Timeline for When are "diagrams of cofibrations" projectively cofibrant?
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10 events
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| Jul 19, 2011 at 2:40 | comment | added | Tom Goodwillie | This condition on a poset $P$ (every "diagram of cofibrations" is a cofibrant diagram) implies that for any element of $P$ the set of all lower bounds is totally ordered. | |
| Jul 18, 2011 at 22:40 | comment | added | D.-C. Cisinski | This will fail when $P$ is the product of two copies of $\{0\to 1\}$, so that to be a "finite Thomason-contractible poset" won't be sufficient. I think the case of spans is quite representative: you will need $P$ to be a directed category which is free (i.e. isomorphic to the free category on a graph). Note that, for a directed category, to be free is a very big constraint; this implies, for instance, that for any model category $M$, the functor $Ho(M^P)\to Ho(M)^P$ is full and essentially surjective. | |
| Jul 18, 2011 at 21:06 | history | edited | Harry Gindi | CC BY-SA 3.0 |
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| Jul 18, 2011 at 20:57 | comment | added | Harry Gindi | @Karol: Sorry, one more question, what if P is a finite Thomason-contractible (asphérique) poset? | |
| Jul 18, 2011 at 18:19 | comment | added | Karol Szumiło | Does "Thomason-contractible" mean "with a contractible nerve"? If so this is not sufficient and my previous comment explains why. There are contractible direct categories with no initial object, for example the direct part of $\Delta$ (i.e. the category of finite nonempty totally ordered sets and injective order-preserving maps). There are also examples among posets. | |
| Jul 18, 2011 at 17:32 | comment | added | Harry Gindi | What if P is Thomason-contractible? Also, even if the object is not projectively cofibrant, can we at least compute hocolims with it under certain circumstances? | |
| Jul 18, 2011 at 14:30 | comment | added | Karol Szumiło | If $P$ is a direct category, then projectively cofibrant diagrams coincide with Reedy cofibrant diagrams. It is easily seen that only very uncomplicated direct categories will satisfy your property. For example a constant diagram is always a "diagram of cofibrations", but constant diagrams are Reedy cofibrant only when $P$ is a coproduct of categories with initial objects. The situation is probably only worse for non-direct categories. | |
| Jul 18, 2011 at 14:03 | comment | added | Tom Goodwillie | It's very rarely true. Note that if this is true for $P$ then it is also true for any subposet $Q\subset P$ such that no element of $Q$ is less than any element of $P-Q$. It also appears to me that, in order for it to hold for $P$, the realization of the nerve of $P$ must be homotopy equivalent to a discrete space. | |
| Jul 18, 2011 at 12:01 | history | edited | Harry Gindi | CC BY-SA 3.0 |
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| Jul 18, 2011 at 11:42 | history | asked | Harry Gindi | CC BY-SA 3.0 |