Applications of Pushouts I’m new to category theory and I’m studying by myself. There’s an exercise from course notes I’m following that asks to find the object $X$ such that the diagram
$$
  \require{AMScd}
  \begin{CD}
    \{ \ast \} @>>> V_1  \\
    @VVV            @VVV \\
    V_2        @>>> X
  \end{CD}
$$
is a pushout (I’m in the category of vector spaces).
I believe that $X = V_{1} \oplus V_{2}$. Also, in $\mathrm{Top}$, pushouts can be used to construct spheres from disks, what’s the ‘meaning’ of this particular pushout?
 A: You are correct, $X=V_1\oplus V_2$. To see this, suppose that for some vector space $V $ we have $f_1$ and $f_2$ morphisms which make the square commute. Then clearly the map $f:X\to V $ given by $f(u)=f_1 (\pi_1 (u))+f_2 (\pi_2 (u)) $, where $\pi_i $ is the projection onto $V_i $, will make the diagram commute.
Suppose that we have another morphism $g $ which makes our diagram commute. Then for $i_j:V_j\to X $ the inclusion map, $g\circ i_j=f_j $. Now, by the definition of $X $ we have that any $u\in X $ is such that $u=i_1 (\pi_1 (u))+i_2 (\pi_2 (u))$. Applying $g $ to both sides and using the fact that $g $ is linear then gives that $g=f $, as desired.
Finally, I am guessing you are using an inductive definition to get spheres, so that $S^k $ mapping into the boundary of $D^k $ pushes out to $S^{k+1} $. This construction just means that we can get a sphere by "gluing" two disks of lower deminsion along their boundary, so that the disks are the hemispheres of the sphere. This way of thinking about spheres is useful when studying algebraic topology, since it gives you a CW structure, and a solid geometric connection to lower deminsional spheres as well.
A: A common occurrence in topology is to take a pushout of two inclusions $Y \hookleftarrow A \hookrightarrow X$.  So, $A$ lives a double life, as a subspace $X$ and as a subspace of $Y$.  In this scenario the pushout is always going to be a quotient space of the coproduct/disjoint union $X \amalg Y$.  After forming the coproduct, start gluing the points from $A$ together (points in $A$ show up twice $X \amalg Y$, so identify them one-by-one).  The resulting quotient space is the pushout.  You can check that the universal mapping properties of the quotient exactly say what the universal mapping property of the pushout should require.  
As one particular case, taking $A$ to be a circle included as the boundary of two disks gives you $S^2$ as the pushout, and this goes through in higher dimensions the same way.   
