Archive of changes made during January 2009. The substantive content of this page should not be altered. For past versions of this page beyond its own history, start here and work backwards.
Finally decided I need to get over my Australian training and make weak things the default, at least when writing on the nLab. So I split up 2-categorical limit into 2-limit for the weak notion and strict 2-limit for the strict one. The exposition here could probably use help, and I have a question about terminology for strict 2-limits.
added the defintiion in terms of adjoints to the constant diagram functor to limit
started giving a discussion of the point of (higher) category theory in physics
added a reference to dualizing object
Started trying to bring some order to the treatment of coverages by creating coverage and working on Grothendieck topology and Lawvere-Tierney topology. I wasn’t quite bold enough to try to excise the word “topology” from this subject entirely yet.
Created directed homotopy theory. This is at present a stub plus an inadequate list of references that do not do justice to the area … as yet. I have built in some links but feel there should be others.
Created group T - complex, but more needs adding here.
somewhat hastily created fundamental infinity-groupoid (have the vague recollection that I wrote something like that before… but can’t see it anymore…)
created cobordism and extended cobordism and included a bunch of literature related to that and to multispan, notably to Marco Grandis’ work on Cospans in Algebraic Topology and to Jeffrey Morton’s work on multispans ans extended cobordisms
further details at multispan (or in the file linke to there)
worked on directed object:
added discussion of degenerate examples of interval objects
suggested (and inserted) a formalization of the reparametrization axiom that Toby had left open
replied to the discussion at directed space asking Toby to suggest how to deal with the slight terminological subtlety we’re running into
added references to Fahrenberg and Raussen at directed space (there must be more literature on directed homotopy theory! somebody should look it up)
Other, more minor, edits.
Since Urs and Eric didn’t respond to my question about deleting the discussion in monoidal category (see 2009-01-19), I went ahead and deleted it — restore it if you like, after reading the comments at the end of that entry!
The picture of the pentagon identity in monoidal category has mysteriously disappeared, though the source code is still present. Help!
I have added a question to homotopy hypothesis asking what criteria should be added to give ‘good’ categorical or groupoidal models for homotopy types.
For directed object I have, similarly, tried to pose question about the criteria that should be ‘directing’ our search for good concepts in this case.
I have also added a question about the ‘optimal’ definition of fibration, as it seems to me that the lifting property is nearer the idea of fibration than the transport? one that Mike has put forward.
following discussion by Eric at directed space I propose in the discussion section a formalization of the notion “an object is directed” and “an object is undirected” for the case that is an object in a category with interval object.
expanded the entry on monoid, giving lots of examples of monoid objects in monoidal categories. I think lists of examples like can be very useful and fun, and I want more! I would like a list of PROPs, for example, saying that is the PROP for commutative monoids, and so on.
If Urs is happy with how the discussion at the end of monoidal category has been incorporated into the body of the article, maybe we can remove that discussion.
slightly expanded the entry on braided monoidal category - but it really needs some diagrams!
replied to discussion at interval object
Toby and I appear to have finally reached a consensus on terminology at extensive category. I did some work on this page and coherent category, pretopos, and Grothendieck topos making them more consistent, and also created disjoint coproduct and coproduct.
Responded to Toby at Lawvere-Tierney topology.
the entry smooth Lorentzian space is currently oscillating a bit between showing a desire to state a new nice definition for (possibly generalized) Lorentzian spaces and the aim to first establish some facts about the usual definition
filled in stuff at causet
added questions to Trimble's notion of weak n-category
expanded on the examples in generalized universal bundle
Toby Bartels: I added stuff that I'm trying to understand to Lawvere-Tierney topology. It is probably correct, but I had some questions. (See also the relevant terminological discussion at Grothendieck topology.)
started generalized universal bundle after all – now I really need to run
said more at the beginning of interval object – and are having a discussion with Todd about examples and the necessity to relax the structure, and his Trimblean definition of -category – which we ought to have an entry about
in the entry category of fibrant objects
categories: reference-entry for Brown -- Abstract Homotopy Theory and Generalized Sheaf Cohomology
added a remark on the definition of kernel and cokernel of morphisms to pointed object – some expert please have a look at that
added pointed category and also a handful of trivial entries which are actually superfluous, since I hadn’t had my first coffee yet and wasn’t paying proper attention
began, at Urs’s request, an entry for Yoneda reduction
finally submitted draft of regular category, after having it locked for a few days and doing very little with it in the end.
Did some rephrasing at infinity-category.
Added the unit axiom to monoidal model category.
reacted to the discussion at path object
created cylinder functor
added my 2cent and my questions to the discussion at homotopy on interval objects – and added the entry interval object but filling it not with a statement but with discussion: I say what I came to think a well behaved interval object should satisfy and am asking for information on what is known in this regard
had the honor of creating Yoneda lemma
created geometric function theory to accompany a guest post by David Ben-Zvi – maybe somebody of the groupoidification team can eventually add an entry on Hecke algebra and its relation to groupoidification
Timothy Porter created
gave the bare definition of crossed module
added details (definition and theorems) to category of fibrant objects
restructured the index page mathematics slightly and tried to complete the lists of keywords at foundations and logic. Can I suggest that everybody who creates a new entry considers adding a link to that entry, if appropriate, to the lists of “related entries” at one of the big index pages? I am thinking it would be good to give readers a chance to browse our material by topic and get an impression for what is there and what not (yet).
added a few remarks at higher category theory. I am not happy with that entry. Clearly we need a more comprehensive discussion there eventually.
Toby Bartels: I reacted to everything that everybody did this year, and I interacted with everything that everybody did today. More specifically:
am reading Mike’s Homotopy limits and colimits and enriched category theory and started adding some central definitions as entries here, such as closed monoidal homotopical category and enriched homotopical category and some related entries such as homotopical category and homotopy coherent category theory. Eventually I would like to see if some of my favorite monoidal homotopical categories are examples. See my request hereonoidal homotopical category).
following the discussion which I now moved to discussion on terminology -- omega-category I moved the original material on that to strict omega-category and kept at omega-category just a pointer to that entry. This means we should go through a bunch of entries and rename links to omega-category into links to strict omega-category.
meant to follow up on Todd’s discussion of sheaf and stack in terms of sieves and relate it to the entry on descent and codescent and infinity-stack but was distracted by other things. But I did write up a quick rough note on this into a LaTeX file, which you can see here for the time being: On descent, sieves and omega-functors..
expanded on the sheaf-condition, offering in parallel to the definition in terms of sieves the definition in terms of orientals with a a few remarks on how both are related and what the issue is with generalizing to stack and infinity-stack.
took the liberty of changing in Toby’s entry (n,r)-category the ”-categories” appearing there to ”-categories”, since I thought we did agree that ”-category” is the generic term while omega-category specifically means strict globular -category.
created a blog entry nLab – General Discussion and added a remark at the beginning of the original wiki entry General Discussion asking to post further general discussion not to the wiki, but to the blog.
there is a discussion going on at subcategory over whether any faithful functor should be called a “subcategory.” Input welcomed.
added a comment on terminological objections to Grothendieck topology.
corrected the example of abelian groups at tensor product.
added examples and comment on non-monoidal closure to closed category
Toby Bartels: I posted periodic table and (n,r)-category, which I wrote today offline. There are probably lots of typos right now; I don't have a lot of time online today. Also, the former one really only contains an appendix that I wanted to write, not so much the article itself. But at least you can read what's there now.
Todd Trimble has expanded on site and sheaf. I have started adding a bit more motivational detail to site and links to descent, stack and infinity-stack. There is still plenty of room here for saying more about this general story.
Apparently, we all took a break from the Lab for the New Year!