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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeMay 2nd 2012
   • (edited May 2nd 2012)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexBasic definitions are covered in this [paper](http://www.intlpress.com/HHA/v6/n1/a12/v6n1a12.pdf) by Steiner. Recall that by a theorem of Steiner, the category of strict ω-categories has a dense subcategory given by those strict ω-categories associated with augmented directed complexes admitting a strongly loop-free unital basis. He then uses this fact to extend the ordinary tensor product of chain complexes first to a tensor product of augmented directed complexes (which has the property that the tensor product of two ADCs admitting a strongly loop-free unital basis admits a strongly loop-free unital basis) and then to the category of all strict ω-categories. However, the combinatorics of this tensor product are an absolute nightmare from the point of view of thinking about decompositions into objects belonging to Θ. The problem is that the procedure generating the new composites is extremely wild, in the sense that structure cells can be whiskered far away from their location in a generating diagram to form composites that can't be "read off" the generating diagram in any meaningful sense. However, if we look at Verity's construction of the lax Gray tensor product for stratified sets, we see that he makes an interesting choice to describe it as a "quotient" (by specifying thin simplices) of the product of the underlying simplicial sets by those simplices satisfying an "Alexander-whitney" condition involving partition operators, but as it so happens, the underlying simplicial set has a rather well-behaved structure. All of the "freely generated" information in the tensor product is obtained after composing with the reflector of the inclusion of complicial sets into stratified sets. Then an interesting way to deal with the combinatorial horribleness of the lax Gray tensor product might involve giving a new "simpler" tensor product analogous to the product of underlying simplicial sets and then describing the lax Gray tensor product as a "quotient" by the "Alexander-Whitney cells" _taken in the category of presheaves on Θ._ Since the quotient is taken in cellular sets rather than strict ω-categories, it (thankfully) doesn't contain all of the wild composites generated freely from the various whiskerings of structure cells. However, the image of this quotient under the realization functor from cellular sets to strict ω-categories is indeed the ordinary lax Gray tensor product. We have a sort of obvious place to start in specifying such a simpler tensor product: By the Dold-Kan correspondence, we have a simplicial tensor product of connective chain complexes (which extends in an obvious way to $\mathbf{Z}$-augmented connective chain complexes) obtained by the formula $A\otimes_\Delta B = N(\Gamma(A)\otimes \Gamma(B))$, where $N$ and $\Gamma$ are the normalized chain complex functor and its quasi-inverse functor. Then if $A$ and $B$ are augmented directed complexes, the question is how to equip $A\otimes_\Delta B$ with distinguished submonoids to specify its "directed" structure. We know that a directed structure determines a partial order on each chain group by saying $a\leq b$ when $b-a$ belongs to the distinguished submonoid. We would like it to be the case that if $A$ and $B$ admit bases, then $A\otimes_\Delta B$ admits a basis, which is given in each degree by the set of minimal nonzero elements $\geq 0$. Moreover, it would be nice to know if this tensor product is closed on the subcategory of augmented directed complexes admitting strongly loop-free unital bases. However, it's not totally clear how the distinguished submonoids should be defined. If we think of $\Gamma(A)_n$ as the group of chain maps $Hom(\mathbf{Z}[\Delta^n],A),$ we might be able to define our submonoid to be the one comprising those equivalence classes (modulo degeneracies) of tensors of chain maps taking the basis elements of $\mathbf{Z}[\Delta^n]$ to elements of the distinguished submonoids. This tensor product has the advantage (I can compute what it "should" look like in simple cases, even though I have no formula yet) of making it so the structure cells can't be pulled around by whiskering and then composed in nearly arbitrary order (to see an example of where this can happen, look at the lax Gray tensor product of $\Delta^2\otimes \Delta^m$ where $\Delta^k$ denotes the strict $\omega$-category associated with the 1-category $[0 \lt \dots \lt k]$. Then the structure cells can be composed in at least the nth catalan number of different orders once they've been whiskered.). If we compare this with the "simplicial" tensor product, the simplicial tensor product has a uniquely specified way of composing structure cells, namely because the structure cells can't really be composed with one another in any reasonable way (except diagonally or whiskered-diagonally). In particular, the simplicial tensor product can in this case be described using certain "generalized shuffles". So to reiterate, the question is how we should define the distinguished submonoids of the chain groups of the simplicial tensor product of augmented directed complexes.

   Basic definitions are covered in this paper by Steiner.

   Recall that by a theorem of Steiner, the category of strict ω-categories has a dense subcategory given by those strict ω-categories associated with augmented directed complexes admitting a strongly loop-free unital basis. He then uses this fact to extend the ordinary tensor product of chain complexes first to a tensor product of augmented directed complexes (which has the property that the tensor product of two ADCs admitting a strongly loop-free unital basis admits a strongly loop-free unital basis) and then to the category of all strict ω-categories.

   However, the combinatorics of this tensor product are an absolute nightmare from the point of view of thinking about decompositions into objects belonging to Θ. The problem is that the procedure generating the new composites is extremely wild, in the sense that structure cells can be whiskered far away from their location in a generating diagram to form composites that can’t be “read off” the generating diagram in any meaningful sense.

   However, if we look at Verity’s construction of the lax Gray tensor product for stratified sets, we see that he makes an interesting choice to describe it as a “quotient” (by specifying thin simplices) of the product of the underlying simplicial sets by those simplices satisfying an “Alexander-whitney” condition involving partition operators, but as it so happens, the underlying simplicial set has a rather well-behaved structure. All of the “freely generated” information in the tensor product is obtained after composing with the reflector of the inclusion of complicial sets into stratified sets.

   Then an interesting way to deal with the combinatorial horribleness of the lax Gray tensor product might involve giving a new “simpler” tensor product analogous to the product of underlying simplicial sets and then describing the lax Gray tensor product as a “quotient” by the “Alexander-Whitney cells” taken in the category of presheaves on Θ. Since the quotient is taken in cellular sets rather than strict ω-categories, it (thankfully) doesn’t contain all of the wild composites generated freely from the various whiskerings of structure cells. However, the image of this quotient under the realization functor from cellular sets to strict ω-categories is indeed the ordinary lax Gray tensor product.

   We have a sort of obvious place to start in specifying such a simpler tensor product: By the Dold-Kan correspondence, we have a simplicial tensor product of connective chain complexes (which extends in an obvious way to $\mathbf{Z}$-augmented connective chain complexes) obtained by the formula $A\otimes_\Delta B = N(\Gamma(A)\otimes \Gamma(B))$, where $N$ and $\Gamma$ are the normalized chain complex functor and its quasi-inverse functor. Then if $A$ and $B$ are augmented directed complexes, the question is how to equip $A\otimes_\Delta B$ with distinguished submonoids to specify its “directed” structure. We know that a directed structure determines a partial order on each chain group by saying $a\leq b$ when $b-a$ belongs to the distinguished submonoid. We would like it to be the case that if $A$ and $B$ admit bases, then $A\otimes_\Delta B$ admits a basis, which is given in each degree by the set of minimal nonzero elements $\geq 0$. Moreover, it would be nice to know if this tensor product is closed on the subcategory of augmented directed complexes admitting strongly loop-free unital bases.

   However, it’s not totally clear how the distinguished submonoids should be defined. If we think of $\Gamma(A)_n$ as the group of chain maps $Hom(\mathbf{Z}[\Delta^n],A),$ we might be able to define our submonoid to be the one comprising those equivalence classes (modulo degeneracies) of tensors of chain maps taking the basis elements of $\mathbf{Z}[\Delta^n]$ to elements of the distinguished submonoids.

   This tensor product has the advantage (I can compute what it “should” look like in simple cases, even though I have no formula yet) of making it so the structure cells can’t be pulled around by whiskering and then composed in nearly arbitrary order (to see an example of where this can happen, look at the lax Gray tensor product of $\Delta^2\otimes \Delta^m$ where $\Delta^k$ denotes the strict $\omega$-category associated with the 1-category $[0 \lt \dots \lt k]$. Then the structure cells can be composed in at least the nth catalan number of different orders once they’ve been whiskered.). If we compare this with the “simplicial” tensor product, the simplicial tensor product has a uniquely specified way of composing structure cells, namely because the structure cells can’t really be composed with one another in any reasonable way (except diagonally or whiskered-diagonally). In particular, the simplicial tensor product can in this case be described using certain “generalized shuffles”.

   So to reiterate, the question is how we should define the distinguished submonoids of the chain groups of the simplicial tensor product of augmented directed complexes.

2. • CommentRowNumber2.
   • CommentAuthorHarry Gindi
   • CommentTimeMay 2nd 2012
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexTyler Lawson also worked out a kind of structure theory for the underlying chain complexes at this [MO answer](http://mathoverflow.net/questions/94606/explicit-description-of-the-simplicial-tensor-product-of-chain-complexes/94640#94640).

   Tyler Lawson also worked out a kind of structure theory for the underlying chain complexes at this MO answer.

3. • CommentRowNumber3.
   • CommentAuthorHarry Gindi
   • CommentTimeMay 29th 2012
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexI just found an example which proves that this thing fails to be functorial on augmented directed complexes. To see this, consider the coface $\delta^1:\Delta^1 \hookrightarrow \Delta^2$ as a map of ADCs, where these simplices are embedded along the full and faithful 1-dimensional embedding of $\Delta$ into the category of augmented directed complexes (rather than the oriental embedding, which is rather far from full). Then taking the simplicial tensor product of this map with the object $\Delta^1$, the resulting map fails to take positive elements to positive elements.

   I just found an example which proves that this thing fails to be functorial on augmented directed complexes.

   To see this, consider the coface $\delta^1:\Delta^1 \hookrightarrow \Delta^2$ as a map of ADCs, where these simplices are embedded along the full and faithful 1-dimensional embedding of $\Delta$ into the category of augmented directed complexes (rather than the oriental embedding, which is rather far from full). Then taking the simplicial tensor product of this map with the object $\Delta^1$, the resulting map fails to take positive elements to positive elements.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeMay 29th 2012
   • (edited May 29th 2012)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI added the above mentioned Steiner's reference to [[strict omega-groupoid]].

   I added the above mentioned Steiner’s reference to strict omega-groupoid.