1
$\begingroup$

In homotopy theory, the mapping cone of a continuous map $f\colon X \to Y$ is the homotopy pushout over the following span:

$$ \require{AMScd} \begin{CD} X @>{f}>> Y\\ @VVV \\ \{*\} \end{CD} $$

I.e., it is universal among all squares of the form $$ \begin{CD} X @>{f}>> Y\\ @VVV @VVV\\ \{*\}@>>> Z \end{CD} $$ where the square commutes up to homotopy.

But what is a good name for such an object $Z$? Normally, I would call it a cocone, but I would rather not use the word cone to mean two different things.

Square and cospan are possibilities, but they seem a bit too general: I want to refer specifically to cocones for the first diagram.

Is there a good alternative word?

$\endgroup$
  • $\begingroup$ 'Span' is probably not good. Of course it's unlikely to conflict with the linear-algebraic terminology, but it already has meaning in the category-theoretic context: n-Lab. $\endgroup$ – LSpice Dec 16 '17 at 0:03
  • $\begingroup$ @LSpice That's my mistake - I meant 'cospan' (over the objects $\{*\}$ and $Y$). $\endgroup$ – John Gowers Dec 16 '17 at 0:41
  • $\begingroup$ Sure, but my impression was that '(co)span' was the name for two arrows among three objects, not four arrows among four objects. If you're familiar with the useage, though, then I trust your sense of it more than mine. $\endgroup$ – LSpice Dec 16 '17 at 1:28
  • 3
    $\begingroup$ Often $Z$ is called a "cofiber" of the map $f$ -- is this terminology not suitable? $\endgroup$ – Tyler Lawson Dec 16 '17 at 2:25
  • $\begingroup$ Why not just cone of (or over) $f$? $\endgroup$ – მამუკა ჯიბლაძე Dec 16 '17 at 6:24

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.