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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 18th 2010
   • (edited May 18th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexHere is some construction that I am thinking about, and of which I would like to know, and hence to ask, if possibly it is being considered elsewhere, possibly even has some name and with luck even some theory attached to it. Or else, if I am just hallucinating. My input is an essential geometric morphism $(f_! \dashv f^* \dashv f_*) : C \leftrightarrow D$ and for any object $c \in C$ I consider the "cokernel" $(f^* f_! c )/c$ of the unit, by wich I mean the pushout $$ \array{ c &\to& * \\ \downarrow && \downarrow \\ f^* f_! c &\to& (f^* f_! c)/c } \,, $$ and similarly the "kernel" $(f^* f_* c)^c$ of the counit, by which I mean the pullback $$ \array{ (f^* f_*c )^c &\to& f^*f_*c \\ \downarrow && \downarrow \\ \coprod_a * &\to& c } \,, $$ where the coproduct in the bottom left runs over the global elements I think these two constructions extend to functors that themselves form an adjoint pair. $$ (f^* f_!(-)/(-) \dashv (f^* f_*(-))^{(-)} : C \to C \,. $$ Does this kind of construction ring a bell with anyone?

   Here is some construction that I am thinking about, and of which I would like to know, and hence to ask, if possibly it is being considered elsewhere, possibly even has some name and with luck even some theory attached to it. Or else, if I am just hallucinating.

   My input is an essential geometric morphism $(f_! \dashv f^* \dashv f_*) : C \leftrightarrow D$ and for any object $c \in C$ I consider the “cokernel” $(f^* f_! c )/c$ of the unit, by wich I mean the pushout

   $$\array{ c &\to& * \\ \downarrow && \downarrow \\ f^* f_! c &\to& (f^* f_! c)/c } \,,$$

   and similarly the “kernel” $(f^* f_* c)^c$ of the counit, by which I mean the pullback

   $$\array{ (f^* f_*c )^c &\to& f^*f_*c \\ \downarrow && \downarrow \\ \coprod_a * &\to& c } \,,$$

   where the coproduct in the bottom left runs over the global elements

   I think these two constructions extend to functors that themselves form an adjoint pair.

   $$(f^* f_!(-)/(-) \dashv (f^* f_*(-))^{(-)} : C \to C \,.$$

   Does this kind of construction ring a bell with anyone?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeMay 18th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexIt says your comment is invalid XML, and of course I can't make heads or tails of the source code display.

   It says your comment is invalid XML, and of course I can’t make heads or tails of the source code display.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 18th 2010
   • (edited May 18th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> It says your comment is invalid XML, and of course I can't make heads or tails of the source code display. Hm, that's strange. For me the entry displays correctly. So that also means I don't know what I might have to fix...

   It says your comment is invalid XML, and of course I can’t make heads or tails of the source code display.

   Hm, that’s strange. For me the entry displays correctly. So that also means I don’t know what I might have to fix…

4. • CommentRowNumber4.
   • CommentAuthorAndrew Stacey
   • CommentTimeMay 18th 2010
   • PermaLink
   Author: Andrew Stacey
   Format: MarkdownItex(That generally means that it's _me_ that has to fix something. Apparently, one isn't allowed to put newlines in the "title" or "alt" attributes of an image. Now I know.)

   (That generally means that it’s me that has to fix something. Apparently, one isn’t allowed to put newlines in the “title” or “alt” attributes of an image. Now I know.)

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 19th 2010
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexIt displays fine for me in Firefox.

   It displays fine for me in Firefox.

6. • CommentRowNumber6.
   • CommentAuthorAndrew Stacey
   • CommentTimeMay 19th 2010
   • PermaLink
   Author: Andrew Stacey
   Format: MarkdownItex(To clarify - with apologies for the distraction - the problem was with the _pictorial_ representation of the mathematics. So those who see the MathML - in particular, those who use firefox - wouldn't have seen anything wrong. People using a [substandard browser](http://www.google.com/chrome) see pictures and then would have been hit by the bug, which was putting invalid text in the "alt" and "title" attributes.)

   (To clarify - with apologies for the distraction - the problem was with the pictorial representation of the mathematics. So those who see the MathML - in particular, those who use firefox - wouldn’t have seen anything wrong. People using a substandard browser see pictures and then would have been hit by the bug, which was putting invalid text in the “alt” and “title” attributes.)

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeMay 19th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexI kept thinking if I should maybe switch to Chrome. Somebody told me that the speed increase compared to Firefox is remarkable.

   I kept thinking if I should maybe switch to Chrome. Somebody told me that the speed increase compared to Firefox is remarkable.

8. • CommentRowNumber8.
   • CommentAuthorAndrew Stacey
   • CommentTimeMay 19th 2010
   • PermaLink
   Author: Andrew Stacey
   Format: MarkdownItexI would wait until it supports MathML. Otherwise, the nLab is going to become a little confusing for you!

   I would wait until it supports MathML. Otherwise, the nLab is going to become a little confusing for you!

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeMay 19th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI use Firefox for MathML sites (the nLab, the nCafe) and Chrome for everything else. That's not just because Chrome is faster (especially with things like Gmail) but because Firefox has a tendency to crash my computer for some reason.

   I use Firefox for MathML sites (the nLab, the nCafe) and Chrome for everything else. That’s not just because Chrome is faster (especially with things like Gmail) but because Firefox has a tendency to crash my computer for some reason.

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 19th 2010
    • PermaLink
    Author: Urs
    Format: MarkdownItex> because Firefox has a tendency to crash my computer for some reason. Yes. It used to wok fine on my system, until one of these "security updates" installed itself. Often I think back to the times when software developers had other aspects than "security" on their mind. I understand that it's necessary, but it has a general bad effect on IT people, I think. I used to have to deal with an IT group who would enjoy covering up incompetence with important-sounding security talk. "No, we can't do that for you, it would be too much of a security risk on our system." Okay, that's off topic. On with something more serious...

    because Firefox has a tendency to crash my computer for some reason.

    Yes. It used to wok fine on my system, until one of these “security updates” installed itself.

    Often I think back to the times when software developers had other aspects than “security” on their mind. I understand that it’s necessary, but it has a general bad effect on IT people, I think. I used to have to deal with an IT group who would enjoy covering up incompetence with important-sounding security talk. “No, we can’t do that for you, it would be too much of a security risk on our system.”

    Okay, that’s off topic. On with something more serious…