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The fibred product of two morphisms $f:\ X\ \longrightarrow\ S$ and $g:\ Y\ \longrightarrow\ S$ is often denoted $X\times_SY$, or sometimes notation like $X{\ }_f\hspace{-2pt}\times_gY$ is used if there is a chance of confusion about the morphisms.

Now I'm working with a fibred product of (finitely) many morphisms of the forms $f_i:\ X\ \longrightarrow\ S_i$ and $g_i:\ Y_i\ \longrightarrow\ S_i$. One way to denote this fibred product would be $$X\times_{S_1}Y_1\times_{S_2}Y_2\times_{S_3}\cdots\times_{S_n}Y_n,$$ but this bothers me because it makes it look like the fibred products $Y_i\times_{S_{i+1}}Y_{i+1}$ exists, which they don't. It also does not convey the idea that the $Y_i$ are all glued onto $X$. Is there a nice way to denote such a fibred product? Are there publications in which such fibred products show up, which have nice notation for them?

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    $\begingroup$ Is is $X\times_{S_1\times...\times S_n}(Y_1\times Y_2\times...\times Y_n)$ ? $\endgroup$ – Roland Jun 28 '17 at 17:26
  • $\begingroup$ I am not sure I understand your notation since $Y_i$ has no map to $S_{i+1}$. Do you mean $(...((X\times_{S_1} Y_1)\times_{S_2} Y_2)\times_{S_3}....)\times_{S_n} Y_n$ where the structural maps for the left factor of the products are given by the projection onto $X$ followed by $f_i:X\rightarrow S_i$ ? If that is the case, my above comment gives notation (which I believe is better) for the same thing. $\endgroup$ – Roland Jun 28 '17 at 21:38
  • $\begingroup$ Here is another one : put $X_i=X\times_{S_i} Y_i$, then the fiber product is $X\times_{X^n} X_i$. But I believe I have seen the first notation more often. $\endgroup$ – Roland Jun 29 '17 at 7:26
  • $\begingroup$ I forgot the \prod, the fiber product is $X\times_{X^n}\prod X_i$. $\endgroup$ – Roland Jun 29 '17 at 7:59
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    $\begingroup$ I just saw your comment and it makes me realize that you could also write it $X_1\times_X X_2\times_X ...\times_X X_n$. Or even if you are willing to change to the category $\mathcal{C}/X$, you can write $\prod X_i$. (As above $X_i=X\times_{S_i} Y_i$). That makes four notations you can choose from ;) I think all have their advantages and draw-backs. $\endgroup$ – Roland Jul 9 '17 at 11:35
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Summarizing the comments, there are a few options. In this case the preferred choice was $$X\times_{S_1\times...\times S_n}(Y_1\times Y_2\times...\times Y_n).$$ A slick alternative is to define $X_i=X\times_{S_i} Y_i$, then the fibred product can be written as $$X\times_{X^n}\prod X_i\qquad\text{ or }\qquad X_1\times_X X_2\times_X ...\times_X X_n.$$ A less elegant alternative is to write $$(...((X\times_{S_1} Y_1)\times_{S_2} Y_2)\times_{S_3}....)\times_{S_n} Y_n.$$

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I don't quite understand what you're looking for, but fiber products are the category-theorists generalization of intersection; in particular, intersection of subsets corresponds to the fiber product of their inclusions. Given this, I think $f \cap g$ is reasonably notation for the fiber product of $f$ and $g$, since it makes the following formula true: $$\mathrm{incl}_A \cap \mathrm{incl}_B = \mathrm{incl}_{A \cap B}.$$ In light of this, why not write $$\bigcap_{i \in I}f_i$$ for the indexed fiber product of $i \mapsto f_i$?

I'm not sure if this quite answers your question. From what I surmise, you're looking for the function $$\prod_{i \in I} f_i \cap g_i$$ where $\prod$, being functorial, can be applied to a family of functions. In particular, define: $$\left(\prod_{i \in I} f_i\right)(a) = \mathop{\pi}_{i \in I}(f_i(a_i))$$ where $\pi$ in this context refers to the infinitary ordered tuple constructor.

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