I'm on the faculty of Wellesley College, and I'm a Research Affiliate at M.I.T.
by Bill Dwyer, Myself, Dan Kan, and Jeff Smith.
This was previously titled ``Model Categories and More General Abstract Homotopy Theory, The Next Generation'', which was itself a rewrite of what was called ``Model Categories and More General Abstract Homotopy Theory''. It discusses homotopy colimits, homotopy limits, and other homotopical universal constructions in a ``homotopical category'', i.e., a category with a class of ``weak equivalences'', but not necessarily having classes of cofibrations or of fibrations.
This has been published by the AMS, and is available directly from the AMS.
This was until recently known as ``Localization of Model Categories'', and before that it was called ``Localization, Cellularization, and Homotopy Colimits''.
We define the class of ``cellular model categories'' (which includes almost all the model categories I know about) and construct the ``Bousfield localization'' of a cellular model category with respect to an arbitrary set of maps.
This requires quite a bit of work with model categories, some of it new, and some of it known to a select few but generally inaccessible to newcomers. We present the definitions and ideas of model categories from first principles, giving complete arguments in an attempt to make this accessible to those with no experience in working with model categories. (Whether we've been successful is another question, but we've made the attempt.) We've tried to separate out the parts of proofs that are really generalities about model categories and added those to Part 2. Thus, Part 2 is a comprehensive reference for many ideas used in doing homotopy theory in model categories.
This has been published by the AMS, and is available directly from the AMS.
Errata, as of August 1, 2018: pdf file.
If M is a model category and Z is an object of M, then there are model category structures on the category of objects of M over Z and the category of objects of M under Z under which a map is a cofibration, fibration, or weak equivalence if and only if its image in M under the forgetful functor is, respectively, a cofibration, fibration, or weak equivalence. It is asserted without proof in "Model Categories and Their Localizations" that if M is cofibrantly generated, cellular, or proper, then so is the category of objects of M over Z. The purpose of this brief note is to fill in the proofs of those assertions and to state and prove the analogous results for undercategories.
Here are:
We show that the functor that takes a multicosimplicial object to its diagonal cosimplicial object is a right Quillen functor. This implies that the diagonal of a Reedy fibrant multicosimplicial object is a Reedy fibrant cosimplicial object, which has applications to the calculus of functors. We also show that, although the diagonal functor is a Quillen functor, it is not a Quillen equivalence for multicosimplicial spaces.
We also discuss total objects and homotopy limits of multicosimplicial objects. We show that the total object of a multicosimplicial object is isomorphic to the total object of the diagonal, and that the diagonal embedding of the cosimplicial indexing category into the multicosimplicial indexing category is homotopy left cofinal, which implies that the homotopy limits are weakly equivalent if the multicosimplicial object is at least objectwise fibrant.
Here's a pdf file for the version of August 28, 2015.
This is a draft of a work in progress.
This is a careful proof that the Quillen model category of topological spaces satisfies the axioms for a model category. We do not assume any familiarity with model categories.
We construct a functorial on the nose (not just up to homotopy) CW-approximation to a space. This construction preserves subspaces, and so it also defines a relative functorial CW-approximation (for pairs of spaces) and commutes with intersctions of subspaces of a fixes space.
This is a careful presentation of an elementary proof of the well known theorem that each homotopy group (or, in degree zero, pointed set) of the inverse limit of a tower of fibrations maps naturally onto the inverse limit of the homotopy groups (or, in degree zero, pointed sets) of the spaces in the tower, with kernel naturally isomorphic to lim1 of the tower of homotopy groups of one dimension higher.
We present an elementary proof of the well known theorem that the inverse limit of a levelwise weak equivalence of towers of fibrations of topological spaces is a weak equivalence. This follows, of course, from the short exact sequence for a homotopy group of the inverse limit, but this proof avoids any mention of lim1.
This is a classroom note giving a careful explanation of the statement that a colimit is a left adjoint to the constant diagram functor, in the special case of pushouts.
This is a classroom note giving a careful proof that in the Puppe sequence constructed from a map f from X to Y by repeatedly taking mapping cones, the map between the spaces that are homotopy equivalent to the suspension of X and the suspension of Y is homotopy equivalent to the negative of the suspension of f.
This is an unpublished manuscript of Chris Reedy from around 1974 that's been circulating as an increasingly faded photocopy. It's been typed into LaTeX, and the author has given permission for it to be posted on the net.
This is a LaTeX documentclass that (along with its user manual, examdoc.tex) tries to make it easy for even a LaTeX novice to prepare exams.
Specifically, exam.cls sets the page layout so that there are one inch margins all around and provides commands that make it easy to format questions (and parts of questions, and subparts of parts, and subsubparts of subparts), assign point values to the questions (or parts, etc.), create grading tables, create very flexible headers and footers, modify the margins, and include solutions that are printed only if you include the command to print them. The user manual was written in an attempt to make this all seem simple, even to the inexperienced LaTeX user.
Latest official version: Version 2.7, dated February 26,
2021.
To run LaTeX on examdoc.tex, you need exam.cls. Either put exam.cls into the LaTeX inputs directory somewhere or just keep it in the current directory. If you've already got an older copy of exam.cls somewhere on your system (e.g., if you have TeXLive installed), then put this newer version into the current directory to try it out.
Betatest version: Version 2.703beta of exam.cls, dated February 12, 2023, and the md5sum of that file.
Changes since version 2.5:
We modify the definition of \colorfbox to eliminate the warning
Package xcolor Warning: Incompatible color definition
when the user has loaded xcolor.sty instead of color.sty. We fixed this by having \colorfbox check whether the user has loaded xcolor.sty or color.sty, and then using the appropriate command when saving the current color.
We modify the changes in version 2.701$\beta$ so that they only have effect if the user gives the command \solutionsreseteqcounter, so that the default behavior is identical to the behavior before version 2.701$\beta$. This can be cancelled at any point by giving the command \nosolutionsreseteqcounter. There is also a new documentclass option ``solutionsreseteqcounter'', which is equivalent to giving the command \solutionsreseteqcounter at the beginning of the file.
All solution environments (solution, solutionorbox, solutionorlines, solutionordottedlines, solutionorgrid, and solutionbox) now save the current value of the equation counter and then reset the equation counter to zero, so that equations numbers start from one within each solution, and then restore the saved value of the equation counter at the end of the environment. This corrects the problemn that if an equation anvironment appeared inside of a solution, it would affect the equation numbers appearing later in the document, even if solutions weren't being printed. Note: This was modified in version 2.701$\beta$ so that it only takes effect if the user gives the command \solutionsreseteqcounter, and so the default behavior is identical to the behavior before version 2.701$\beta$.
No longer betatest.
We fixed a bug, apparently introduced in version 2.602, that affected vertical point tables (both regular and bonus, but not combined), shifting the total points in the table towards the right of the cell, rather than having the total points centered. The fix was to delete an \hspace*{\fill} that had no business being there.
We introduced a new command, \unstarredvspace, that changes the solution environment so that, when solutions are not being printed, any optional space called for is inserted with \vspace{amount}, instead of \vspace*{amount}. There is also a new command \nounstarredvpsace that reverts to the default.
We corrected the bug introduced in Version 2.604$\beta$ that caused a \ref to a \correctchoice in a choices environment to have the wrong value. The correction was to change \stepcounter to \refstepcounter.
We changed the code for the \correctchoice command in both the checkboxes environment and the choices environment to correct a bug that caused the item label for the second item to appear in the wrong place when the first item is a \correctchoice and there's no text in between the \begin{checkboxes} and the \correctchoice (or in between the \begin{choices} and the \correctchoice).
No longer betatest.
We changed the code for multicolumn grade and point tables to remove the incompatibility with colortbl.sty and other packages that load colortbl.sty (e.g., xcolor.sty with the "table" option).
We changed some command and environment names so that exam.cls is now compatible with framed.sty and packages that load framed.sty (e.g., minted.sty).
No longer betatest.
Bugfix: We changed \@setheadheight and \@setfootheight to fix a bug that was introduced by the bugfix in version 2.306beta, 2009/03/28: If the second page has a different \textheight (because of a change in either headheight or footheight between pages 1 and 2), then page 2 would use the \textheight of page 1. Pages 3 and beyond would get the correct \textheight. The original version of this set \@colroom and \vsize to the new \textheight, but that had a bug in that if a float appeared at the top of a page, there would be no notice taken of the space lost to the float, and so the text would overrun the bottom of the page. The bugfix in version 2.306beta eliminated the changes to \@colht and \vsize. In this bugfix, we adjust \@colroom, \@colht, and \vsize in the same way that we adjust \textheight.
Multicolumn grade and point tables, and a new syntax for multirow grade and point tables (which were introduced in version 2.508beta).
Multicolumn tables are vertically oriented, while multirow tables are horizontally oriented, and so they do not take the optional argument choosing between horizontal and vertical. They all take one required argument specifying the number of columns (for multicolumn) or the number of rows (for multirow).
The tables can be:
grade tables or point tables,
plain, bonus, or combined,
indexed by questions or by pages,
complete or partial.
As usual, if you omit the optional argument that chooses between questions and pages, you get questions.
The new commands are:
\def\multirowgradetable{numrows}[questions or pages]
\def\multirowpointtable{numrows}[questions or pages]
\def\multirowbonusgradetable{numrows}[questions or pages]
\def\multirowbonuspointtable{numrows}[questions or pages]
\def\multirowcombinedgradetable{numrows}[questions or pages]
\def\multirowcombinedpointtable{numrows}[questions or pages]
\def\multirowpartialgradetable{numrows}{rangename}[questions or pages]
\def\multirowpartialpointtable{numrows}{rangename}[questions or pages]
\def\multirowpartialbonusgradetable{numrows}{rangename}[questions or pages]
\def\multirowpartialbonuspointtable{numrows}{rangename}[questions or pages]
\def\multirowpartialcombinedgradetable{numrows}{rangename}[questions or pages]
\def\multirowpartialcombinedpointtable{numrows}{rangename}[questions or pages]
\def\multicolumngradetable{numcols}[questions or pages]
\def\multicolumnpointtable{numcols}[questions or pages]
\def\multicolumnbonusgradetable{numcols}[questions or pages]
\def\multicolumnbonuspointtable{numcols}[questions or pages]
\def\multicolumncombinedgradetable{numcols}[questions or pages]
\def\multicolumncombinedpointtable{numcols}[questions or pages]
\def\multicolumnpartialgradetable{numcols}{rangename}[questions or pages]
\def\multicolumnpartialpointtable{numcols}{rangename}[questions or pages]
\def\multicolumnpartialbonusgradetable{numcols}{rangename}[questions or pages]
\def\multicolumnpartialbonuspointtable{numcols}{rangename}[questions or pages]
\def\multicolumnpartialcombinedgradetable{numcols}{rangename}[questions or pages]
\def\multicolumnpartialcombinedpointtable{numcols}{rangename}[questions or pages]
The older grade and point table commands can still be used. For example, the commands
\gradetable[h][questions]
\multirowgradetable{1}[questions]
are equivalent.
The distance between the rows of a multirow table and between the columns of a multicolumn table is \doublerulesep, the default value of which is 2.0pt. You can change that using a \setlength command, as in
\setlength{\doublerulesep}{0.5in}
New commands: Multirow grade and point tables.
These are all horizontally oriented tables, and so do not take the optional argument choosing between horizontal and vertical. They all take one required argument specifying the number of columns, which is the number of columns used for the point values (including the total), but not counting the column of row headings.
Note: The syntax was changed in version 2.509beta, so that you now specify the number of *rows* rather than the number of *columns*! For example, the first command below should now be
\multirowgradetable{numrows}[questions or pages]
The tables can be:
grade tables or point tables,
plain, bonus, or combined,
indexed by questions or by pages,
complete or partial.
The new commands are:
\multirowgradetable{numcols}[questions or pages]
\multirowpointtable{numcols}[questions or pages]
\multirowbonusgradetable{numcols}[questions or pages]
\multirowbonuspointtable{numcols}[questions or pages]
\multirowcombinedgradetable{numcols}[questions or pages]
\multirowcombinedpointtable{numcols}[questions or pages]
\multirowpartialgradetable{numcols}{rangename}[questions or pages]
\multirowpartialpointtable{numcols}{rangename}[questions or pages]
\multirowpartialbonusgradetable{numcols}{rangename}[questions or pages]
\multirowpartialbonuspointtable{numcols}{rangename}[questions or pages]
\multirowpartialcombinedgradetable{numcols}{rangename}[questions or pages]
\multirowpartialcombinedpointtable{numcols}{rangename}[questions or pages]
New commands:
\pointstwosided
\pointstwosidedreversed
The first causes points to be in the right margin on odd numbered pages and in the left margin on even numbered pages.
The second causes points to be in the left margin on odd numbered pages and in the right margin on even numbered pages.
We fixed a bug in the choices and checkboxes environments that arose only when \CorrectChoiceEmphasis used color. If it did, and if the text of a correct choice exactly filled a line, and if there was no blank line in the latex file separating this correct choice from the following choice, there would be an extra blank line inserted after the correct choice. We did this by inserting \color@begingroup and \color@endgroup as needed. (We're pretty sure the actual fix was the \endgraf in the expansion of \color@endgroup.)
We fixed a bug in the solutionbox environment that caused enumerate, itemize, or description environments to have their text stick into the right margin. We did this by resetting \@totalleftmargin and \linewidth in the box containing the solution.
Both the lines created with the \fillwithlines command and the dotted lines created with the \fillwithdottedlines command can now be printed in color. To do this, you must give the command
\usepackage{color}
in your preamble, and then you can use the new commands below.
New commands:
\colorfillwithlines
\colorfillwithdottedlines
The first causes the lines drawn by the \fillwithlines command to be drawn in color. The default color is set by the command
\definecolor{FillWithLinesColor}{gray}{0.8}
and the color can be changed by giving a new \definecolor command. You can return to black lines by giving the command
\nocolorfillwithlines
\colorfillwithdottedlines causes the lines drawn by the \fillwithdottedlines command to be drawn in color. The default color is set by the command
\definecolor{FillWithDottedLinesColor}{gray}{0.8}
and the color can be changed by giving a new \definecolor command. You can return to black dotted lines by giving the command
\nocolorfillwithdottedlines
The command
\colorsolutionboxes
that was created in version 2.501beta now affects not only the boxes created by \solutionbox, but also by \makeemptybox, \solutionorbox, and all of the boxes printed by all of the various solution environments when solutions are being printed surrounded by a box.
Changed the \solutionbox environment so that it works correctly inside a tabular.
Also: The \solutionbox frame can now be printed in color, as long as you load color.sty in the preamble.
Usage: Say
\usepackage{color}
in the preamble, and then give the command
\colorsolutionboxes
to have the frame around a solutionbox in color. The default color was created by the command
\definecolor{SolutionBoxColor}{gray}{0.8}
and you can change the color by giving a new \definecolor command (which must be done *after* the \colorsolutionboxes command).
To cancel color solutionbox frames and return to black, give the command
\nocolorsolutionboxes
"Getting up and running with AMS-LaTeX" (contained in the file amshelp.tex) is a primer on using AMS-LaTeX. It's intended for people with at least some experience using TeX, AMS-TeX, or LaTeX.
This is an attempt to get you up and running with AmS-LaTeX as quickly as possible. These instructions (along with the template file template.tex) won't be a substitute for the full documentation, but they may get you started quickly enough so that you'll only need to refer to the main documentation occasionally.
In addition to descriptions of the basics of AMS-LaTeX, there are sections with careful descriptions of the various environments for displayed mathematics, how to use Xy-pic to draw commutative diagrams, and how to use amsrefs to create a bibliography.
This is the rewrite for the current version of AMS-LaTeX (version 2.2) of the instructions I originally wrote for an early version of AMS-LaTeX. This is version 2.3, dated January 28, 2013.
Here are