I am currently taking a course in algebraic topology, which also covers a lot of category theory. My question is pretty straightforward:
How do you visualize the (homotopy) pull back of a diagram $B\to C\leftarrow A$ ?
In class the professor usually just kind of says "Well if you think about it you get....". I have figured out that for a (homotopy) push out you want to glue the two spaces together along their common points but the description of the pullback has eluded me.
The standard construction of the homotopy pullback involves the total path space $C^I$ (i.e. the space of all continuous maps $I \to C$, with the appropriate topology). More precisely, if $f : A \to C$ and $g : B \to C$ are the given maps, then: $$A \times^\mathrm{h}_C B = \{ (a, p, b) \in A \times C^I \times B : p(0) = f(a), p(1) = g(b) \}$$
The path space $C^I$ is rather difficult to visualise, as it is generally infinite dimensional. But the idea should be clear enough: a point of $A \times^\mathrm{h}_C B$ is a point in $A$ and a point in $B$ together with a path in $C$ connecting the images of those two points.
Notice that there is a canonical map $A \times_C B \to A \times^\mathrm{h}_C B$. Under good conditions, this is a homotopy equivalence. Indeed, if $g : B \to C$ is a Hurewicz fibration, then the homotopy extension/lifting property gives us a map $(A \times^\mathrm{h}_C B) \times I \to B$ extending the canonical map $(A \times^\mathrm{h}_C B) \times \{ 1 \} \to B$ and lifting the canonical map $(A \times^\mathrm{h}_C B) \times I \to C$; restricting to $(A \times^\mathrm{h}_C B) \times \{ 0 \} \to B$ we then get a commutative square $$\begin{array}{ccc} (A \times^\mathrm{h}_C B) & \rightarrow & B \\ \downarrow & & \downarrow \\ A & \rightarrow & C \end{array}$$ and hence a map $A \times^\mathrm{h}_C B \to A \times_C B$ that is left inverse to the canonical map $A \times_C B \to A \times^\mathrm{h}_C B$; in fact, this map $A \times^\mathrm{h}_C B \to A \times_C B$ is also a homotopy right inverse. Thus, when $g : B \to C$ is a Hurewicz fibration, the homotopy pullback coincides with the ordinary pullback (up to homotopy equivalence).