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.
Created retract and idempotent.
Clarified homotopy limits at model 2-category and strict 2-limit.
added more examples to homotopy limit and span trace (see blog comment)
added based loop object to free loop object at loop space object and linked that with generalized universal bundle
Tim:
Mike:
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.
Redirected Gray category to the already-existing Gray-category. Should we have an official policy on the use or non-use of hyphens? (-:
I haven’t done much here recently because I’ve been working in my own web on a project developing 2-categorical logic.
Eric:
Tim:
Continued work on crossed module, crossed n-cube and related entries.
Linked the Dold-Kan correspondence to the entry on the Moore complex where there was already some discussion of this topic.
have a question/request at 2-limit about “homotopical enrichment”. Can anyone say more about this? Tim?
created loop space object and included at span trace as an example the statement that the loop space object is the homotopy trace of the identity span on the object
have a request at the end of 2-categorical limit that we eventually give a formulation in terms of homotopy limits.
added the defintiion in terms of adjoints to the constant diagram functor to limit
replied to Eric at generalized universal bundle
created Dold-Kan correspondence and also Dold-Kan theorem as a redirect
created simplicial presheaf, SSet
have a remark/question at one point in David’s text on philosophy
started giving a discussion of the point of (higher) category theory in physics
added examples and a bit more to distributor and implemented Todd’s suggestion there
have a question at homotopy theory on Loday’s result
created Gray category
created 2-vector space, bimodule, distributor (and profunctor as a redirect)
Tim:
Continued adding stuff to various entries on $cat^n$-groups and references for Loday’s theorem at cat-n-group.
Created 2-crossed module.
Urs:
have a question for Andrew at Froelicher space
added a reference to dualizing object
created cat-n-group
added a bunch of references to directed homotopy theory which Tim Porter kindly provided
started adding/searching for references on $n$-fold groupoids to homotopy hypothesis, added a respective remark to n-fold category
Tim:
Joined the $n$-Community. Hello.
Modified HomePage to point to the $n$-Forum.
Made list of generalised smooth spaces on the generalised smooth spaces page (with the intention of starting pages on the various types).
Tim:
Created Moore complex which contains the definition of the homotopy groups of a simplicial group.
Added query to homotopy theory: should a summary of the Baues approach to abstract homotopy theory be included somewhere?
Mike:
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.
Added relevant comments to anafunctor and folk model structure.
Todd:
added more to geometric shapes for higher structures;
gave an answer to Eric’s question at globe
took some issue with Urs about whether span traces (and implicitly, cospan traces) have or haven’t appeared in the literature
Tim:
Created pospace to help the entry on directed space, at which I gave a link through to the new page.
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)
created an entry on Nils Baas’ concept of hyperstructures, mainly to point out how close it is at least in spirit to spans appearing in groupoidification – and in fact to multispans
created span trace and co-span co-trace (I am not sure about the (best) spelling convention!) after I had checked with John that this has not previously been said explicitly in the literature
Tim:
Mike:
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)
replied to Toby’s remark on the need for directed homotopies at directed space.
Wrote pushout in the same gentle style as the previous article pullback. I want nice easy introductions to all our favorite limits and colimits!
Wrote brief stubs for colimit and totally ordered set.
Content:
Discussion:
Wiki structure:
Other, more minor, edits.
Mike:
Mike:
Changed enriched homotopy theory, to reflect my feeling that it is more general than homotopy coherent category theory.
Created enriched factorization system, orthogonality, and Galois connection.
Created split epimorphism, strong epimorphism, and extremal epimorphism just to satisfy links.
created complete Segal space and (infinity,n)-category
added diagrams to weak factorization system and created a stub entry monomorphism to saturate all the wanted links to it
John:
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 added endofunctor and strict monoidal category.
Mike:
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.
Mike:
separated directed object from directed space and included the definition of directed topological space by Grandis
further reacted at directed space and created homotopy hypothesis
following discussion by Eric at directed space I propose in the discussion section a formalization of the notion “an object $X$ is directed” and “an object $X$ is undirected” for the case that $X$ is an object in a category with interval object.
John:
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 $FinSet$ 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!
Mike:
Added detail, examples, and terminological comments to bicategory.
Created 2-categorical limit.
Replied at finite set.
Eric:
Mike:
Added another argument in favor of using $B$ at category algebra.
Expanded the topos-theoretic discussion at finite set, including some examples.
continued fiddling with interval object
added references to Trimble's notion of weak n-category
Added the internal version to group.
Created well-powered category.
Jim Stasheff added a word of caution at Hopf algebra
added references to interval object which should be moved to Trimble's notion of weak n-category (which is however locked at the moment and I have to run now)
meant to further develop interval object but got distracted – the little I could get done is on pages 5,6,7 here
replied to discussion at interval object
Toby Bartels: Wrote algebraic theory, mostly to distinguish it from Lawvere theory.
Mike:
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.
Tim Porter I have added in a new entry on the notion of a simplicial T-complex, which should help provide background for the still needed entry on complicial sets.
to saturate links from Trimble's notion of weak n-category and interval object I created A-infinity algebra and A-infinity category and filled in a little bit of information
started proposing a full definition at interval object following the discussion at that entry and at Trimble's notion of weak n-category – please check!
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.)
Mike:
Added detail about internal-homs and smash products to pointed object, in response to Todd’s comment.
Created adjunct.
Urs:
started generalized universal bundle after all – now I really need to run…
created pointed set and under category, am planning to expand on the bigger story indicated at pointed set, but not today
expanded the entry bi-brane in order to accompany some discussion about geometric function theory at Ben-Zvi on geometric function theory
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 $\infty$-category – which we ought to have an entry about
in the entry category of fibrant objects
added comments on relation to pointed object and pointed category
made the factorization lemma more explicit and related it to the notion of Morita equivalence
created a 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…
Mike:
Incorporated the discussion into k-tuply monoidal n-category and added more examples. Perhaps some of this page would better go at delooping hypothesis?
Created pointed object and spectrum.
category: reference
for them.Mike:
Created bijection and subsingleton.
Answered Toby’s three questions, and continued discussion at inhabited set.
Mike:
Joined the discussion at category algebra.
Removed the comments at constructivism that have now been incorporated into the main text (thanks for writing that detailed section, Toby).
Created finite set and choice object.
added some discussion to classifying topos, linking up to the entry Lawvere theory
created entry on Cauchy complete category
Mike:
Removed the discussion at field, since it had converged and been incorporated into the entry.
Incorporated the apparent conclusion of the discussion at extensive category into the entry.
Added some details to excluded middle.
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.
would like to point out that material relevant to entries like universal property, universal construction and Yoneda lemma exists at representable functor; this could use further expansion
Urs:
Toby:
Mike:
Created coherent category, Boolean category, Heyting category, equivalence relation, kernel pair, exact category, balanced category, and pretopos.
Finally got fed up with the nonexistent links to it everywhere and created product. Also created generalized the.
Did some rephrasing at infinity-category.
Urs:
created directed space
added a bit to David’s entry on the generalized tangle hypothesis
reacted to Mike’s comments at infinity-stack homotopically
Continued my discussions with Toby at Grothendieck topology, extensive category, and constructivism, including one that should be moved to predicativism.
Started new discussions at infinity-stack homotopically, k-tuply monoidal n-category, and inhabited set.
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
created pushout-product axiom and linked to it from monoidal model category
tried to provide a more unified perspective on infinity-stack and infinity-category and nonabelian cohomology at infinity-stack homotopically
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
Toby Bartels: I wrote about constructivism and the empty set.
Created geometric morphism, locale, and sober space.
Added a more classical version to homotopy.
created regular category
created image
created Rel
created Lawvere theory
started some discussion at simplex category
Timothy Porter created
nerve (but see below)
gave the bare definition of crossed module
finally created algebraic definition of higher category and n-fold category. Just stubs so far, this deserves much (much) more discussion, clearly.
started filling in nerve after I found an empty entry of that name – now Timothy Porter sends me by email his version. Either he or I should merge the material…
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 references to Timothy Porter and Cordier at homotopy coherent category theory
Expanded slightly on Timothy Porter’s entry on simplicially enriched category.
added a few remarks at higher category theory. I am not happy with that entry. Clearly we need a more comprehensive discussion there eventually.
Mike: Created constructivism and imported the relevant discussion from apartness relation.
Toby Bartels: Mike Shulman and I are having terminological discussions. Also, he fixed my theorem at extensive category (which will go in our paper, John).
Toby Bartels: I reacted to everything that everybody did this year, and I interacted with everything that everybody did today. More specifically:
Split off internal category from internalization. Probably a lot of links need to be updated.
Summarized the discussion at subcategory into a section called “Non-evil variants.”
Created the entry red herring principle.
created the entry CW complex
added an alternative definition of cube category, linking it to $\Delta$ and string diagrams
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).
started replying to Todd’s remarks at stack by addding a section Descent in terms of pseudo-functors to descent and codescent.
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.
Added disambiguation comments to simplicial category.
Created Gray tensor product and Gray-category.
Added a terminological suggestion to omega-category.
Refactored tensor product, removing the discussion which prompted the refactoring. Created fc-multicategory.
Urs:
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.
added, following email discussion with Todd, a discussion to sieve of which sieves are the presheaf incarnations of which covers, and why
adapted n-category and n-groupoid to Toby’s (n,r)-category (which is a great entry – we should flesh that out further).
took the liberty of changing in Toby’s entry (n,r)-category the “$\omega$-categories” appearing there to “$\infty$-categories”, since I thought we did agree that “$\infty$-category” is the generic term while omega-category specifically means strict globular $\infty$-category.
separated infinity-category from higher category theory and added some brief stub entries such as n-category, n-groupoid and Kan complex.
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.
rearranged tensor product following the comments and corrections by Toby and Mike (thanks!).
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
added to (and edited) examples under closed category
created dinatural transformation
added examples to extensive category
added to the discussion on Day convolution
added some detail to the definition of Lawvere-Tierney topology
did closed category and linked to it from Mike’s cartesian monoidal category
expanded on Day convolution
split the information previously contained in site into several separate entries: cover, sieve, Grothendieck topology, Lawvere-Tierney topology and, of course, site.
Mike Shulman created extensive category, cartesian monoidal category
Todd Trimble expanded on stack and connected it to the discussion of covering sieves at site
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 created operad
Urs: started adding remarks to opposite category about the big story hidden here related to duality.
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.
Urs: motivated by some discussion we had on the blog, in the entry About I started giving some indications on what the $n$Lab is and what it is not.
Owen Biesel: added diagram, global element, terminal object, limit.
Apparently, we all took a break from the $n$Lab for the New Year!
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