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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeOct 18th 2011
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexLet P be the strict omega-category generated as follows: Let A_0=∂O^1, and let E_n=Hom(∂O^n,A_{n-1}), where A_n is obtained by the cocartesian square: E_n × ∂O^n -> A_{n-1} |                        | |                        | v                        v E_n × O^n ----> A_n This gives an inductive system A_0 -> A_1 -> ... -> A_n ->... And we call the inductive limit of this system P. Then P gives a factorization of ∂O^1->O^0 into a natural cofibration followed by a natural trivial fibration. Recall that the Crans-Gray tensor product is a biclosed monoidal structure on str-ω-cat induced by the Day convolution product on the category of cubical sets extending the canonical product of cubes $\Box_i \times \Box_j=\Box_{i+j}$ inherited from the category of sets. Recall that in a monoidal category $(V,\otimes, I, \ell, r)$ (here, $\ell:I\otimes X \cong X$ and $r:X\otimes I \cong X$ are the natural isomorphisms for the unity object), an object X is called _quasi-central_ in V if there exists a natural isomorphism $\alpha: X\otimes - \to - \otimes X$ such that for any objects A and B in V, the following coherence conditions are satisfied: $\alpha_{A\otimes B}=(1_{A} \otimes \alpha_B)(\alpha_A \otimes 1_B$ and $\alpha_{I}=\ell^{-1}_X r_X$ Then my question: Is the object P defined as above quasicentral in the Crans-Gray monoidal category of strict ω-categories? If not, can it at least be shown that it is quasicentral in the monoidal subcategory of cofibrant objects? The intuition here is that it seems like for any fixed arbitrary strict $\omega$-category $C$ and any cofibrant object $A$, you can manufacture a natural bijection $$Hom(P\otimes A, C)\cong Hom(P,RHom(A,C))\cong Hom(P,LHom(A,C)) \cong Hom(A\otimes P,C)$$ since a P-shaped left-lax transfor seems like it can be reversed to a P-shaped right-lax transfor by composing with a fixed automorphism of P (since P classifies equivalences).

   Let P be the strict omega-category generated as follows:

   Let A_0=∂O^1, and let E_n=Hom(∂O^n,A_{n-1}), where A_n is obtained by the cocartesian square:

   E_n × ∂O^n -> A_{n-1}
   |                        |
   |                        |
   v                        v
   E_n × O^n —-> A_n

   This gives an inductive system A_0 -> A_1 -> … -> A_n ->…

   And we call the inductive limit of this system P.

   Then P gives a factorization of ∂O^1->O^0 into a natural cofibration followed by a natural trivial fibration.

   Recall that the Crans-Gray tensor product is a biclosed monoidal structure on str-ω-cat induced by the Day convolution product on the category of cubical sets extending the canonical product of cubes $\Box_i \times \Box_j=\Box_{i+j}$ inherited from the category of sets.

   Recall that in a monoidal category $(V,\otimes, I, \ell, r)$ (here, $\ell:I\otimes X \cong X$ and $r:X\otimes I \cong X$ are the natural isomorphisms for the unity object), an object X is called quasi-central in V if there exists a natural isomorphism $\alpha: X\otimes - \to - \otimes X$ such that for any objects A and B in V, the following coherence conditions are satisfied:

   $\alpha_{A\otimes B}=(1_{A} \otimes \alpha_B)(\alpha_A \otimes 1_B$

   and

   $\alpha_{I}=\ell^{-1}_X r_X$

   Then my question: Is the object P defined as above quasicentral in the Crans-Gray monoidal category of strict ω-categories?

   If not, can it at least be shown that it is quasicentral in the monoidal subcategory of cofibrant objects?

   The intuition here is that it seems like for any fixed arbitrary strict $\omega$-category $C$ and any cofibrant object $A$, you can manufacture a natural bijection

   $$Hom(P\otimes A, C)\cong Hom(P,RHom(A,C))\cong Hom(P,LHom(A,C)) \cong Hom(A\otimes P,C)$$

   since a P-shaped left-lax transfor seems like it can be reversed to a P-shaped right-lax transfor by composing with a fixed automorphism of P (since P classifies equivalences).

2. • CommentRowNumber2.
   • CommentAuthorHarry Gindi
   • CommentTimeOct 19th 2011
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexUhh... Can you do something silly like this: A map $P\otimes D_n\to X$ corresponds by adjunction to a map $D_n\to LHom(P,X)$, but there exists a natural isomorphism $P\to P^{op}$, so we can compose $D_n\to LHom(P^{op},X)$, but then by duality, this corresponds to a map $D_n\to LHom(P,X^{op}).$ Passing back along the adjoint, we obtain a map $P\otimes D_n \to X^{op}$, which gives, by duality, a map $(P\otimes D_n)^{op}\to X$, but due to the interaction of duality with the tensor product, this gives us a map $D_n^{op}\otimes P^{op} \to X$. Then precomposing with the isomorphism $D_n\otimes P \to D_n^{op} \otimes P^{op}$, this gives a map $D_n\otimes P \to X$. Obviously, we can do this in reverse, so we see that we have commutativity of $P$ with the n-globes and their boundaries... if I didn't screw up!

   Uhh…

   Can you do something silly like this:

   A map $P\otimes D_n\to X$ corresponds by adjunction to a map $D_n\to LHom(P,X)$, but there exists a natural isomorphism $P\to P^{op}$, so we can compose $D_n\to LHom(P^{op},X)$, but then by duality, this corresponds to a map $D_n\to LHom(P,X^{op}).$ Passing back along the adjoint, we obtain a map $P\otimes D_n \to X^{op}$, which gives, by duality, a map $(P\otimes D_n)^{op}\to X$, but due to the interaction of duality with the tensor product, this gives us a map $D_n^{op}\otimes P^{op} \to X$. Then precomposing with the isomorphism $D_n\otimes P \to D_n^{op} \otimes P^{op}$, this gives a map $D_n\otimes P \to X$.

   Obviously, we can do this in reverse, so we see that we have commutativity of $P$ with the n-globes and their boundaries… if I didn’t screw up!