As I mentioned in my “new member” post (I don’t know how to do fancy references to other posts yet), I am looking for a notion of “remainder” in category theory.

I’m describing a “system behavior” and a map of that behavior to the behavior of a “component” of the system.
The question at hand is: what is the best description I can find of the behavior of the **other** components?

Because a composition of components (and synchronization between) them, in my category leads to a “jointly monic family” of maps from the system behavior to the behavior of the components, I could therefore say that I’m looking for the “smallest” component that allows me to form a “jointly monic pair” with the map into the “known” component.

I got to this definition at first:

- given a map $f : X \rightarrow Y$, a “remainder” of $f$ is any object $Z$ for which there exists a map $g : X \rightarrow Z$ such that $(f,g)$ is jointly monic, and in addition it holds that for any other $g' : X \rightarrow Z'$ such that $(f,g')$ is jointly monic, there is a unique $k : Z \rightarrow Z'$ such that $kg = g'$.

I though it was a rather natural notion, but have not found references to it yet.

Mike Shulman and David Roberts already responded with two questions:

1) can I give an example of whether such a remainder can exist at all? He didn’t think it would exist for Set.

2) Isn’t it a trivial notion? If you take $Y = X$, the identity might do the job.

As to 1)… I’m going to think about it. I think I have an example in the category of prefix orders, on which I just posted a page, but I still need to check it. (Apologies for asking the question before doing my homework… but sometimes you can spend hours figuring out what other people already know ;-)

As to 2)… shoot… that is a suggestion I overlooked. In my own examples there are often “smaller” candidates for $Y$ than $X$, but they do not seem to come out this way. Intuitively, in Set, the map $g$ should not be injective at points where $f$ is already injective. But the definition still allows that apparently. Back to the drawing board… Any suggestions?

]]>I’ve noticed that the page on projection does not mention the more generic definition of projection used on wikipedia: https://en.wikipedia.org/wiki/Projection_(mathematics)

Is there a particular reason for this?

]]>being new to the forum and the nLab, I would like to make sure my intentions on using this align with yours.

I'm a researcher in theoretical computer science, interested in formalizing the semantics of models of execution of real-time and cyber-physical systems.

While working on a set of lecture notes on the topic, I noticed I like to use category theory "as a compass".

I am developing my own behavioral models, and by considering them as objects and morphisms in a category, I have found many small `tweaks' that make my definitions work better.

I go by the rule: "if the standard category theoretic definitions give me what I hoped to get in the first place, I'm on the right track", and that has helped me a lot (even though it also slows me down terribly).

Having done this for a while, I'm now coming to a point where notions like product, limit, and monomorphism indeed reflect the intuitions I would like them to reflect.

Furthermore, recently I have found out that I should actually be looking at my categories as concrete categories, and then embeddings turn out to be useful.

Together with a colleague we even found 'upgrades' of known categorical definitions, which seem helpful.

And that's the point when it all of a sudden stopped... or at least changed...

Now, whenever I meet a new topic I would like to investigate, I try to think of the 'natural' way to represent it in terms of objects and morphisms,

rather than thinking about the best way to do it "within" my own category (categories) of choice first. And while this is fun, it leaves me with the

problem that I invariably come up with definitions that sound very plausible, but that I cannot find (quickly) in the basic literature on category theory.

For example, yesterday I decided I need a notion of "remainder". The practical problem at hand requires me to study a morphism from the behavior

of a complex system to the behavior of a component, and then "divide out" the behavior of the component to obtain the behavior of "the rest of the system".

I consider the complex system as something that defines a "relation" between components, by synchronizing them in some unknown way.

I think that jointly monic families (the dual of [[jointly epimorphic family]]) are a nice way to capture such relations categorically (something that is not really mentioned on that page, by the way...),

so my way of "dividing" comes down to (I don't know how to make diagrams here yet...):

* given a map f : X --> Y, find (the smallest) object Z (which I'll call the remainder of f) for which there exists a map g : X --> Z that makes the family (f,g) jointly monic.

where "smallest" means that any other object Z' with map g' : X --> Z' that makes (f,g') jointly monic has a unique map k : Z --> Z' such that kg = g'.

I think it is a rather natural notion, but I can't find it anywhere (at least not quickly, and my colleagues are getting on with the problem, so this is a 'side thing' and should not cost hours of library research).

Now, what I would like to ask (finally) is:

1) Is this the right forum to get suggestions on references to (anything like) the definition above? (If so, please comment!!)

2) Is nLab the right place to start a page and write down my notes on researching the above definition, and others that may come up?

3) How do you/we deal with "original work" on topics (even before it is published elsewhere)?

Can I just start a page on a topic, name it whatever I see fit, and "see what happens"?

How can I indicate that pages are "standard", "pretty standard", or "have not passed the test of peer-review anywhere yet"?

A page may be "needing review" because it is "standard stuff that may not have been explained correctly", or because it is "new stuff, and the definitions may not make sense really yet".

4) Should I ask permission first here, before making changes to other peoples pages (for example, adding the typical use of jointly monic families as a generalization of relations to the page [[jointly epimorphic family]])?

Or do you just do it first, and hope for forgiveness (which is much quicker, but it is also a bit rude I suppose).

Kind regards,

Pieter ]]>

I’ve been thinking about generalizing the Cech-Delign double complex to the case where $U(1)$ is replaced by some Lie group, $G$, and $\mathbb{R}$ is replaced with $\mathfrak{g}$. I came across this post on Nonabelian Weak Deligne Hypercohomology by Urs a while back and was wondering if his musing was ever fully considered/resolved?

For full context of why I’m considering this: I was working on a project during my PhD that I’d like to eventually publish, but I constructed an element in a (Hochschild-like) curved dga with a Chen map to holonomy (path or surface) which is a chain map and map of algebras. It was suggested that this is not enough of a “result” unless I could find the right notion of equivalence to fully flesh out this map. I was hoping that some resolution of the referenced article above could allow me to put my work in that context.

P.S. This is my first post here so my apologies if I made a cultural error.

]]>I discovered from here that anonymous posting to the nForum had been switched off back in 2013. It has been requested several times over the past months that we should allow anonymous posting to the nForum, since it is allowed on the nLab. Thus I have now switched anonymous posting back on for the most commonly used categories.

I have also removed admin rights from the ’mathforge admin’ account, since those who had the password to it are not involved with the admin/development of the nLab currently. Adeel still has admin rights. I gave myself the rights temporarily to switch on anonymous posting, but have removed them again, because I find the user interface too cluttered!

By the way, if any of the other of you would like an admin account, so that spam can be deleted, etc, let me know. You could have a separate one from your main account, if you too find the UI too cluttered.

]]>Every now and then some spam gets through the filters on the forum. Toby and I can remove it from public view (it stays in the database so if it turns out that it wasn't spam it can easily be deleted). Also, I can set the user account which was used to "banned" to prevent it being used again. Other useful information, such as IPs and email addresses, is also logged. But to keep the place clean, we need to be aware of the spam occurring. I, in particular, don't read every post made here on the forum. So if you spot some spam, please alert someone. Use this discussion if you like.

]]>I was just wondering why there was so little on “Institution independent Model Theory” or Absrtact Model Theory in the wiki. I found this short entry for Abstract Model Theory, and a link to yet non existing page on institutions.

I am trying to use this to see if this can help me extend the semantic Web semantics to modal logic. The reason is that institutions have been used to show the coherence between the different RDF logics - RDFS, OWL, … and so it seems that it should be helpful to go beyond that.

Some papers on semantic web and institutions are listed below. These are great because the semantic web is quite simple, useful, - and I understand it well - and these show in a practical way how to think about institutions, which would be otherwise much more difficult to get into. Also the basics of Abstract Model theory are quite intuitive

- Lucanu, D., Li, Y. F., & Dong, J. S. (2006). Semantic web languages–towards an institutional perspective. In Algebra, Meaning, and Computation (pp. 99-123). Springer, Berlin, Heidelberg.
- Bao, J., Tao, J., McGuinness, D. L., & Smart, P. (2010). Context representation for the semantic web.

The last one ties rdf to Contexts and to Institutions.

The RDF model is actually really simple btw. See the question and answer “What kind of Categorical object is an RDF Model?”

It is nearly self evident from using it that RDF already contains modal logic (see my short example on semweb mailing list), especially as for RDF1.0 xml syntax one can have relations to RDF/XML literals, whose interpretations are of course sets of models, and in RDF1.1 this is made clearer with the notion of DataSets which are sets of graphs. But they have not given a semantics for it… But self evidence does not make for a proof. (and by the way, RDF/XML is really the ugliest syntax existing. Much better to consider N3 which is Tim Berners-Lee’s neat notation for doing logic on the web.

- Berners-Lee, T., Connolly, D., Kagal, L., Scharf, Y., & Hendler, J. (2008). N3logic: A logical framework for the world wide web. Theory and Practice of Logic Programming, 8(3), 249-269.

Btw, as an extra part the discussion on modal logic in RDF is tied up with the notion of context, which may just be another way of thinking of modal logic (I am working to see if there is a difference)

- Guha, R. V. (1991). Contexts: a formalization and some applications (Vol. 101). Stanford, CA: Stanford University.
- Hayes, P. (1997, November). Contexts in context. In Context in knowledge representation and natural language, AAAI Fall Symposium.
- Bizer, C., Carroll, J. J., Hayes, P., & Stickler, P. (2005). Named Graphs, Provenance and Trust. In Proceedings of the 14th international conference on World Wide Web.
- Hayes, P. (2007). Context Mereology. In AAAI Spring Symposium: Logical Formalizations of Commonsense Reasoning (pp. 59-64). This is I thought a really neat paper.
- Bao, J., Tao, J., McGuinness, D. L., & Smart, P. (2010). Context representation for the semantic web.
- Klarman, S. (2013). Reasoning with contexts in description logics.

So because there was little on the wiki on abstract model theory I was wondering if that was not quite thought of as good Category Theory, or if there just had not been time to complete that page. And for Contexts I was wondering if this was the right place to look at. In the book “Institution independent Model Theory” R Diaconescu has a chapter on Kripke frames, but I think we actually need neighborhood semantics, that is not relations between one world and another but between one world and a set of worlds. So that one can represent inconsistent sets of ideas. (which the web really is a big example of)

]]>There is a page nForum membership ranks and privileges which is a bit out of date (It was edited last in 2011.) Perhaps with the changes that have been happening it could be looked at again some time soon.

]]>A *lot* of new visitors have trouble entering math on the forum. It ought to be easy to add some more descriptive help text to the formatting options below the post. Could it say something like

To use (La)TeX mathematics in your post, make sure “Markdown+Itex” is selected and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted; see here for a list.

?

]]>Hopefully, the discussion topic says it all. In particular, this is about

- whether Cat already has a “Latest Changes” thread

an article whose title is too short to make the search functionality work (conveniently, for me).

]]>How do I subscribe to a discussion?

If I click on the link on the left, some blinking green dots appear, and they seem to never stop (and I get no emails).

How does it work, am I missing anything obvious? (In which case I apologize!)

]]>When I try to look at the Science of Logic discussion here, I’m getting

page contains the following errors:

error on line 93 at column 163: Extra content at the end of the document Below is a rendering of the page up to the first error.

and

]]>Fatal error: Maximum execution time of 30 seconds exceeded in /home/nlab/www/nforum/extensions/XHTMLValidator/default.php on line 2949

Hello,

In account->Notification options->Notify me, when comments are posted in my own discussions, when the checkbox is activated the green diamonds start to indicate that the request is sent but it continues in this state indefinitely long (for me, win 10, chrome 49).

]]>I tried to change my forum name from fastlane69 (the default username I use for everything) to my real name Ricardo_Rademacher but I could not find the option.

]]>Could we add a note next to the comment-entry box telling people that they can use the syntax `[[page name]]`

to link to pages from the nLab from the nForum? Newcomers always have to be told that, and of course, because how else would they know?

When I go to “Search -> Advanced -> Discussion comment search” and try to search by author I get the following error message:

XML Parsing Error: junk after document element Location: http://nforum.ncatlab.org/search/?PostBackAction=Search&Advanced=1&Type=Comments&Keywords=convex&Categories%5B%5D=&AuthUsername=Karol&btnSubmit=Search Line Number 2, Column 1:

Notice: Array to string conversion in/home/nlab/www/nforum/extensions/FeedThis/default.phpon line229

^

It would be great if somebody could look into that.

]]>If it isn't possible to change my username, can I delete my account so that I can reopen one with my desired username? ]]>

When signing into the nForum today with my google account, I was pointed to this page: OpenID 2.0 for Google Accounts is going away.

]]>I noticed some changes in the way the rss feed come to my reader. I am not sure whether something changed. In any case, this link that I got by copy-and-paste does not seem right:

http://nforum.mathforge.org/http%3A%2F%2Fnforum.mathforge.org%2Fsearch%2F%3FPostBackAction%3DSearch%26amp%3BType%3DTopics%26amp%3BPage%3D1%26amp%3BFeed%3DRSS2%26amp%3BFeedTitle%3DAll%2BDiscussions%2BFeed

Is this a problem on my side?

]]>Here’s something that confuses me about the “tags” on the nForum. Can a tag contain spaces? The comment “(comma separated)” on the tags input box suggests that spaces in a tag would be okay (in particular, wouldn’t break it into two tags), but I don’t think I see any tags containing spaces in the tag cloud.

]]>Have you all considered making an nlab chatroom? It seems that there’s enough frequent interaction on the forum that such a thing would be decently populated. I know I’d certainly spend some time in there! I have no idea what the technical challenges associated to this might be, so perhaps it’s beyond the scope of the people running the nlab and nforum, but it seems a natural next step.

Anyway… just an idea. If there was a chat room I would have just mentioned this in there. Or I guess I wouldn’t have. Uhoh, I’ve confused myself again.

-Jon

]]>Hi,

I happened to notice today that there are problems with the proof of Lemma 1. Unfortunately, I do not have time to fix this myself, but I thought I’d let you know.

It can be fixed as follows.

1) Define $\hat{X}$ to be the pullback of $f \times id : X \times Y \rightarrow Y \times Y$ and the map $Y^{I} \rightarrow Y \times Y$ which is currently denoted (I would not use this notation myself!) by $(d_0,d_1)$.

2) The required map $p : Z \rightarrow Y$ is the composite of the map $Z \rightarrow X \times Y$ which is part of the pullback of 1), and the projection map $X \times Y \rightarrow Y$.

3) The required fibration $Z \rightarrow X$ arises in the same way as in 2), but composing with the the other projection map instead,

4) The required map $X \rightarrow Z$ arises via the universal property of the pullback by using the map $f \circ c : X \rightarrow Y^I$, where $c$ is the map $Y \rightarrow Y^I$ appearing in the factorisation which defines $Y^I$, and the map $id \times f : X \rightarrow X \times Y$.

]]>