Timeline for Equivalence of categories of abelian presheaves reflects isomorphisms of rigid categories?
Current License: CC BY-SA 3.0
18 events
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| May 21, 2012 at 15:39 | comment | added | Benjamin Steinberg | That does sound interesting. I ran into some papers mentioning $\Theta$ once or twice. I will look into it. | |
| May 21, 2012 at 15:32 | comment | added | Harry Gindi | So if the embedding of $\Delta$ into chain complexes can be extended to $\Theta$ and this extends to a Dold-Kan equivalence between abelian presheaves on $\Theta$ and chain complexes, this gives us a way to "freely assign orientations" to arbitrary commuting diagrams of shapes belonging to $\Theta$ in a strict $\omega$-category. However, by a theorem of C. Berger, $\Theta$ is dense in strict omega-cats, so every $\omega$-cat $X$ is canonically the colimit of $\Theta\downarrow X$, which would extend the orientation procedure to al $\omega$-cats. Then morphisms out of that will be the lax ones. | |
| May 21, 2012 at 15:24 | comment | added | Harry Gindi | Well, strict $\omega$-categories are much easier to handle than bicategories, since all coherence diagrams are trivial (no pentagons or anything nasty). The category $\Theta$ plays the role that $\Delta$ plays for $1$-categories, that is to say free objects on appropriate pushouts of source-target alternating diagrams of simple (globular) cells. Steiner's theory then shows that these things are just finite chain complexes of free abelian groups equipped with bases. Moreover, "orientalization" is just the (normalized) embedding of $\Delta$ into chain complexes (with the obvious basis). | |
| May 21, 2012 at 15:21 | history | edited | Benjamin Steinberg | CC BY-SA 3.0 |
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| May 21, 2012 at 15:06 | comment | added | Benjamin Steinberg | I am glad this helps. I must confess my category theory knowledge ends with toposes and 2/bi-categories. I still haven't made it into the world of $\omega$-categories. One of these days... In any event, my rule number 1 for category theory counter examples is look at finite monoids with trivial groups of units. | |
| May 21, 2012 at 14:54 | comment | added | Harry Gindi | Great! This helps a great deal. Thanks a lot. Ultimately, I'm hoping that presheaves of abelian groups on Joyal's category $\Theta$ admit a Dold-Kan theorem (i.e. an equivalence of categories with connective chain complexes of abelian groups) that extends the Dold-Kan theorem on $\Delta$ (via the canonical embedding of $\Delta$ into $\Theta$). Using some machinery of Steiner, such an equivalence should give an extension of Street's "orientalization" functor to all cellular sets (and therefore to all strict $\omega$-categories). This would ultimately give a defn of a higher lax functor. | |
| May 21, 2012 at 14:52 | history | edited | Benjamin Steinberg | CC BY-SA 3.0 |
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| May 21, 2012 at 14:45 | comment | added | Benjamin Steinberg | The Cauchy completions of my monoids are finite with $2^n$ objects. | |
| May 21, 2012 at 14:45 | history | edited | Benjamin Steinberg | CC BY-SA 3.0 |
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| May 21, 2012 at 14:45 | comment | added | Harry Gindi | What if your categories are locally finite but have multiple objects? | |
| May 21, 2012 at 14:43 | comment | added | Benjamin Steinberg | Of course the Cauchy completions are rigid. | |
| May 21, 2012 at 14:43 | comment | added | Benjamin Steinberg | The Cauchy completions of the monoids I gave above are not isomorphic. Finite monoids are determined up to isomorphism by Morita equivalence of presheaf toposes. In general, two monoids $M$ and $N$ have equivalent classifying toposes iff there is an idempotent $e\in M$ such that $MeM=M$ and $eMe\cong N$. For finite monoids, this $e$ will have to be the identity. | |
| May 21, 2012 at 14:40 | comment | added | Harry Gindi | One other question, I guess. What if you require that the rigid categories are cauchy complete (i.e. all idempotents split)? | |
| May 21, 2012 at 14:38 | history | edited | Benjamin Steinberg | CC BY-SA 3.0 |
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| May 21, 2012 at 14:35 | vote | accept | Harry Gindi | ||
| May 21, 2012 at 14:34 | history | edited | Harry Gindi | CC BY-SA 3.0 |
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| May 21, 2012 at 14:34 | comment | added | Harry Gindi | Great! I was hoping it was false, since it it were true, I'd have zero chance of extending the Dold-Kan theorem to a new context. | |
| May 21, 2012 at 14:33 | history | answered | Benjamin Steinberg | CC BY-SA 3.0 |