# projection functors [closed]

For categories $\mathcal C$ and $\mathcal D$, there are functors $\Pi_1 : \mathcal C \times \mathcal D \to \mathcal C$ and $\Pi_2 : \mathcal C \times \mathcal D \to \mathcal D$ defined as follows:\begin{align}\Pi_1(A, B) & = A & \Pi_2(A, B) & = B\\\Pi_1(f, g) & = f & \Pi_2(f, g) & = g\end{align}If $\mathcal C$ and $\mathcal D$ are small, $\Pi_1$ and $\Pi_2$ are projections in $\mathbf{Cat}$.

My question is whether the notation $\Pi_1$/$\Pi_2$ is standard, and whether there is an established name for these functors. So far, I have called them projection functors.

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"Projection functor" is the standard name, yes. The $\Pi_i$ notation is very common, but even so I'd probably say something like "the projection functor $\Pi_1$" on first use, to make it clear. Question is probably more suitable for math.stackexchange.com, but anyway, that's it answered. –  Tom Leinster Sep 5 '12 at 9:59
Thanks a lot. Could you maybe repost this as an answer instead of a comment, so that I can “accept” it? –  Wolfgang Jeltsch Sep 5 '12 at 11:44
I agree this is more suitable for math.SE. But Tom is right, although I haven't seen capital pis used very often for projections of any sort, more often I see lowercase ones. –  Mike Shulman Sep 6 '12 at 14:20
On reflection I agree, Mike. With capital pis, there's a risk of confusion with the big-pi-like symbol used for a product. –  Tom Leinster Sep 15 '12 at 16:12