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Given a group, there is another way to define its "(co-)homology" using a classifying space. Specifically, one takes the partially ordered set of its proper non-trivial subgroups (if they exist), and defines the (co-)homology of the group to be the singular (co-)homology of the classifying space of the partially ordered set viewed as a small category. There are many different way to apply this method, for example, one can take the $p$-subgroups instead; or one could take the poset of subgroups which containing some particular subgroup. And there has been some known interesting results, see for example

Brown, K. Euler characteristics of groups: the $p$-fractional part. (1975)

Quillen, D. Homotopy Properties of the Poset of Nontrivial $p$-Subgroups of a Group (1978)

Group (co-)homology obtained this way, in general is quite different from the usual ones (for discrete groups or topological groups). The classifying spaces here tend to be contractible frequently. Typically, people use this for finite groups, which sometimes provides links to homological theory of rings (see, e.g. Stanley–Reisner rings).

This method could be easily generalized to well-powered categories (i.e. categories whose objects have small collections of sub-objects). Yet, I always thought this is not the "right" kind of (co-)homology theory.

Is my feeling heuristically correct or wrong, or is it just some random feeling? I would also like to know if there is any interesting/useful application of this approach (other than groups). Thanks!

Addendum Dan and Ralph's comment make comments made me realize that it was my own wishful thinking to name this "group (co-)homology". Quillen certainly defined these for group and used these in his above-cited article, but he never explicitly called them group (co-)homology. Of course, everything is useful in some sense; I guess by "not right" I meant that they can not provide much information about the group itself. But then you'll ask me what I mean by "much information"....

So let me only ask this: What are the other interesting applications of this kind of (co-)homology theory using posets of sub-objects in other areas, excluding applications to general posets and similar applications to groups? (Apologies for this messy post.)

2 added 666 characters in body; edited title

TheWrongkindofSubobject-poset (co-)homology?co-)homology

Given a group, there is another way to define its "(co-)homology" using a classifying space. Specifically, one takes the partially ordered set of its proper non-trivial subgroups (if they exist), and defines the (co-)homology of the group to be the singular (co-)homology of the classifying space of the partially ordered set viewed as a small category. There are many different way to apply this method, for example, one can take the $p$-subgroups instead; or one could take the poset of subgroups which containing some particular subgroup. And there has been some known interesting results, see for example

Brown, K. Euler characteristics of groups: the $p$-fractional part. (1975)

Quillen, D. Homotopy Properties of the Poset of Nontrivial $p$-Subgroups of a Group (1978)

Group (co-)homology obtained this way, in general is quite different from the usual ones (for discrete groups or topological groups). The classifying spaces here tend to be contractible frequently. Typically, people use this for finite groups, which sometimes provides links to homological theory of rings (see, e.g. Stanley–Reisner rings).

This method could be easily generalized to well-powered categories (i.e. categories whose objects have small collections of sub-objects). Yet, I always thought this is not the "right" kind of (co-)homology theory.

Is my feeling heuristically correct or wrong, or is it just some random feeling? I would also like to know if there is any interesting/useful application of this approach (other than groups). Thanks!

Addendum Dan and Ralph's comment make me realize that it was my own wishful thinking to name this "group (co-)homology". Quillen certainly defined these for group and used these in his above-cited article, but he never explicitly called them group (co-)homology. Of course, everything is useful in some sense; I guess by "not right" I meant that they can not provide much information about the group itself. But then you'll ask me what I mean by "much information"....

So let me only ask this: What are the other interesting applications of this kind of (co-)homology theory using posets of sub-objects in other areas? (Apologies for this messy post.)

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The Wrong kind of (co-)homology?

Given a group, there is another way to define its "(co-)homology" using a classifying space. Specifically, one takes the partially ordered set of its proper non-trivial subgroups (if they exist), and defines the (co-)homology of the group to be the singular (co-)homology of the classifying space of the partially ordered set viewed as a small category. There are many different way to apply this method, for example, one can take the $p$-subgroups instead; or one could take the poset of subgroups which containing some particular subgroup. And there has been some known interesting results, see for example

Brown, K. Euler characteristics of groups: the $p$-fractional part. (1975)

Quillen, D. Homotopy Properties of the Poset of Nontrivial $p$-Subgroups of a Group (1978)

Group (co-)homology obtained this way, in general is quite different from the usual ones (for discrete groups or topological groups). The classifying spaces here tend to be contractible frequently. Typically, people use this for finite groups, which sometimes provides links to homological theory of rings (see, e.g. Stanley–Reisner rings).

This method could be easily generalized to well-powered categories (i.e. categories whose objects have small collections of sub-objects). Yet, I always thought this is not the "right" kind of (co-)homology theory.

Is my feeling heuristically correct or wrong, or is it just some random feeling? I would also like to know if there is any interesting/useful application of this approach (other than groups). Thanks!