Robert Bruner: Okay, this is part two of the DEMO we'll talk today in this part about the.
Robert Bruner: computation of chain maps.
Robert Bruner: between my bills okay so remember I J and E I includes the six suspension of the module and one into em those top T cells, you see there and the co fiber quotient of that is m for the bottom two cells here in em.
Robert Bruner: And we have I nj here, and then we have one more map, which is the extension co cycle, which gives the boundary map from him went around to him for and that's going to compute.
Robert Bruner: The boundary map from suspension six of co fiber to around business suspension of co fiber new So if you look down here, you see that the geometric degree here s comma T minus six is.
Robert Bruner: T minus s minus six and then we go to T minus X minus one and so that's a degree five map here.
Robert Bruner: So.
Robert Bruner: The first two are easy the map definition files, for I J and J are simple and then humble he takes a tiny bit more work.
Robert Bruner: Once we've created these files will install them using command new map will use do lists to compute the lists of the chain maps.
Robert Bruner: And then we use collect collect all that data into form that we can read easily OK, so the code cycles defining I nj ever easy first Jay.
Robert Bruner: Jay maps mdm for and the co cycle, corresponding to that is just the projection from the three zero part of the resolution of em.
Robert Bruner: At composed with the map from mdm for now module definition files have the format.
Robert Bruner: home logical logical degree internal degree the main co domain name of the map and the number of generators that are mapped now remember emma's sickly so Caesar as a single copy of the steamer now there's only one generator CESAR to map So this has got to be one here.
Robert Bruner: Now the remainder of the file then tells where the the generators, then, are which generators are mapped I should say, and where they go, so the only generator that is mapped is the zeros generator the beginning, the first generator and the list, and it goes to.
Robert Bruner: The bottom class in em for now, remember, if we go back to him for up here remember generate the bottom class in him for here is generator number zero okay so that's where it goes, and so this says, then that generator number zero C zero goes to.
Robert Bruner: A psalm of one term which is generator zero and acted upon acted upon by the only element of the scene or non zero element of the scene right algebra and degrees zero, namely the identity map the way to read this format is explained in that archive paper.
Robert Bruner: That john wrongness and I posted when we put up the database on the NI RD data our database archive okay so that's the module definition file for.
Robert Bruner: J now, for I, what are we doing that includes well actually i've done i've taken the edge joint here the map, I actually went from the suspension, six of them one to em.
Robert Bruner: I don't want to create a new module and more suspension, six of them one, so instead I will go from one day suspension minus six event that works, just as well.
Robert Bruner: Again, the Eco cycle defining this map is simply the composition of the first step of the resolution, the APP and morph ISM from.
Robert Bruner: The zero m word he stars, the resolution of em one I mean, these are the in one or the stars, the resolution of em one with the homomorphic.
Robert Bruner: OK, so the coma logical degree is zero we're starting from the zero here the internal degrees minus sorry, yes, the internal degree is minus six that's the suspension minus six of em.
Robert Bruner: Okay, so zero minus six is to here says, and then m one m says we're going from coal much degree zero of em one two suspension minus six of them.
Robert Bruner: And then I is the name we're giving this map and again, only one generation map because it's cyclical that generator number zero goes to one term this time it's generator noon or two in the module m okay so let's go back briefly and see that.
Robert Bruner: here's generator number two in the module m this for so complex or for dimension for module is M and generator number two's where the bottom sell them one gets mapped to the magpie.
Robert Bruner: So that's what that definition files going to look like now, the next one is the Co cycle, which defines the extension here.
Robert Bruner: Okay sorry i've reversed your arrow direction of the maps but and forth the right here in the middle suspension, six of them one left, this is the extension we've got the extension co cycle is simply what you get by lifting.
Robert Bruner: The map from him for to him for.
Robert Bruner: Over the resolution out to stage one here okay so what's going to happen here easier oh again em for sickly easier, I was going to be a single copy of the screen right algebra.
Robert Bruner: that's going to have to map to the bottom class in him, because the bottom class in maps the bottom class name for that's where the generator and easier oh goes in for.
Robert Bruner: Okay, so then think about he won he won mapping the E zero here is going to have to kill off square one, and square to because those are zero in em for.
Robert Bruner: It won't kill us square for because for 49 04 and, in fact, at this point, maybe want to go look at the definition at the differentials for em for here so let's do that.
Robert Bruner: Now.
Robert Bruner: As I look directly at the differentials in home logical degree for that the machine created their little ugly to read.
Robert Bruner: here.
Robert Bruner: For that's way too far out I want one okay.
Robert Bruner: There we go that's more like diff that one.
Robert Bruner: Alright, so these are still a little bit ugly to read so there's a nice little program called convert.
Robert Bruner: It it'll take these differential files and convert them to one that's written in terms of the milner basis it's a little easier to read so.
Robert Bruner: I like to call this H diff dot one for human readable diff dot one the format of these files, and this is explained in the archive paper that I mentioned is 211 there's a header at the top, that has two entries in it.
Robert Bruner: That says there are seven generators and it's complete through Internet agree 40, then it has blocks.
Robert Bruner: which have one header for each block there's a one here there's two down then here there's a six down here, each of these blocks has.
Robert Bruner: A header consisting of one term, and then the third one says there's one element and each block okay so that's.
Robert Bruner: This element see there aren't any elements, more than one term, but if we look up here, you see here's a block that has one element, but it's got many terms in it, it has seven terms, there okay so.
Robert Bruner: that's what we're going to do, and then the last term I here says write it and internal that is milner basis terms now if I look at each one.
Robert Bruner: I can see what this map from each one to e zero down here, it looks like the map the differential from end zero is display here.
Robert Bruner: Getting it highlighted doing beamer is almost impossible, so we won't bother with that.
Robert Bruner: generator number zero is in degree one it goes to square one okay well that's we mentioned already, we had to kill square one generator number one is in degree to it kills square two okay that we kind of saw to.
Robert Bruner: Now the next generator goes to square six and then after that the generators are in degree at a higher if you think about him it's top cells and dimension seven so they're clearly.
Robert Bruner: and suspension, six of them one its top so isn't dimension seven so clearly generators and degrees higher aren't going to matter.
Robert Bruner: So in fact here this generator number 2012 that hit square six is the one that matters okay so generate around with to you, and he one.
Robert Bruner: maps to score six and zero where's that in em well if we go back up to the exact sequence here square six on the bottom sell of em is exactly generator number two there okay so let's go back to the extension co cycle here that says.
Robert Bruner: This generator.
Robert Bruner: goes to the degree six class in em now the extension co cycle from a one to one.
Robert Bruner: As to actually get the class in one that hits that and that's the bottom class so that's generator number zero in at one okay so here's how you say all of that, in a map definition file.
Robert Bruner: Okay coma logical degree one okay that's going from he won the first stage of the resolution, not the zeros.
Robert Bruner: Internal degrees six Okay, so that is all visible in e one mapping to suspension six.
Robert Bruner: And then the domain is m for well that's because we're looking at the resolution of em for here and mapping to am one, the Co domain is m one we're going to call the map, he.
Robert Bruner: What does it do it matched generator number two in he won in the resolution of him for Okay, so that we saw that back here generator zero on one map trivially generator number two is the one that maps non trivially.
Robert Bruner: And that goes to the bottom class in one, so one term and generator number zero, and then the identity on the screen right algebra acting on that okay so that's all we have to do to put in there okay so let's now create those and create those maps.
Robert Bruner: Okay, so we go back up to a and then let's do I first, and let me go back up here, so I actually have the file in front of me, so I don't miss type.
Robert Bruner: OK OK so VI dot death i'll just call it is death okay So the first thing now, this is important.
Robert Bruner: The first line of the map definition file needs to have these six things in an s T domain co domain map name and number of generators.
Robert Bruner: Okay, so in this case it's zero minus six m one m maps name is I maps one term non trivially that term is generator number zero, and it goes to something with one.
Robert Bruner: Some of one term which is generator to acted upon by the identity on the news.
Robert Bruner: So now.
Robert Bruner: New map.
Robert Bruner: High def here in the new map man, all you need to do is to give it the map definition, because everything it needs to knows in the map definition itself.
Robert Bruner: Okay So here we go it says it created m one slash maps put the map, I in it it's ready to compute lifts your change directory to m one.
Robert Bruner: m add it to your maps files necessary will we're going to use the maps file it generated so we don't have to do that, and then we run the script do lists okay let's do that.
Robert Bruner: To me one dot slash new lifts now this runs by Colin logical degrees we're just going to do them up 03 60 and.
Robert Bruner: files, we want the maps, we want to lift or the maps and maps let's look at file maps here, briefly, is the only element in it okay so we're just doing I if you sometimes want to.
Robert Bruner: Often you'll have a map file, it has many more things in the third video will see an example maps files could have a lot more things in Okay, so do lists.
Robert Bruner: Zero to 60.
Robert Bruner: For the maps the maps file now i'm not going to put this in the background, because this is very talkative and we wouldn't be able to do anything else in the meantime and it's very fast.
Robert Bruner: Okay, so it's talking a lot here he's a little things is to error messages popping up back here, those are done, those are happening because we're trying to map.
Robert Bruner: Call degree 42 two degree 42 but the resolution has only been created outside degree 40 and so it's.
Robert Bruner: not able to do that, so it computes everything it can compute, so this is one thing you have to watch out for if you extend your resolutions.
Robert Bruner: Then you need to extend the maps, so you have to go back rerun do list again it doesn't.
Robert Bruner: When you extend the resolution it doesn't look around and say what are all the maps that were involved here and extend those automatically for you that's lift up to you.
Robert Bruner: Okay, so now let's collect all this oh let's actually just briefly look at maps file here the maps file is just called I and actually.
Robert Bruner: let's look at it with a bit more resolution Okay, so in here, you see.
Robert Bruner: The definition that's the one we gave it.
Robert Bruner: Is that's the maps file we we handed it the map itself.
Robert Bruner: Is has the entire chain map that it computed in here so there's a lot of stuff here.
Robert Bruner: The.
Robert Bruner: map.org that is created by composing with the argument that by Marty on the augmentation idea, so all of this, all map.org keeps track of is the action by the square to to the eyes and that's actually useful in computing.
Robert Bruner: bracket and the like in these files brackets brackets that Sim contain information about brackets there and map.org also contains information.
Robert Bruner: i'm sorry map only contains information about what's happening to the resolution F after you might out by.
Robert Bruner: The augmentation ideas so it's telling you the induced map indexed and it's brackets that will tell you the induced.
Robert Bruner: tell you about the bracket to read about the brackets go look at the archive posting that has full detail in there.
Robert Bruner: On that Okay, so this we don't want to read this map.org, though, even though it's not a terrible thing because it's really just kind of a bunch of numbers here, so instead what we'll do is reused this program collect.
Robert Bruner: were collected into all I this is telling us what the map, it does.
Robert Bruner: Okay, so now let's go back and compare this to what we see remember what I did it included the.
Robert Bruner: m one this bottom.
Robert Bruner: module and here's The co fiber of to into.
Robert Bruner: em and then computed the chain map and display that what is what happens when you apply ext That gives you the the induced map from X of em to extra them one.
Robert Bruner: Okay, so what we're going to see here is I applied to generate or number two for an m is going to go to generator to to in co fiber two.
Robert Bruner: Okay, so, two, four and m is at here comb logical degree to generate or number four Okay, so this is in degree 12 here and that's going to go to to to in one.
Robert Bruner: To two and then one is back here okay that's an agree six well that's just right, this was a map of internal degree six something of internal degree.
Robert Bruner: will actually is internal degree 14 going to something you've entered agree eight but geometric degree also changes by six so degrees 12 was founded three six.
Robert Bruner: And then generate H two times that that's going to be generated or number four in home logical degree three up here must then go to.
Robert Bruner: Three three here let's see yes, there we go, we see it.
Robert Bruner: Using the key there we go, three, four and m, I sent 34233 and then one okay so that's what that map looks like now go back up to the.
Robert Bruner: screen right algebra directory and.
Robert Bruner: create the definition file for J okay so what's that look like.
Robert Bruner: 00 am to four will call J term 0100 text at.
Robert Bruner: The map definition file for Jay and we install this new map.
Robert Bruner: OK, so now we change directory down to em notice it's kind of remind you, where you want to be.
Robert Bruner: Okay, very good end to em and run do lists.
Robert Bruner: We did before.
Robert Bruner: Okay now this doesn't do any this does it's, this is a home, this is an interim degrees zero map, so we didn't get those.
Robert Bruner: Errors saying that we couldn't map something here, the reason we got them back here for em one was that we'd computer them once you're interested in or degrees 60 but suspension minus six of them was through degree.
Robert Bruner: Three degree 40 I mean it was computed through degree 40 minus six only 34 so things in in one of intro degree beyond 34 had no place to go to an m we get and computed that yet Okay, so this is done so that will collect the results collect maps all J.
Robert Bruner: And then have a look at that.
Robert Bruner: And we see now.
Robert Bruner: make this a little smaller to spread it out okay there we go now, this is going from.
Robert Bruner: em for up here in X it's going from him for up here at the top 2am in the middle okay so 00 NM for goes to 00 in.
Robert Bruner: And m here okay so Jay have 00 zeros this generator here, he goes to 00 in here Okay, so this does pretty much the things you'd like to see but notice generator number one two here.
Robert Bruner: haha one one guy in him for got mapped one, three and m forgot map, but one, two and then for that map trivially it's not mentioned, because it went to zero, so this only list of things going on to.
Robert Bruner: notice one, three and then four goes to one two in em okay that makes sense that here one, three and four is this Sigma and degree seven and it goes to Sigmund degree seven here but that's January or number two because it's an intervening, so it came by the boundary map from.
Robert Bruner: The co fiber to isn't here in him Okay, so you can read through here and see.
Robert Bruner: Exactly what's happening in particularly helpful and by degrees like appear where he had two things mapping and.
Robert Bruner: Well it's pretty clear what's going on, but you know getting out into higher degrees it's it's very helpful to have this to tell you exactly what the map looks like alright so that's that one now let's go create the extension coast backup to a.
Robert Bruner: VI E that death and rather than re imagine how to do this now just read what I had typed here okay.
Robert Bruner: One six okay cool much good agree one internal degree six it was from him for two m one m the maps cold he maps one generator non true really that was January or number two in it maps to one term consistently generator zero acted upon by.
Robert Bruner: The identity map steam right algebra.
Robert Bruner: OK, so now new map.
Robert Bruner: He def in OK, so now it's set everything up we change directory to em for remember that's the domain.
Robert Bruner: map files are always create the The sub directory containing a map is a subdirectory of the directory, which contains the domain module okay so.
Robert Bruner: And it tells what it's co domain is of course okay so go down to em for dot slash do lifts zero to 60.
Robert Bruner: Now, this has got a coma logical degree one shift in it so that'll generate a few error messages because.
Robert Bruner: One of these resolutions has been computed a bit for the other doesn't hurt anything you should always compute your resolutions, a little further than you know examine the chain maps to see you don't have to be too fussy about that alright So there we go.
Robert Bruner: Okay, so.
Robert Bruner: Okay, the shape file was only good through filtration 60 and mediated filtration 61 well there we go, we started over here and filtration 61.
Robert Bruner: And it tried to map to filtration 61 and m for and there isn't one we haven't completed that yet so if we'd wanted one we'd have had to run them for out through one home logical degree greater OK, so now that's nice collect.
Robert Bruner: All that he in this look at all that he to see what we've got OK, so now go back and look at the resolution we had here.
Robert Bruner: Okay, so now, now we might want to roll this around a little bit more okay so 00 and m one here first entry 00 and then one goes to one two in under the map, he okay so 00 and one.
Robert Bruner: K down here in em 100 is.
Robert Bruner: The generator and degree by degree 001 logical degrees zero generator zero and it's going to go to.
Robert Bruner: One to up in co fiber new okay on logical degree one generator number two ah, there we go that's that little lightning flash here that we saw did not map into Bo one okay so it's exactly it by this lightning flash and co fiber to that came around and hit it OK, so now.
Robert Bruner: generator one zero and then one okay so that's what that each one on the bottom sell here that goes around and hit generate your number two in important.
Robert Bruner: Okay, to to me okay so call much will be re to generate our number two there it is each one on the beginning of that lightning flash.
Robert Bruner: And so you can run through here and see exactly which what the map, but the boundary map in ext from cover to up to CAFE renew looks like defining bill one here.
Robert Bruner: Okay, in the next section, we will see how to also compute the action of.
Robert Bruner: Extra over a of F F to this.