Let $f:A\to C\leftarrow B:g$ be morphisms in a category. There exists in literature a useful notation for the morphisms $\bar f:A\times_C B\to B$ and $\bar g:A\times_C B\to A$ in terms of $f$ and $g$?
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top
Sometimes these morphisms are called projections, and denoted $p_A$ and $p_B.$ This is motivated by the example of the category of sets and set maps where we can describe the pullback as follows:
Here the map $A\times_CB\to A$ takes $(a,b)$ to $a$, and $A\times_CB\to B$ takes $(a,b)$ to $b$, so in fact the two maps are projections.
Well, $A \times_C B \to B$ is the pullback of $f$ along $g$. Therefore it is often denoted as $f^*$ or (especially in algebraic geometry) as $f_B$.