## Looping and delooping, suspension, the stabilization theorem

### Sjoerd Crans – 21 August 1996

1. basics of \omega-teisi, don't have precise definition. In particular, dimension raising of composition.
2. Delooping: define k-monoidal \omega-tas as \omega-tas with one (k-1)-arrow, and k-monoidal nD tas as (n+k)D tas with one (k-1)-arrow. View as nD tas with extra structure. Inclusion is \Sigma^k.
3. Looping: from pointed \omega-tas C, make C(id^k-1_c, id^k-1_c), which is k-monoidal, call this \Omega^k. \Sigma^k -| \Omega^k.
4. Suspension: underlying functor forgetting k-monoidal structure. Left adjoint to this, giving free k-monoidal structure, is suspension.
5. Baez-Dolan's stabilization hypothesis (rephrased for teisi): S: (k+2)-monoidal kD teisi --> (k+3)-monoidal kD teisi is iso. proof: follows immediately by checking the dimension raising of (k+3)-monoidal structure. So stabilization *theorem* is easy consequence of basics of \omega-teisi.
6. This will be sections 2.6.1 - 2.6.3 of the forthcoming 80+ page paper "Central observations on \omega-teisi".

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