Timeline for Calculating limits progressively
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Nov 28, 2017 at 3:16 | vote | accept | CommunityBot | ||
| Nov 27, 2017 at 7:56 | answer | added | Mike Shulman | timeline score: 5 | |
| Nov 27, 2017 at 1:52 | answer | added | Dylan Wilson | timeline score: 14 | |
| Nov 27, 2017 at 1:29 | comment | added | user13113 | @Dylan: I'll have to reflect upon it, but yes I think that answers my question! | |
| Nov 27, 2017 at 1:15 | history | edited | user13113 | CC BY-SA 3.0 |
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| Nov 27, 2017 at 1:14 | comment | added | Dylan Wilson | (that also works for homotopy limits). | |
| Nov 27, 2017 at 1:14 | history | edited | user13113 | CC BY-SA 3.0 |
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| Nov 27, 2017 at 1:13 | comment | added | Dylan Wilson | explicitly: suppose you have a map $K \to L$, then right Kan extending will have value at $x \in L$ given by the limit of the values in $K_{x/}$. This will be the same as one of the values you already have in $K$ if $K_{x/}$ has an initial object. So the `local' phenomena you want occurs if all the categories $K_{x/}$ have an initial object except possibly one of them. | |
| Nov 27, 2017 at 1:13 | history | edited | user13113 | CC BY-SA 3.0 |
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| Nov 27, 2017 at 1:12 | comment | added | Dylan Wilson | ah- so you're right Kan extending but in a different way, by 'collapsing' pieces of the diagram as you specified (though I think there's a typo in your target diagram there). Okay, well you can figure out exactly when you get the 'local' change you want from the formula for a right Kan extension as a limit of some undercategory. | |
| Nov 27, 2017 at 1:07 | comment | added | Dylan Wilson | subdiagrams (which doesn't affect computation of the next Kan extension, etc.) | |
| Nov 27, 2017 at 1:07 | comment | added | user13113 | @Dylan: I mean to consider the functor of diagrams from $\bullet \to \bullet \leftarrow \bullet \to \bullet \leftarrow \bullet$ to $\bullet \leftarrow \bullet \to \bullet$ that collapses the middle three vertices to a single vertex. Then given a diagram indexed by the first category, to consider the analog of a "limit cone" from a diagram indexed by the second category. | |
| Nov 27, 2017 at 1:06 | comment | added | Dylan Wilson | @Hurkyl: I don't know what it means to `collapse $B \leftarrow C \rightarrow D$ to a point, since that's not what you did: you instead 'collapsed' the top and bottom spans, i.e. computed their pullback. Like, what do you mean in general by taking some sub-diagram and 'collapsing' it? It seems more like what's happening is you're choosing some way of taking a diagram $K$ and subdividing the cone $K^{\triangleleft}$, then computing the limit (evaluation of right Kan extension at cone point) by iteratively right Kan extending along your subdivision and simultaneously restricting to initial | |
| Nov 27, 2017 at 0:59 | comment | added | Vladimir Sotirov | It seems like you're asking for some kind of analysis of when a (full) inclusion of a subdiagram $K\hookrightarrow D_1$ equalizes any functor $D_1\xrightarrow{J} C$ and its Kan extension along $D_1\xrightarrow{F}D_2$ (since a Kan extension along $D_1\to\mathbf 1$ is a (co)limit). | |
| Nov 27, 2017 at 0:52 | comment | added | user13113 | @Dylan: The point about locality is that manipulations on the $C \to D \leftarrow E$ part can be done in isolation (or, at least, the specific manipulation of substituting a cospan with its limit, but I imagine much more is possible) -- the modification to that subdiagram has no effect on the rest of the diagram, and conversely the rest of the diagram doesn't effect how the modification is performed. Operating on the $B \leftarrow C \to D$ subdiagram, however, doesn't have that property: collapsing it down to a point affected vertices that weren't part of the subdiagram. | |
| Nov 27, 2017 at 0:37 | comment | added | Dylan Wilson | What precisely is the complaint about the second way of computing the limit? I don't understand what you mean by 'local'. | |
| Nov 27, 2017 at 0:14 | history | asked | user13113 | CC BY-SA 3.0 |