# nLab 2009 January changes

### Archive

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.

# 2009-01-26

• 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?

• Urs

• incorporated Todd’s remark into span trace and created a general entry on trace in monoidal categories; tried to add some clarification that the point of span trace and co-span co-trace was not to describe the concept of trace in general, but to describe how for spans it harmonizes with the interpretation of spans as linear maps in groupoidification and nicely matches with the fact that on co-spans regarded as cobordisms it realizes the idea that one glues the two ends of a cobordism together to get the trace – I found that simple observation noteworthy in the context of the cobordisms hypothesis, in that all extended cobordisms seem to be generated by the interval just under cartesian product and co-span co-trace, so any map from extended cobordisms to multispans which sends pushouts to pullbacks and regards the interval as weakly equivalent to the point should be fixed by its value on the point (that last statement should be scrutinized, but it is what made me want to make the (obvious) notion of co-span co-trace explicit)
• Mike:

# 2009-01-25

• 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.

# 2009-01-20

• 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:

• Urs

• commented in the discussion at fibration on the use of the word transport
• Tim

• 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:

# 2009-01-19

• Urs

• 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!

# 2009-01-06

• 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).

• Urs

• further expanded on a bit and harmonized a bit more the circle of entries globe category, simplex category, cube category and globe, simplex, cube linked to and summarized in geometric shapes for higher structures. At globe I give a reply to a question by Eric on how to think of globes as “pointed spheres” by stating a claim that the $(n+1)$-globe is the double cone over the $n$-globe in a precise sense. I believe this is true, but am not entirely happy with the proof I have. Would be great if somebody could check this.
• Mike

• Created power and copower. Possibly these should be just one page?

I think it is good to have many separate entries for sub-concepts if they all link to each other, maybe with a brief comment. –Urs

I agree. —Toby

# 2009-01-01

Apparently, we all took a break from the $n$Lab for the New Year!

category: meta

Revised on August 23, 2015 22:29:43 by Toby Bartels (98.16.168.238)