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?

  • $\begingroup$ '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

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