Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Suppose we have a group $G$ then one can construct $BG$ and one of the essential part of the construction is the co-unit map. Now suppose we have a ring spectrum $R$, then having a co-unit splits the ring spectrum as $R = S \vee R'$, which is never an interesting case.

So my question is what is the best one can do? I know its a vague question. But is there an alternative thing that one can do, which closely resembles bar construction that one does for a group?

share|improve this question
1  
I don't see an analogy in your proposal since you don't take any bar construction on $R$. The 'bar construction' on $R$ would be the topological Hochschild homology spectrum $THH(R)$, which has a 'counit' $TTH(R)\rightarrow R$. –  Fernando Muro Feb 19 '13 at 14:48
1  
Prasit: Such a thing does work in a good category of spectra, and in fact it is the case that you can split of the unit in a similar way to get a reduced bar construction. I will look around for the right diagram, I can't recall it off the top of my head. –  Sean Tilson Feb 19 '13 at 15:06
2  
To add to Sean's comments, the two-sided bar construction B(M,A,N) can be formed in myriads of contexts in which one has a monoid A in a monoidal category with right action on M and left action on N. Topological groups G and ring spectra R are just two examples. The fact that you can't generally take M=S in general in the ring spectra example is not so important. The precise relationship with THH is on pp 173-174 of EKMM. –  Peter May Feb 19 '13 at 15:25
    
These are all really helpful comments! I will look into the references. –  Prasit Feb 19 '13 at 15:38
1  
Edit: Fernando: There is a bar construction for spectra and it does not necessarily give you $THH$. The naive two sided bar construction $B(M,A,N)$ frequently models $M \wedge_A N$, there are some cellularity (is that a word) hypotheses. The cyclic bar construction is an example of the two sided bar construction. This can all be found in EKMM. –  Sean Tilson Feb 19 '13 at 17:27

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.