# nLab 2009 March changes

### Archive

Archive of changes made during March 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.

# 2009-03-28

• Mike: Prompted by discussion with Zoran, created strict epimorphism and added a lengthier discussion of types of epimorphism to epimorphism.

• Zoran Škoda: Created etale space. The order of exposition is important, particularly in view of anticipated additional details. In Kan extension added a detailed paragraph on an example how left Kan extension pointwise formula has intuitive meaning in the case of constructing pullback for (pre)sheaves on topological spaces. Created torsor with structure category following the version in Moerdijk’s book.

• Finn Lawler: Created linear logic – just a short stub with basic ideas on motivation and models, plus a couple of references. Comments effusively welcomed. (Edit: also removed Thursday’s query box from context).

# 2009-03-27

• Mike:

• Urs:

• created exact functor

• created filtrant category

• added to Higher Topos Theory more introductory/overview remarks which are supposed to be helpful for the newbie

• created Yoneda extension

• added section to Kan extension on formulas in terms of limits and colimits over comma categories;

• added a section on the “local” computation of adjoint functors at adjoint functor and point out how this induces the local/global dichotomy at limit, homotopy limit and Kan extension (see my previous modification below)

• if I noticed correctly, Mike had changed my original notation $p^* := - \circ p : [C',D] \to [C,D]$ for precomposition with a functor $p : C \to C'$ (pullback notation) at Kan extension to $p_*$ (pushforward notation). I have now added a section Remark on terminology: pushforward vs. pullback which is supposed to clarify this terminology issue.

Mike: That wasn’t me. I’m not sure that such a discussion belongs at Kan extension; it might belong somewhere but I would rather than the page Kan extension just pick one notation and possibly link to a discussion.

Zoran Škoda: It was me who changed, though I better did not. I am happy with the original notation as well. For as your discussion on pushfowards I am less happy. Namely, if one is not happy with the direction of maps between open sets, one just redefines what is a morphism of sites (opposite to the functor direction), so that the morphism of sites is always correct direction. So, unless one does not have strong feeling on the choice of pushfoward pullback meaning, what is not in this case, mayeb original notation just caring about covariant vs contravariant was better.

• addressed Zoran’s and Tim’s remarks at Kan extension: I have added now to Kan extension as well as to limit – in analogy to what we already had at homotopy limit – an explicit discussion of the difference between local and global definitions of the universal constructions

• created universal construction – but filled in just a question/query

• Tim: I have raised a query at Kan extension.

• Zoran Škoda: Created abelian category with multiple equivalent definitions.

• If people don't like having several entries in one day per Mike's request, another option (hopefully good enough for Mike) is to move your entire list up to the top when you add to it (being sure to add to the top of your list too).
• Finn has nothing to apologise for at context.
• Zoran and I are discussing terminology at projective limit.
• Zoran and Mike are discussing terminology at representable functor (I only made a more philosophical comment).
• Compare nice category of spaces with convenient category of topological spaces.
• I accept Mike's terminology at set theory.
• I refactored kernel to use primarily the equaliser definition in any pointed-enriched category.

# 2009-03-22

• Toby Bartels: I added a section on morphisms between contexts (the substitutions, or interpretations), including (as an example) a complete description of the category of contexts of the theory of a group. There is an exercise (to describe that category in group-theoretic terms) whose formatting all authors might want to look at.

# 2009-03-21

• Tim:
• I added a link to p-adic solenoid in shape theory as that example gives insights on the links between this area and dynamical systems.

# 2009-03-10

• Toby Bartels: I've written several more articles on very basic topics, such as those that used to be ‘?’-links below. You can see them on Recently Revised; I don't think that anything merits great attention.

# 2009-03-09

• Toby Bartels: I finally wrote relation, which makes me realise that there is no subset yet …. Also order, but that's just a list of links to more specific pages.

# 2009-03-08

• Toby Bartels: I tried to clarify the difference between a preorder (a structure on a given set that satisfies certain properties) and a proset (a set equipped with such a structure). I need to finish that for partial order/poset and total order/toset, although I would also entertain the idea that these should all be redirected one way or the other. But I got sidetracked writing linear order and loset instead. (And then there's quasiorder; I don't think that quoset is necessary for reasons that I don't want to get into here.)

• Attempted to answer Eric’s plea for a category-theoretic definition of ‘Hasse diagram’, in the discussion at the bottom of preorder. Unfortunately I don’t know the official definition of ‘Hasse diagram’ — though I know one when I see one.

• Made a short page on proset, since Toby seems to be using this as a synonym for preorder.

# 2009-03-05

• Continued debate on what to call “generalised object” since there seem to be some generalised objections (sorry).

# 2009-03-01

category: meta

Revised on August 23, 2015 22:29:59 by Toby Bartels (98.16.168.238)