call a fusion system here is what Puig calls a divisible Frobenius system.

If script upper F is a fusion system over upper S and upper P comma upper Q less-than-or-equal-to upper S , then we write upper H o m Subscript script upper F Baseline left-parenthesis upper P comma upper Q right-parenthesis equals upper M o r Subscript script upper F Baseline left-parenthesis upper P comma upper Q right-parenthesis to emphasize that morphisms in the category script upper F are all homomorphisms, and upper I s o Subscript script upper F Baseline left-parenthesis upper P comma upper Q right-parenthesis for the subset of isomorphisms in script upper F . Thus upper I s o Subscript script upper F Baseline left-parenthesis upper P comma upper Q right-parenthesis equals upper H o m Subscript script upper F Baseline left-parenthesis upper P comma upper Q right-parenthesis if StartAbsoluteValue upper P EndAbsoluteValue equals StartAbsoluteValue upper Q EndAbsoluteValue , and upper I s o Subscript script upper F Baseline left-parenthesis upper P comma upper Q right-parenthesis equals normal empty-set otherwise. Also, upper A u t Subscript script upper F Baseline left-parenthesis upper P right-parenthesis equals upper I s o Subscript script upper F Baseline left-parenthesis upper P comma upper P right-parenthesis and upper O u t Subscript script upper F Baseline left-parenthesis upper P right-parenthesis equals upper A u t Subscript script upper F Baseline left-parenthesis upper P right-parenthesis slash upper I n n left-parenthesis upper P right-parenthesis . Two subgroups upper P comma upper P prime less-than-or-equal-to upper S are called script upper F -conjugate if upper I s o Subscript script upper F Baseline left-parenthesis upper P comma upper P Superscript prime Baseline right-parenthesis not-equals normal empty-set .

The fusion systems we consider here will all satisfy the following additional condition. Here, and throughout the rest of the paper, we write upper S y l Subscript p Baseline left-parenthesis upper G right-parenthesis for the set of Sylow p -subgroups of upper G . Also, for any upper P less-than-or-equal-to upper G and any g element-of upper N Subscript upper G Baseline left-parenthesis upper P right-parenthesis , c Subscript g Baseline element-of upper A u t left-parenthesis upper P right-parenthesis denotes the automorphism c Subscript g Baseline left-parenthesis x right-parenthesis equals g x g Superscript negative 1 .

Definition 1.2

Let script upper F be a fusion system over a p -group upper S .

A subgroup upper P less-than-or-equal-to upper S is fully centralized in script upper F if StartAbsoluteValue upper C Subscript upper S Baseline left-parenthesis upper P right-parenthesis EndAbsoluteValue greater-than-or-equal-to StartAbsoluteValue upper C Subscript upper S Baseline left-parenthesis upper P prime right-parenthesis EndAbsoluteValue for all upper P prime less-than-or-equal-to upper S that are script upper F -conjugate to upper P .

A subgroup upper P less-than-or-equal-to upper S is fully normalized in script upper F if StartAbsoluteValue upper N Subscript upper S Baseline left-parenthesis upper P right-parenthesis EndAbsoluteValue greater-than-or-equal-to StartAbsoluteValue upper N Subscript upper S Baseline left-parenthesis upper P prime right-parenthesis EndAbsoluteValue for all upper P prime less-than-or-equal-to upper S that are script upper F -conjugate to upper P .

script upper F is a saturated fusion system if the following two conditions hold:

(I)

Any upper P less-than-or-equal-to upper S which is fully normalized in script upper F is fully centralized in script upper F , and upper A u t Subscript upper S Baseline left-parenthesis upper P right-parenthesis element-of upper S y l Subscript p Baseline left-parenthesis upper A u t Subscript script upper F Baseline left-parenthesis upper P right-parenthesis right-parenthesis .

(II)

If upper P less-than-or-equal-to upper S and phi element-of upper H o m Subscript script upper F Baseline left-parenthesis upper P comma upper S right-parenthesis are such that phi upper P is fully centralized, and if we setupper N Subscript phi Baseline equals StartSet g element-of upper N Subscript upper S Baseline left-parenthesis upper P right-parenthesis vertical-bar phi c Subscript g Baseline phi Superscript negative 1 Baseline element-of upper A u t Subscript upper S Baseline left-parenthesis phi upper P right-parenthesis EndSet comma

then there is phi overbar element-of upper H o m Subscript script upper F Baseline left-parenthesis upper N Subscript phi Baseline comma upper S right-parenthesis such that phi overbar vertical-bar Subscript upper P Baseline equals phi .

The above definition is slightly different from the definition of a “full Frobenius system” as formulated by Lluís Puig ReferencePu2, §2.5, but is equivalent to his definition by the remarks after Proposition A.2. Condition (I) can be thought of as a “Sylow condition”. It says that upper O u t Subscript script upper F Baseline left-parenthesis upper S right-parenthesis has order prime to p (just as upper N Subscript upper G Baseline left-parenthesis upper S right-parenthesis slash upper S has order prime to p if upper S element-of upper S y l Subscript p Baseline left-parenthesis upper G right-parenthesis ); and more generally it reflects the fact that for any p -subgroup upper P less-than-or-equal-to upper G , there is some upper S element-of upper S y l Subscript p Baseline left-parenthesis upper G right-parenthesis such that upper N Subscript upper S Baseline left-parenthesis upper P right-parenthesis element-of upper S y l Subscript p Baseline left-parenthesis upper N Subscript upper G Baseline left-parenthesis upper P right-parenthesis right-parenthesis . Another way of interpreting this condition is that if StartAbsoluteValue upper N Subscript upper S Baseline left-parenthesis upper P right-parenthesis EndAbsoluteValue greater-than-or-equal-to StartAbsoluteValue upper N Subscript upper S Baseline left-parenthesis upper P prime right-parenthesis EndAbsoluteValue for upper P prime script upper F -conjugate to upper P , then upper C Subscript upper S Baseline left-parenthesis upper P right-parenthesis and upper A u t Subscript upper S Baseline left-parenthesis upper P right-parenthesis approximately-equals upper N Subscript upper S Baseline left-parenthesis upper P right-parenthesis slash upper C Subscript upper S Baseline left-parenthesis upper P right-parenthesis must also be maximal in the same sense. As for condition (II), it is natural to require that some extension property hold for morphisms in script upper F , and upper N Subscript phi is by definition the largest subgroup of upper N Subscript upper S Baseline left-parenthesis upper P right-parenthesis to which phi could possibly extend.

The motivating example for this definition is the fusion system of a finite group upper G . For any upper S element-of upper S y l Subscript p Baseline left-parenthesis upper G right-parenthesis , we let script upper F Subscript upper S Baseline left-parenthesis upper G right-parenthesis be the fusion system over upper S defined by setting upper H o m Subscript script upper F Sub Subscript upper S Subscript left-parenthesis upper G right-parenthesis Baseline left-parenthesis upper P comma upper Q right-parenthesis equals upper H o m Subscript upper G Baseline left-parenthesis upper P comma upper Q right-parenthesis for all upper P comma upper Q less-than-or-equal-to upper S .

Proposition 1.3

Let upper G be a finite group, and let upper S be a Sylow p -subgroup of upper G . Then the fusion system script upper F Subscript upper S Baseline left-parenthesis upper G right-parenthesis over upper S is saturated. Also, a subgroup upper P less-than-or-equal-to upper S is fully centralized in script upper F Subscript upper S Baseline left-parenthesis upper G right-parenthesis if and only if upper C Subscript upper S Baseline left-parenthesis upper P right-parenthesis element-of upper S y l Subscript p Baseline left-parenthesis upper C Subscript upper G Baseline left-parenthesis upper P right-parenthesis right-parenthesis , while upper P is fully normalized in script upper F Subscript upper S Baseline left-parenthesis upper G right-parenthesis if and only if upper N Subscript upper S Baseline left-parenthesis upper P right-parenthesis element-of upper S y l Subscript p Baseline left-parenthesis upper N Subscript upper G Baseline left-parenthesis upper P right-parenthesis right-parenthesis .

Proof.

Fix some upper P less-than-or-equal-to upper S , and choose g element-of upper G such that g Superscript negative 1 Baseline upper S g contains a Sylow p -subgroup of upper N Subscript upper G Baseline left-parenthesis upper P right-parenthesis . Then g upper P g Superscript negative 1 Baseline less-than-or-equal-to upper S and upper N Subscript g Sub Superscript negative 1 Subscript upper S g Baseline left-parenthesis upper P right-parenthesis element-of upper S y l Subscript p Baseline left-parenthesis upper N Subscript upper G Baseline left-parenthesis upper P right-parenthesis right-parenthesis , and so upper N Subscript upper S Baseline left-parenthesis g upper P g Superscript negative 1 Baseline right-parenthesis element-of upper S y l Subscript p Baseline left-parenthesis upper N Subscript upper G Baseline left-parenthesis g upper P g Superscript negative 1 Baseline right-parenthesis right-parenthesis . This clearly implies that StartAbsoluteValue upper N Subscript upper S Baseline left-parenthesis g upper P g Superscript negative 1 Baseline right-parenthesis EndAbsoluteValue greater-than-or-equal-to StartAbsoluteValue upper N Subscript upper S Baseline left-parenthesis upper P prime right-parenthesis EndAbsoluteValue for all upper P prime less-than-or-equal-to upper S that are upper G -conjugate to upper P . Thus g upper P g Superscript negative 1 is fully normalized in script upper F Subscript upper S Baseline left-parenthesis upper G right-parenthesis , and upper P is fully normalized in script upper F Subscript upper S Baseline left-parenthesis upper G right-parenthesis if and only if StartAbsoluteValue upper N Subscript upper S Baseline left-parenthesis upper P right-parenthesis EndAbsoluteValue equals StartAbsoluteValue upper N Subscript upper S Baseline left-parenthesis g upper P g Superscript negative 1 Baseline right-parenthesis EndAbsoluteValue , if and only if upper N Subscript upper S Baseline left-parenthesis upper P right-parenthesis element-of upper S y l Subscript p Baseline left-parenthesis upper N Subscript upper G Baseline left-parenthesis upper P right-parenthesis right-parenthesis . A similar argument proves that upper P is fully centralized in script upper F Subscript upper S Baseline left-parenthesis upper G right-parenthesis if and only if upper C Subscript upper S Baseline left-parenthesis upper P right-parenthesis element-of upper S y l Subscript p Baseline left-parenthesis upper C Subscript upper G Baseline left-parenthesis upper P right-parenthesis right-parenthesis .

If upper P is fully normalized in script upper F Subscript upper S Baseline left-parenthesis upper G right-parenthesis , then since upper N Subscript upper S Baseline left-parenthesis upper P right-parenthesis element-of upper S y l Subscript p Baseline left-parenthesis upper N Subscript upper G Baseline left-parenthesis upper P right-parenthesis right-parenthesis , the obvious counting argument shows that

upper A u t Subscript upper S Baseline left-parenthesis upper P right-parenthesis element-of upper S y l Subscript p Baseline left-parenthesis upper A u t Subscript upper G Baseline left-parenthesis upper P right-parenthesis right-parenthesis and upper C Subscript upper S Baseline left-parenthesis upper P right-parenthesis element-of upper S y l Subscript p Baseline left-parenthesis upper C Subscript upper G Baseline left-parenthesis upper P right-parenthesis right-parenthesis period

In particular, upper P is fully centralized in script upper F Subscript upper S Baseline left-parenthesis upper G right-parenthesis , and this proves condition (I) in Definition 1.2.

To see condition (II), let upper P less-than-or-equal-to upper S and g element-of upper G be such that g upper P g Superscript negative 1 Baseline less-than-or-equal-to upper S and is fully centralized in script upper F Subscript upper S Baseline left-parenthesis upper G right-parenthesis , and write upper P prime equals g upper P g Superscript negative 1 for short. Set

upper N equals StartSet x element-of upper N Subscript upper S Baseline left-parenthesis upper P right-parenthesis vertical-bar c Subscript g Baseline ring c Subscript x Baseline ring c Subscript g Superscript negative 1 Baseline element-of upper A u t Subscript upper S Baseline left-parenthesis upper P Superscript prime Baseline right-parenthesis EndSet and upper N Subscript upper G Baseline equals upper N dot upper C Subscript upper G Baseline left-parenthesis upper P right-parenthesis semicolon

and similarly

upper N prime equals StartSet x element-of upper N Subscript upper S Baseline left-parenthesis upper P Superscript prime Baseline right-parenthesis vertical-bar c Subscript g Superscript negative 1 Baseline ring c Subscript x Baseline ring c Subscript g Baseline element-of upper A u t Subscript upper S Baseline left-parenthesis upper P right-parenthesis EndSet and upper N prime Subscript upper G Baseline equals upper N prime dot upper C Subscript upper G Baseline left-parenthesis upper P Superscript prime Baseline right-parenthesis period

In particular, g upper N Subscript upper G Baseline g Superscript negative 1 Baseline equals upper N prime Subscript upper G ( upper N and upper N prime are conjugate modulo centralizers), and thus g upper N g Superscript negative 1 and upper N prime are two p -subgroups of upper N prime Subscript upper G . Furthermore,

left-bracket upper N prime Subscript upper G Baseline colon upper N Superscript prime Baseline right-bracket equals left-bracket upper C Subscript upper G Baseline left-parenthesis upper P Superscript prime Baseline right-parenthesis colon upper C Subscript upper S Baseline left-parenthesis upper P prime right-parenthesis right-bracket

is prime to p (since upper P prime is fully centralized), so upper N prime element-of upper S y l Subscript p Baseline left-parenthesis upper N prime Subscript upper G right-parenthesis . Since upper C Subscript upper G Baseline left-parenthesis upper P prime right-parenthesis normal-subgroup-of upper N prime Subscript upper G has p -power index, all Sylow p -subgroups of upper N prime Subscript upper G are conjugate by elements of upper C Subscript upper G Baseline left-parenthesis upper P prime right-parenthesis , and hence there is h element-of upper C Subscript upper G Baseline left-parenthesis upper P prime right-parenthesis such that h left-parenthesis g upper N g Superscript negative 1 Baseline right-parenthesis h Superscript negative 1 Baseline less-than-or-equal-to upper N prime . Thus c Subscript h g Baseline element-of upper H o m Subscript script upper F Sub Subscript upper S Subscript left-parenthesis upper G right-parenthesis Baseline left-parenthesis upper N comma upper S right-parenthesis extends c Subscript g Baseline element-of upper H o m Subscript script upper F Sub Subscript upper S Subscript left-parenthesis upper G right-parenthesis Baseline left-parenthesis upper P comma upper S right-parenthesis .

Puig’s original motivation for defining fusion systems came from block theory. Let k be an algebraically closed field of characteristic p not-equals 0 . A block in a group ring k left-bracket upper G right-bracket is an indecomposable 2-sided ideal which is a direct summand. Puig showed ReferencePu that the Brauer pairs associated to a block b (the “ b -subpairs”), together with the inclusion and conjugacy relations defined by Alperin and Broué ReferenceAB, form a saturated fusion system over the defect group of b . See, for example, ReferenceAB or ReferenceAlp, Chapter IV, for definitions of defect groups and Brauer pairs of blocks.

In practice, when proving that certain fusion systems are saturated, it will be convenient to replace condition (I) by a modified version of the condition, as described in the following lemma.

Lemma 1.4

Let script upper F be a fusion system over a p -group upper S which satisfies condition (II) in Definition 1.2, and also satisfies the condition

(I prime )

Each subgroup upper P less-than-or-equal-to upper S is script upper F -conjugate to a fully centralized subgroup upper P prime less-than-or-equal-to upper S such that upper A u t Subscript upper S Baseline left-parenthesis upper P prime right-parenthesis element-of upper S y l Subscript p Baseline left-parenthesis upper A u t Subscript script upper F Baseline left-parenthesis upper P prime right-parenthesis right-parenthesis .

Then script upper F is a saturated fusion system.

Proof.

We must prove condition (I) in Definition 1.2. Assume that upper P less-than-or-equal-to upper S is fully normalized in script upper F . By (I prime ), there is upper P prime less-than-or-equal-to upper S which is script upper F -conjugate to upper P , and such that upper P prime is fully centralized in script upper F and upper A u t Subscript upper S Baseline left-parenthesis upper P prime right-parenthesis element-of upper S y l Subscript p Baseline left-parenthesis upper A u t Subscript script upper F Baseline left-parenthesis upper P prime right-parenthesis right-parenthesis . In particular,

StartLayout 1st Row with Label left-parenthesis 1 right-parenthesis EndLabel StartAbsoluteValue upper C Subscript upper S Baseline left-parenthesis upper P Superscript prime Baseline right-parenthesis EndAbsoluteValue greater-than-or-equal-to StartAbsoluteValue upper C Subscript upper S Baseline left-parenthesis upper P right-parenthesis EndAbsoluteValue and StartAbsoluteValue upper A u t Subscript upper S Baseline left-parenthesis upper P Superscript prime Baseline right-parenthesis EndAbsoluteValue greater-than-or-equal-to StartAbsoluteValue upper A u t Subscript upper S Baseline left-parenthesis upper P right-parenthesis EndAbsoluteValue period EndLayout

On the other hand, since upper P is fully normalized,

StartAbsoluteValue upper C Subscript upper S Baseline left-parenthesis upper P right-parenthesis EndAbsoluteValue dot StartAbsoluteValue upper A u t Subscript upper S Baseline left-parenthesis upper P right-parenthesis EndAbsoluteValue equals StartAbsoluteValue upper N Subscript upper S Baseline left-parenthesis upper P right-parenthesis EndAbsoluteValue greater-than-or-equal-to StartAbsoluteValue upper N Subscript upper S Baseline left-parenthesis upper P Superscript prime Baseline right-parenthesis EndAbsoluteValue equals StartAbsoluteValue upper C Subscript upper S Baseline left-parenthesis upper P Superscript prime Baseline right-parenthesis EndAbsoluteValue dot StartAbsoluteValue upper A u t Subscript upper S Baseline left-parenthesis upper P Superscript prime Baseline right-parenthesis EndAbsoluteValue comma

and hence the inequalities in (1) are equalities. Thus upper P is fully centralized and upper A u t Subscript upper S Baseline left-parenthesis upper P right-parenthesis element-of upper S y l Subscript p Baseline left-parenthesis upper A u t Subscript script upper F Baseline left-parenthesis upper P right-parenthesis right-parenthesis . This proves (I).

For any pair of fusion systems script upper F 1 over upper S 1 and script upper F 2 over upper S 2 , let script upper F 1 times script upper F 2 be the obvious fusion system over upper S 1 times upper S 2 :

upper H o m Subscript script upper F 1 times script upper F 2 Baseline left-parenthesis upper P comma upper Q right-parenthesis equals StartSet left-parenthesis alpha 1 comma alpha 2 right-parenthesis StartAbsoluteValue element-of upper H o m left-parenthesis upper P comma upper Q right-parenthesis EndAbsoluteValue Subscript upper P Baseline upper P less-than-or-equal-to upper P 1 times upper P 2 comma alpha Subscript i Baseline element-of upper H o m Subscript script upper F Baseline left-parenthesis upper P Subscript i Baseline comma upper S Subscript i Baseline right-parenthesis EndSet

for all upper P comma upper Q less-than-or-equal-to upper S 1 times upper S 2 . The following technical result will be needed in Section 5.

Lemma 1.5

If script upper F 1 and script upper F 2 are saturated fusion systems over upper S 1 and upper S 2 , respectively, then script upper F 1 times script upper F 2 is a saturated fusion system over upper S 1 times upper S 2 .

Proof.

For any upper P less-than-or-equal-to upper S 1 times upper S 2 , let upper P 1 less-than-or-equal-to upper S 1 and upper P 2 less-than-or-equal-to upper S 2 denote the images of upper P under projection to the first and second factors. Thus upper P less-than-or-equal-to upper P 1 times upper P 2 , and this is the smallest product subgroup which contains upper P . Similarly, for any upper P less-than-or-equal-to upper S 1 times upper S 2 and any phi element-of upper H o m Subscript script upper F 1 times script upper F 2 Baseline left-parenthesis upper P comma upper S 1 times upper S 2 right-parenthesis , phi 1 element-of upper H o m Subscript script upper F 1 Baseline left-parenthesis upper P 1 comma upper S 1 right-parenthesis and phi 2 element-of upper H o m Subscript script upper F 2 Baseline left-parenthesis upper P 2 comma upper S 2 right-parenthesis denote the projections of phi .

We apply Lemma 1.4, and first check condition (I prime ). Fix upper P less-than-or-equal-to upper S 1 times upper S 2 ; we must show that upper P is script upper F 1 times script upper F 2 -conjugate to a subgroup upper P prime which is fully centralized and satisfies upper A u t Subscript upper S 1 times upper S 2 Baseline left-parenthesis upper P prime right-parenthesis element-of upper S y l Subscript p Baseline left-parenthesis upper A u t Subscript script upper F 1 times script upper F 2 Baseline left-parenthesis upper P prime right-parenthesis right-parenthesis . We can assume that upper P 1 and upper P 2 are both fully normalized; otherwise replace upper P by an appropriate subgroup in its script upper F 1 times script upper F 2 -conjugacy class. Since upper C Subscript upper S 1 times upper S 2 Baseline left-parenthesis upper P right-parenthesis equals upper C Subscript upper S 1 Baseline left-parenthesis upper P 1 right-parenthesis times upper C Subscript upper S 2 Baseline left-parenthesis upper P 2 right-parenthesis , and since (by (I) applied to the saturated fusion systems script upper F Subscript i ) the upper P Subscript i are fully centralized, upper P is also fully centralized. Also, by (I) again, upper A u t Subscript upper S Sub Subscript i Baseline left-parenthesis upper P Subscript i Baseline right-parenthesis element-of upper S y l Subscript p Baseline left-parenthesis upper A u t Subscript script upper F Sub Subscript i Subscript Baseline left-parenthesis upper P Subscript i Baseline right-parenthesis right-parenthesis , and hence upper A u t Subscript upper S 1 times upper S 2 Baseline left-parenthesis upper P 1 times upper P 2 right-parenthesis element-of upper S y l Subscript p Baseline left-parenthesis upper A u t Subscript script upper F 1 times script upper F 2 Baseline left-parenthesis upper P 1 times upper P 2 right-parenthesis right-parenthesis . Thus, if we regard upper A u t Subscript script upper F 1 times script upper F 2 Baseline left-parenthesis upper P right-parenthesis as a subgroup of upper A u t Subscript script upper F 1 times script upper F 2 Baseline left-parenthesis upper P 1 times upper P 2 right-parenthesis , there is an element alpha element-of upper A u t Subscript script upper F 1 times script upper F 2 Baseline left-parenthesis upper P 1 times upper P 2 right-parenthesis such that upper A u t Subscript upper S 1 times upper S 2 Baseline left-parenthesis upper P 1 times upper P 2 right-parenthesis contains a Sylow p -subgroup of alpha upper A u t Subscript script upper F 1 times script upper F 2 Baseline left-parenthesis upper P right-parenthesis alpha Superscript negative 1 Baseline equals upper A u t Subscript script upper F 1 times script upper F 2 Baseline left-parenthesis alpha upper P right-parenthesis . Then

upper A u t Subscript upper S 1 times upper S 2 Baseline left-parenthesis alpha upper P right-parenthesis equals upper A u t Subscript upper S 1 times upper S 2 Baseline left-parenthesis upper P 1 times upper P 2 right-parenthesis intersection upper A u t Subscript script upper F 1 times script upper F 2 Baseline left-parenthesis alpha upper P right-parenthesis element-of upper S y l Subscript p Baseline left-parenthesis upper A u t Subscript script upper F 1 times script upper F 2 Baseline left-parenthesis alpha upper P right-parenthesis right-parenthesis comma

alpha upper P is still fully centralized in script upper F 1 times script upper F 2 , and this finishes the proof of (I prime ).

To prove condition (II), fix upper P less-than-or-equal-to upper S 1 times upper S 2 and phi element-of upper H o m Subscript script upper F 1 times script upper F 2 Baseline left-parenthesis upper P comma upper S 1 times upper S 2 right-parenthesis , and assume phi left-parenthesis upper P right-parenthesis is fully centralized in script upper F 1 times script upper F 2 . Since

upper C Subscript upper S 1 times upper S 2 Baseline left-parenthesis phi left-parenthesis upper P right-parenthesis right-parenthesis equals upper C Subscript upper S 1 Baseline left-parenthesis phi 1 left-parenthesis upper P 1 right-parenthesis right-parenthesis times upper C Subscript upper S 2 Baseline left-parenthesis phi 2 left-parenthesis upper P 2 right-parenthesis right-parenthesis comma

we see that phi Subscript i Baseline left-parenthesis upper P Subscript i Baseline right-parenthesis is fully centralized in script upper F Subscript i for i equals 1 comma 2 . Set

upper N Subscript phi Baseline equals StartSet g element-of upper N Subscript upper S 1 times upper S 2 Baseline left-parenthesis upper P right-parenthesis vertical-bar phi c Subscript g Baseline phi Superscript negative 1 Baseline element-of upper A u t Subscript upper S 1 times upper S 2 Baseline left-parenthesis phi left-parenthesis upper P right-parenthesis right-parenthesis EndSet less-than-or-equal-to upper S 1 times upper S 2

and

upper N Subscript phi Sub Subscript i Subscript Baseline equals StartSet g element-of upper N Subscript upper S Sub Subscript i Subscript Baseline left-parenthesis upper P Subscript i Baseline right-parenthesis vertical-bar phi Subscript i Baseline c Subscript g Baseline phi Subscript i Superscript negative 1 Baseline element-of upper A u t Subscript upper S Sub Subscript i Subscript Baseline left-parenthesis phi Subscript i Baseline left-parenthesis upper P Subscript i Baseline right-parenthesis right-parenthesis EndSet less-than-or-equal-to upper S Subscript i Baseline period

Then phi extends to phi 1 times phi 2 element-of upper H o m Subscript script upper F 1 times script upper F 2 Baseline left-parenthesis upper P 1 times upper P 2 comma upper S 1 times upper S 2 right-parenthesis by definition of script upper F 1 times script upper F 2 , and hence to upper N Subscript phi 1 Baseline times upper N Subscript phi 2 by condition (II) applied to the saturated fusion systems script upper F 1 and script upper F 2 . So (II) holds for the fusion system script upper F 1 times script upper F 2 ( upper N Subscript phi Baseline less-than-or-equal-to upper N Subscript phi 1 Baseline times upper N Subscript phi 2 ), and this finishes the proof that script upper F 1 times script upper F 2 is saturated.

In order to help motivate the next constructions, we recall some definitions from ReferenceBLO. If upper G is a finite group and p is a prime, then a p -subgroup upper P less-than-or-equal-to upper G is p -centric if upper C Subscript upper G Baseline left-parenthesis upper P right-parenthesis equals upper Z left-parenthesis upper P right-parenthesis times upper C prime Subscript upper G Baseline left-parenthesis upper P right-parenthesis , where upper C prime Subscript upper G Baseline left-parenthesis upper P right-parenthesis has order prime to p . For any upper P comma upper Q less-than-or-equal-to upper G , let upper N Subscript upper G Baseline left-parenthesis upper P comma upper Q right-parenthesis denote the transporter: the set of all g element-of upper G such that g upper P g Superscript negative 1 Baseline less-than-or-equal-to upper Q . For any upper S element-of upper S y l Subscript p Baseline left-parenthesis upper G right-parenthesis , script upper L Subscript upper S Superscript c Baseline left-parenthesis upper G right-parenthesis denotes the category whose objects are the p -centric subgroups of upper S , and where upper M o r Subscript script upper L Sub Subscript upper S Sub Superscript c Subscript left-parenthesis upper G right-parenthesis Baseline left-parenthesis upper P comma upper Q right-parenthesis equals upper N Subscript upper G Baseline left-parenthesis upper P comma upper Q right-parenthesis slash upper C prime Subscript upper G Baseline left-parenthesis upper P right-parenthesis . By comparison, upper H o m Subscript upper G Baseline left-parenthesis upper P comma upper Q right-parenthesis approximately-equals upper N Subscript upper G Baseline left-parenthesis upper P comma upper Q right-parenthesis slash upper C Subscript upper G Baseline left-parenthesis upper P right-parenthesis . Hence there is a functor from script upper L Subscript upper S Superscript c Baseline left-parenthesis upper G right-parenthesis to script upper F Subscript upper S Baseline left-parenthesis upper G right-parenthesis which is the inclusion on objects, and which sends the morphism corresponding to g element-of upper N Subscript upper G Baseline left-parenthesis upper P comma upper Q right-parenthesis to c Subscript g Baseline element-of upper H o m Subscript upper G Baseline left-parenthesis upper P comma upper Q right-parenthesis .

Definition 1.6

Let script upper F be any fusion system over a p -group upper S . A subgroup upper P less-than-or-equal-to upper S is script upper F -centric if upper P and all of its script upper F -conjugates contain their upper S -centralizers. Let script upper F Superscript c denote the full subcategory of script upper F whose objects are the script upper F -centric subgroups of upper S .

We are now ready to define “centric linking systems” associated to a fusion system.

Definition 1.7

Let script upper F be a fusion system over the p -group upper S . A centric linking system associated to script upper F is a category script upper L whose objects are the script upper F -centric subgroups of upper S , together with a functor

pi colon script upper L right-arrow Overscript Endscripts script upper F Superscript c Baseline comma

and “distinguished” monomorphisms upper P right-arrow Overscript delta Subscript upper P Baseline Endscripts upper A u t Subscript script upper L Baseline left-parenthesis upper P right-parenthesis for each script upper F -centric subgroup upper P less-than-or-equal-to upper S , which satisfy the following conditions.

(A)

pi is the identity on objects and surjective on morphisms. More precisely, for each pair of objects upper P comma upper Q element-of script upper L , upper Z left-parenthesis upper P right-parenthesis acts freely on upper M o r Subscript script upper L Baseline left-parenthesis upper P comma upper Q right-parenthesis by composition (upon identifying upper Z left-parenthesis upper P right-parenthesis with delta Subscript upper P Baseline left-parenthesis upper Z left-parenthesis upper P right-parenthesis right-parenthesis less-than-or-equal-to upper A u t Subscript script upper L Baseline left-parenthesis upper P right-parenthesis ), and pi induces a bijection upper M o r Subscript script upper L Baseline left-parenthesis upper P comma upper Q right-parenthesis slash upper Z left-parenthesis upper P right-parenthesis ModifyingAbove right-arrow With approximately-equals upper H o m Subscript script upper F Baseline left-parenthesis upper P comma upper Q right-parenthesis period

(B)

For each script upper F -centric subgroup upper P less-than-or-equal-to upper S and each g element-of upper P , pi sends delta Subscript upper P Baseline left-parenthesis g right-parenthesis element-of upper A u t Subscript script upper L Baseline left-parenthesis upper P right-parenthesis to c Subscript g Baseline element-of upper A u t Subscript script upper F Baseline left-parenthesis upper P right-parenthesis .

(C)

For each f element-of upper M o r Subscript script upper L Baseline left-parenthesis upper P comma upper Q right-parenthesis and each g element-of upper P , the following square commutes in script upper L :

One easily checks that for any upper G and any upper S element-of upper S y l Subscript p Baseline left-parenthesis upper G right-parenthesis , script upper L Subscript upper S Superscript c Baseline left-parenthesis upper G right-parenthesis is a centric linking system associated to the fusion system script upper F Subscript upper S Baseline left-parenthesis upper G right-parenthesis . Condition (C) is motivated in part because it always holds in script upper L Subscript upper S Superscript c Baseline left-parenthesis upper G right-parenthesis for any upper G . Conditions (A) and (B) imply that upper P acts freely on upper M o r Subscript script upper L Baseline left-parenthesis upper P comma upper Q right-parenthesis . Together with (C), they imply that the upper Q -action on upper M o r Subscript script upper L Baseline left-parenthesis upper P comma upper Q right-parenthesis is free, and describe how it determines the action of upper P . Condition (C) was also motivated by the proof of Proposition 2.2 below, where we show that the nerve of any centric linking system is equivalent to the homotopy colimit of a certain functor.

Throughout the rest of this paper, whenever we refer to conditions (A), (B), or (C), it will mean the conditions in the above Definition 1.7.

Definition 1.8

A p -local finite group is a triple left-parenthesis upper S comma script upper F comma script upper L right-parenthesis , where script upper F is a saturated fusion system over the p -group upper S and script upper L is a centric linking system associated to script upper F . The classifying space of the p -local finite group left-parenthesis upper S comma script upper F comma script upper L right-parenthesis is the space StartAbsoluteValue script upper L EndAbsoluteValue Subscript p Superscript logical-and .

Thus, for any finite group upper G and any upper S element-of upper S y l Subscript p Baseline left-parenthesis upper G right-parenthesis , the triple left-parenthesis upper S comma script upper F Subscript upper S Baseline left-parenthesis upper G right-parenthesis comma script upper L Subscript upper S Superscript c Baseline left-parenthesis upper G right-parenthesis right-parenthesis is a p -local finite group. Its classifying space is StartAbsoluteValue script upper L Subscript upper S Superscript c Baseline left-parenthesis upper G right-parenthesis EndAbsoluteValue Subscript p Superscript logical-and Baseline asymptotically-equals StartAbsoluteValue script upper L Subscript p Superscript c Baseline left-parenthesis upper G right-parenthesis EndAbsoluteValue Subscript p Superscript logical-and , which by ReferenceBLO, Proposition 1.1 is homotopy equivalent to upper B upper G Subscript p Superscript logical-and .

The following notation will be used when working with p -local finite groups. For any group upper G , let script upper B left-parenthesis upper G right-parenthesis denote the category with one object o Subscript upper G , and one morphism denoted ModifyingAbove g With ˇ for each g element-of upper G .

Notation 1.9

Let left-parenthesis upper S comma script upper F comma script upper L right-parenthesis be a p -local finite group, where pi colon script upper L right-arrow Overscript Endscripts script upper F Superscript c denotes the projection functor. For each script upper F -centric subgroup upper P less-than-or-equal-to upper S , and each g element-of upper P , we write

ModifyingAbove g With caret equals delta Subscript upper P Baseline left-parenthesis g right-parenthesis element-of upper A u t Subscript script upper L Baseline left-parenthesis upper P right-parenthesis comma

and let

theta Subscript upper P Baseline colon script upper B left-parenthesis upper P right-parenthesis right-arrow Overscript Endscripts script upper L

denote the functor which sends the unique object o Subscript upper P Baseline element-of upper O b left-parenthesis script upper B left-parenthesis upper P right-parenthesis right-parenthesis to upper P and which sends a morphism ModifyingAbove g With ˇ (for g element-of upper P ) to ModifyingAbove g With caret equals delta Subscript upper P Baseline left-parenthesis g right-parenthesis . If f is any morphism in script upper L , we let left-bracket f right-bracket equals pi left-parenthesis f right-parenthesis denote its image in script upper F .

The following lemma lists some easy properties of centric linking systems associated to saturated fusion systems.

Lemma 1.10

Fix a p -local finite group left-parenthesis upper S comma script upper F comma script upper L right-parenthesis , and let pi colon script upper L right-arrow Overscript Endscripts script upper F Superscript c be the projection. Fix script upper F -centric subgroups upper P comma upper Q comma upper R in upper S . Then the following hold.

(a)

Fix any sequence upper P right-arrow Overscript phi Endscripts upper Q right-arrow Overscript psi Endscripts upper R of morphisms in script upper F Superscript c , and let psi overTilde element-of pi Subscript upper Q comma upper R Superscript negative 1 Baseline left-parenthesis psi right-parenthesis and ModifyingAbove psi phi With tilde element-of pi Subscript upper P comma upper R Superscript negative 1 Baseline left-parenthesis psi phi right-parenthesis be arbitrary liftings. Then there is a unique morphism phi overTilde element-of upper M o r Subscript script upper L Baseline left-parenthesis upper P comma upper Q right-parenthesis such that StartLayout 1st Row with Label left-parenthesis 1 right-parenthesis EndLabel psi overTilde ring phi overTilde equals ModifyingAbove psi phi With tilde comma EndLayout

and furthermore pi Subscript upper P comma upper Q Baseline left-parenthesis phi overTilde right-parenthesis equals phi .

(b)

If phi overTilde comma phi overTilde prime element-of upper M o r Subscript script upper L Baseline left-parenthesis upper P comma upper Q right-parenthesis are such that the homomorphisms phi equals Overscript def Endscripts pi Subscript upper P comma upper Q Baseline left-parenthesis phi overTilde right-parenthesis and phi prime equals Overscript def Endscripts pi Subscript upper P comma upper Q Baseline left-parenthesis phi overTilde prime right-parenthesis are conjugate (differ by an element of upper I n n left-parenthesis upper Q right-parenthesis ), then there is a unique element g element-of upper Q such that phi overTilde prime equals ModifyingAbove g With caret ring phi overTilde in upper M o r Subscript script upper L Baseline left-parenthesis upper P comma upper Q right-parenthesis .

Proof.

(a) Fix any element alpha element-of pi Subscript upper P comma upper Q Superscript negative 1 Baseline left-parenthesis phi right-parenthesis . By (A), there is a unique element g element-of upper Z left-parenthesis upper P right-parenthesis such that ModifyingAbove psi phi With tilde equals psi overTilde ring alpha ring ModifyingAbove g With caret . Hence equation (1) holds if we set phi overTilde equals alpha ring ModifyingAbove g With caret , and clearly pi Subscript upper P comma upper Q Baseline left-parenthesis phi overTilde right-parenthesis equals phi . Conversely, for any phi overTilde prime element-of upper M o r Subscript script upper L Baseline left-parenthesis upper P comma upper Q right-parenthesis such that psi overTilde ring phi overTilde Superscript prime Baseline equals ModifyingAbove psi phi With tilde we have pi Subscript upper P comma upper Q Baseline left-parenthesis phi overTilde prime right-parenthesis equals phi since they are equal after composing with psi , and so phi overTilde prime equals phi overTilde since (by (A) again) the same group upper Z left-parenthesis upper P right-parenthesis acts freely and transitively on pi Subscript upper P comma upper Q Superscript negative 1 Baseline left-parenthesis phi right-parenthesis and on pi Subscript upper P comma upper R Superscript negative 1 Baseline left-parenthesis psi phi right-parenthesis .

(b) If x element-of upper Q is such that

pi Subscript upper P comma upper Q Baseline left-parenthesis phi overTilde Superscript prime Baseline right-parenthesis equals phi prime equals c Subscript x Baseline ring phi equals pi Subscript upper P comma upper Q Baseline left-parenthesis ModifyingAbove x With caret ring phi overTilde right-parenthesis comma

then by (A) and (C), there is a unique element y element-of upper Z left-parenthesis upper P right-parenthesis such that

phi overTilde prime equals ModifyingAbove x With caret ring phi overTilde ring ModifyingAbove y With caret equals ModifyingAbove x With caret ring ModifyingAbove phi left-parenthesis y right-parenthesis With caret ring phi overTilde period

This proves the existence of g equals x dot phi left-parenthesis y right-parenthesis element-of upper Q such that phi overTilde prime equals ModifyingAbove g With caret ring phi overTilde . Conversely, if

phi overTilde prime equals ModifyingAbove g With caret ring phi overTilde equals ModifyingAbove h With caret ring phi overTilde

for g comma h element-of upper Q , then phi overTilde equals ModifyingAbove g Superscript negative 1 Baseline h With caret ring phi overTilde , so g Superscript negative 1 Baseline h element-of upper C Subscript upper Q Baseline left-parenthesis phi left-parenthesis upper P right-parenthesis right-parenthesis , and g Superscript negative 1 Baseline h element-of phi left-parenthesis upper P right-parenthesis since upper P (and hence phi left-parenthesis upper P right-parenthesis ) is script upper F -centric. Write g Superscript negative 1 Baseline h equals phi left-parenthesis y right-parenthesis for y element-of upper P ; then phi overTilde equals phi overTilde ring ModifyingAbove y With caret by (C), hence ModifyingAbove y With caret equals upper I d by (a), and so y equals 1 (and g equals h ) by (A).

Lemma 1.10(a) implies in particular that all morphisms in script upper L are monomorphisms in the categorical sense.

The next proposition describes how an associated centric linking system script upper L over a p -group upper S contains the category with the same objects and whose morphisms are the sets upper N Subscript upper S Baseline left-parenthesis upper P comma upper Q right-parenthesis .

Proposition 1.11

Let left-parenthesis upper S comma script upper F comma script upper L right-parenthesis be a p -local finite group, and let pi colon script upper L right-arrow Overscript Endscripts script upper F Superscript c be the associated projection. For each upper P less-than-or-equal-to upper S , fix a choice of “inclusion” morphism iota Subscript upper P Baseline element-of upper M o r Subscript script upper L Baseline left-parenthesis upper P comma upper S right-parenthesis such that left-bracket iota Subscript upper P Baseline right-bracket equals i n c l element-of upper H o m left-parenthesis upper P comma upper S right-parenthesis (and iota Subscript upper S Baseline equals upper I d Subscript upper S ). Then there are unique injections

delta Subscript upper P comma upper Q Baseline colon upper N Subscript upper S Baseline left-parenthesis upper P comma upper Q right-parenthesis right-arrow Overscript Endscripts upper M o r Subscript script upper L Baseline left-parenthesis upper P comma upper Q right-parenthesis comma

defined for all script upper F -centric subgroups upper P comma upper Q less-than-or-equal-to upper S , which have the following properties.

(a)

For all script upper F -centric upper P comma upper Q less-than-or-equal-to upper S and all g element-of upper N Subscript upper S Baseline left-parenthesis upper P comma upper Q right-parenthesis , left-bracket delta Subscript upper P comma upper Q Baseline left-parenthesis g right-parenthesis right-bracket equals c Subscript g Baseline element-of upper H o m Subscript script upper F Baseline left-parenthesis upper P comma upper Q right-parenthesis .

(b)

For all script upper F -centric upper P less-than-or-equal-to upper S we have delta Subscript upper P comma upper S Baseline left-parenthesis 1 right-parenthesis equals iota Subscript upper P , and delta Subscript upper P comma upper P Baseline left-parenthesis g right-parenthesis equals delta Subscript upper P Baseline left-parenthesis g right-parenthesis for g element-of upper P .

(c)

For all script upper F -centric upper P comma upper Q comma upper R less-than-or-equal-to upper S and all g element-of upper N Subscript upper S Baseline left-parenthesis upper P comma upper Q right-parenthesis and h element-of upper N Subscript upper S Baseline left-parenthesis upper Q comma upper R right-parenthesis we have delta Subscript upper Q comma upper R Baseline left-parenthesis h right-parenthesis ring delta Subscript upper P comma upper Q Baseline left-parenthesis g right-parenthesis equals delta Subscript upper P comma upper R Baseline left-parenthesis h g right-parenthesis .

Proof.

For each script upper F -centric upper P and upper Q and each g element-of upper N Subscript upper S Baseline left-parenthesis upper P comma upper Q right-parenthesis , there is by Lemma 1.10(a) a unique morphism delta Subscript upper P comma upper Q Baseline left-parenthesis g right-parenthesis element-of upper M o r Subscript script upper L Baseline left-parenthesis upper P comma upper Q right-parenthesis such that left-bracket delta Subscript upper P comma upper Q Baseline left-parenthesis g right-parenthesis right-bracket equals c Subscript g , and such that the following square commutes: Conditions (b) and (c) above also follow from the uniqueness property in Lemma 1.10(a). The injectivity of delta Subscript upper P comma upper Q follows from condition (A), since left-bracket delta Subscript upper P comma upper Q Baseline left-parenthesis g right-parenthesis right-bracket equals left-bracket delta Subscript upper P comma upper Q Baseline left-parenthesis h right-parenthesis right-bracket in upper H o m Subscript script upper F Baseline left-parenthesis upper P comma upper Q right-parenthesis if and only if h Superscript negative 1 Baseline g element-of upper C Subscript upper S Baseline left-parenthesis upper P right-parenthesis equals upper Z left-parenthesis upper P right-parenthesis .

We finish the section with the following proposition, which shows that the classifying space of any p -local finite group is p -complete, and also provides some control over its fundamental group.

Proposition 1.12

Let left-parenthesis upper S comma script upper F comma script upper L right-parenthesis be any p -local finite group. Then StartAbsoluteValue script upper L EndAbsoluteValue is p -good. Also, the composite

upper S right-arrow Overscript pi 1 left-parenthesis StartAbsoluteValue theta Subscript upper S Baseline EndAbsoluteValue right-parenthesis Endscripts pi 1 left-parenthesis StartAbsoluteValue script upper L EndAbsoluteValue right-parenthesis right-arrow Overscript Endscripts pi 1 left-parenthesis StartAbsoluteValue script upper L EndAbsoluteValue Subscript p Superscript logical-and Baseline right-parenthesis comma

induced by the inclusion script upper B left-parenthesis upper S right-parenthesis right-arrow Overscript theta Subscript upper S Baseline Endscripts script upper L , is surjective.

Proof.

For each script upper F -centric subgroup upper P less-than-or-equal-to upper S , fix a morphism iota Subscript upper P Baseline element-of upper M o r Subscript script upper L Baseline left-parenthesis upper P comma upper S right-parenthesis which lifts the inclusion (and set iota Subscript upper S Baseline equals upper I d Subscript upper S ). By Lemma 1.10(a), for each upper P less-than-or-equal-to upper Q less-than-or-equal-to upper S , there is a unique morphism iota Subscript upper P Superscript upper Q Baseline element-of upper M o r Subscript script upper L Baseline left-parenthesis upper P comma upper Q right-parenthesis such that iota Subscript upper Q Baseline ring iota Subscript upper P Superscript upper Q Baseline equals iota Subscript upper P .

Regard the vertex upper S as the basepoint of StartAbsoluteValue script upper L EndAbsoluteValue . Define

omega colon upper M o r left-parenthesis script upper L right-parenthesis right-arrow Overscript Endscripts pi 1 left-parenthesis StartAbsoluteValue script upper L EndAbsoluteValue right-parenthesis

by sending each phi element-of upper M o r Subscript script upper L Baseline left-parenthesis upper P comma upper Q right-parenthesis to the loop formed by the edges iota Subscript upper P , phi , and iota Subscript upper Q (in that order). Clearly, omega left-parenthesis psi ring phi right-parenthesis equals omega left-parenthesis psi right-parenthesis dot omega left-parenthesis phi right-parenthesis whenever psi and phi are composable, and omega left-parenthesis iota Subscript upper P Superscript upper Q Baseline right-parenthesis equals omega left-parenthesis iota Subscript upper P Baseline right-parenthesis equals 1 for all upper P less-than-or-equal-to upper Q less-than-or-equal-to upper S . Also, pi 1 left-parenthesis StartAbsoluteValue script upper L EndAbsoluteValue right-parenthesis is generated by upper I m left-parenthesis omega right-parenthesis , since any loop in StartAbsoluteValue script upper L EndAbsoluteValue can be split up as a composite of loops of the above form.

By Alperin’s fusion theorem for saturated fusion systems (Theorem A.10), each morphism in script upper F , and hence each morphism in script upper L , is (up to inclusions) a composite of automorphisms of fully normalized script upper F -centric subgroups. Thus pi 1 left-parenthesis StartAbsoluteValue script upper L EndAbsoluteValue right-parenthesis is generated by the subgroups omega left-parenthesis upper A u t Subscript script upper L Baseline left-parenthesis upper P right-parenthesis right-parenthesis for all fully normalized script upper F -centric upper P less-than-or-equal-to upper S .

Let upper K normal-subgroup-of pi 1 left-parenthesis StartAbsoluteValue script upper L EndAbsoluteValue right-parenthesis be the subgroup generated by all elements of finite order prime to p . For each fully normalized script upper F -centric upper P less-than-or-equal-to upper S , upper A u t Subscript script upper L Baseline left-parenthesis upper P right-parenthesis is generated by its Sylow subgroup upper N Subscript upper S Baseline left-parenthesis upper P right-parenthesis together with elements of order prime to p . Hence pi 1 left-parenthesis StartAbsoluteValue script upper L EndAbsoluteValue right-parenthesis is generated by upper K together with the subgroups omega left-parenthesis upper N Subscript upper S Baseline left-parenthesis upper P right-parenthesis right-parenthesis ; and omega left-parenthesis upper N Subscript upper S Baseline left-parenthesis upper P right-parenthesis right-parenthesis less-than-or-equal-to omega left-parenthesis upper S right-parenthesis for each upper P . This shows that omega sends upper S surjectively onto pi 1 left-parenthesis StartAbsoluteValue script upper L EndAbsoluteValue right-parenthesis slash upper K , and in particular that this quotient group is a finite p -group.

Set pi equals pi 1 left-parenthesis StartAbsoluteValue script upper L EndAbsoluteValue right-parenthesis slash upper K for short. Since upper K is generated by elements of order prime to p , the same is true of its abelianization, and hence upper H 1 left-parenthesis upper K semicolon double-struck upper F Subscript p Baseline right-parenthesis equals 0 . Thus, upper K is p -perfect. Let upper X be the cover of StartAbsoluteValue script upper L EndAbsoluteValue with fundamental group upper K . Then upper X is p -good and upper X Subscript p Superscript logical-and is simply connected since pi 1 left-parenthesis upper X right-parenthesis is p -perfect ReferenceBK, VII.3.2. Also, upper H Subscript i Baseline left-parenthesis upper X semicolon double-struck upper F Subscript p Baseline right-parenthesis is finite for all i since StartAbsoluteValue script upper L EndAbsoluteValue and hence upper X has finite skeleta. Hence upper X Subscript p Superscript logical-and Baseline right-arrow Overscript Endscripts StartAbsoluteValue script upper L EndAbsoluteValue Subscript p Superscript logical-and Baseline right-arrow Overscript Endscripts upper B pi is a fibration sequence and StartAbsoluteValue script upper L EndAbsoluteValue Subscript p Superscript logical-and is p -complete by ReferenceBK, II.5.2(iv). So StartAbsoluteValue script upper L EndAbsoluteValue is p -good, and pi 1 left-parenthesis StartAbsoluteValue script upper L EndAbsoluteValue Subscript p Superscript logical-and Baseline right-parenthesis approximately-equals pi is a quotient group of upper S . (Alternatively, this follows directly from a “mod p plus construction” on StartAbsoluteValue script upper L EndAbsoluteValue : there are a space upper Y and a mod p homology equivalence StartAbsoluteValue script upper L EndAbsoluteValue right-arrow Overscript Endscripts upper Y such that pi 1 left-parenthesis upper Y right-parenthesis approximately-equals pi 1 left-parenthesis StartAbsoluteValue script upper L EndAbsoluteValue right-parenthesis slash upper K equals pi , and StartAbsoluteValue script upper L EndAbsoluteValue is p -good since upper Y is.)

2. Homotopy decompositions of classifying spaces

We now consider some homotopy decompositions of the classifying space StartAbsoluteValue script upper L EndAbsoluteValue Subscript p Superscript logical-and of a p -local finite group left-parenthesis upper S comma script upper F comma script upper L right-parenthesis . The first, and most important, is taken over the orbit category of script upper F .

Definition 2.1

The orbit category of a fusion system script upper F over a p -group upper S is the category script upper O left-parenthesis script upper F right-parenthesis whose objects are the subgroups of upper S , and whose morphisms are defined by

upper M o r Subscript script upper O left-parenthesis script upper F right-parenthesis Baseline left-parenthesis upper P comma upper Q right-parenthesis equals upper R e p Subscript script upper F Baseline left-parenthesis upper P comma upper Q right-parenthesis equals Overscript def Endscripts upper I n n left-parenthesis upper Q right-parenthesis minus upper H o m Subscript script upper F Baseline left-parenthesis upper P comma upper Q right-parenthesis period

We let script upper O Superscript c Baseline left-parenthesis script upper F right-parenthesis denote the full subcategory of script upper O left-parenthesis script upper F right-parenthesis whose objects are the script upper F -centric subgroups of upper S . If script upper L is a centric linking system associated to script upper F , then pi overTilde denotes the composite functor

pi overTilde colon script upper L right-arrow Overscript pi Endscripts script upper F Superscript c Baseline two headed right-arrow Overscript Endscripts script upper O Superscript c Baseline left-parenthesis script upper F right-parenthesis period

More generally, if script upper F 0 subset-of-or-equal-to script upper F is any full subcategory, then script upper O left-parenthesis script upper F 0 right-parenthesis denotes the full subcategory of script upper O left-parenthesis script upper F right-parenthesis whose objects are the objects of script upper F 0 . Thus, script upper O Superscript c Baseline left-parenthesis script upper F right-parenthesis equals script upper O left-parenthesis script upper F Superscript c Baseline right-parenthesis .

Note the difference between the orbit category of a fusion system and the orbit category of a group. If upper G is a group and upper S element-of upper S y l Subscript p Baseline left-parenthesis upper G right-parenthesis , then script upper O Subscript upper S Baseline left-parenthesis upper G right-parenthesis is the category whose objects are the orbits upper G slash upper P for all upper P less-than-or-equal-to upper S , and where upper M o r Subscript script upper O Sub Subscript upper S Subscript left-parenthesis upper G right-parenthesis Baseline left-parenthesis upper G slash upper P comma upper G slash upper Q right-parenthesis is the set of all upper G -maps between the orbits. If script upper F equals script upper F Subscript upper S Baseline left-parenthesis upper G right-parenthesis is the fusion system of upper G , then morphisms in the orbit categories of upper G and script upper F can be expressed in terms of the set upper N Subscript upper G Baseline left-parenthesis upper P comma upper Q right-parenthesis of elements which conjugate upper P into upper Q :

upper M o r Subscript script upper O Sub Subscript upper S Subscript left-parenthesis upper G right-parenthesis Baseline left-parenthesis upper G slash upper P comma upper G slash upper Q right-parenthesis approximately-equals upper Q minus upper N Subscript upper G Baseline left-parenthesis upper P comma upper Q right-parenthesis

while

upper M o r Subscript script upper O left-parenthesis script upper F right-parenthesis Baseline left-parenthesis upper P comma upper Q right-parenthesis approximately-equals upper Q minus upper N Subscript upper G Baseline left-parenthesis upper P comma upper Q right-parenthesis slash upper C Subscript upper G Baseline left-parenthesis upper P right-parenthesis period

If upper P is p -centric in upper G , then these sets differ only by the action of the group upper C prime Subscript upper G Baseline left-parenthesis upper P right-parenthesis of order prime to p .

We next look at the homotopy type of the nerve of a centric linking system. Here, denotes the category of spaces.

Proposition 2.2

Fix a saturated fusion system script upper F and an associated centric linking system script upper L , and let pi overTilde colon script upper L right-arrow Overscript Endscripts script upper O Superscript c Baseline left-parenthesis script upper F right-parenthesis be the projection functor. Let

upper B overTilde colon script upper O Superscript c Baseline left-parenthesis script upper F right-parenthesis right-arrow Overscript Endscripts

be the left homotopy Kan extension over pi overTilde of the constant functor script upper L ModifyingAbove right-arrow With asterisk . Then upper B overTilde is a homotopy lifting of the homotopy functor upper P right-arrow from bar upper B upper P , and

StartLayout 1st Row with Label left-parenthesis 1 right-parenthesis EndLabel StartAbsoluteValue script upper L EndAbsoluteValue asymptotically-equals ModifyingBelow normal h times normal o times normal c times normal o times normal l times normal i times normal m With long right-arrow Underscript script upper O Superscript c Baseline left-parenthesis script upper F right-parenthesis Endscripts left-parenthesis upper B overTilde right-parenthesis period EndLayout

More generally, if script upper L 0 subset-of-or-equal-to script upper L is any full subcategory, and script upper F 0 subset-of-or-equal-to script upper F Superscript c is the full subcategory with upper O b left-parenthesis script upper F 0 right-parenthesis equals upper O b left-parenthesis script upper L 0 right-parenthesis , then

StartLayout 1st Row with Label left-parenthesis 2 right-parenthesis EndLabel StartAbsoluteValue script upper L 0 EndAbsoluteValue asymptotically-equals ModifyingBelow normal h times normal o times normal c times normal o times normal l times normal i times normal m With long right-arrow Underscript script upper O left-parenthesis script upper F 0 right-parenthesis Endscripts left-parenthesis upper B overTilde right-parenthesis period EndLayout

Proof.

Recall that we write upper R e p Subscript script upper F Baseline left-parenthesis upper P comma upper Q right-parenthesis to denote morphisms in script upper O Superscript c Baseline left-parenthesis script upper F right-parenthesis . By definition, for each script upper F -centric subgroup upper P less-than-or-equal-to upper S , ModifyingAbove upper B With tilde left-parenthesis upper P right-parenthesis is the nerve (homotopy colimit of the point functor) of the overcategory pi overTilde down-arrow upper P , whose objects are pairs left-parenthesis upper Q comma alpha right-parenthesis for alpha element-of upper R e p Subscript script upper F Baseline left-parenthesis upper Q comma upper P right-parenthesis , and where

upper M o r Subscript pi overTilde down-arrow upper P Baseline left-parenthesis left-parenthesis upper Q comma alpha right-parenthesis comma left-parenthesis upper R comma beta right-parenthesis right-parenthesis equals StartSet phi element-of upper M o r Subscript script upper L Baseline left-parenthesis upper Q comma upper R right-parenthesis vertical-bar alpha equals beta ring ModifyingAbove pi With tilde Subscript upper Q comma upper R Baseline left-parenthesis phi right-parenthesis EndSet period

Since StartAbsoluteValue script upper L EndAbsoluteValue approximately-equals ModifyingBelow normal h times normal o times normal c times normal o times normal l times normal i times normal m With long right-arrow Subscript script upper L Baseline left-parenthesis asterisk right-parenthesis , (1) holds by ReferenceHV, Theorem 5.5 (and the basic idea is due to Segal ReferenceSe, Proposition B.1). Similarly, if upper B overTilde Subscript 0 denotes the left homotopy Kan extension over script upper L 0 right-arrow Overscript pi overTilde Subscript 0 Baseline Endscripts script upper O left-parenthesis script upper F 0 right-parenthesis of the constant functor script upper L 0 ModifyingAbove right-arrow With asterisk , then

StartLayout 1st Row with Label left-parenthesis 2 prime right-parenthesis EndLabel StartAbsoluteValue script upper L 0 EndAbsoluteValue asymptotically-equals ModifyingBelow normal h times normal o times normal c times normal o times normal l times normal i times normal m With long right-arrow Underscript script upper O left-parenthesis script upper F 0 right-parenthesis Endscripts left-parenthesis upper B overTilde Subscript 0 Baseline right-parenthesis period EndLayout

It remains only to show that upper B overTilde is a lifting of the homotopy functor upper P right-arrow from bar upper B upper P , and that the inclusion ModifyingAbove upper B With tilde Subscript 0 Baseline left-parenthesis upper P right-parenthesis right-arrow with hook ModifyingAbove upper B With tilde left-parenthesis upper P right-parenthesis is a homotopy equivalence when upper P element-of upper O b left-parenthesis script upper F 0 right-parenthesis .

Let script upper B prime left-parenthesis upper P right-parenthesis subset-of-or-equal-to pi overTilde down-arrow upper P be the subcategory with one object left-parenthesis upper P comma upper I d right-parenthesis and with morphisms StartSet ModifyingAbove g With caret vertical-bar g element-of upper P EndSet . In particular, StartAbsoluteValue script upper B prime left-parenthesis upper P right-parenthesis EndAbsoluteValue asymptotically-equals upper B upper P . We claim that