nLab wide pullback (Rev #5, changes)

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For $f : A \to C$ and $g : B \to C$ two morphisms in a category $C$ , the fiber product $A \times_C B$ of $A$ with $B$ over $C$ is, if it exists, the pullback

\begin{matrix} A \times_{C} B & \to & B \\ ↓ & ↓ ↓^{g} \\ A & \to \overset{f}{\to} & C \end{matrix} .

\array{ A \times_C B &\to& B \\ \downarrow && ~~\downarrow~~ \downarrow^g \\ A ~~&\to&~~ &\stackrel{f}{\to}& C } \,.

This term comes from thinking of $A$ and $B$ as bundles over $C$ ; then the fiber of $A \times_C B$ over a generalized element $x$ of $C$ is the product of the fibers of $A$ and $B$ over $x$ . In other words, the fiber product is the product taken fiber-wise.

Of course, the fiber of $A$ at the generalized element $x: I \to C$ is itself a pullback $I \times_C A$ ; the terminology depends on your point of view.

Examples

In $C =$ Set the fiber product is given by the usual formula

A \times_C B = \left\{ (a,b) \in A \times B \;|\; f(a) = g(b) \right\} \,.

Chris: I’d appreciate it if someone could add an explanation of fiber products as limits of compsimplicial diagrams $A \times B, A \times C \times B, A \times C \times C \times A,...$ , making the coface maps explicit.

Urs: I think it should work like this:

the first two coface maps are in full detail given by

A\times B \stackrel{(Id_A \circ p_1) \times (f\circ p_1) \times (Id_B \circ p_2)}{\to} A \times C \times B

and

A\times B \stackrel{(Id_A \circ p_1) \times (g\circ p_2) \times (Id_B \circ p_2)}{\to} A \times C \times B

With that it is clear that the fiber product $A \times_C B$ is the equalizer of these two maps, i.e. the limit over the diagram

A \times B \stackrel{\to}{\to} A \times C \times B \,.

I am guessing, but haven’t really checked in detail that the further coface maps continue this: the “inner” ones come from the diagonal $C \to C \times C$ and the outer ones from $f$ and $g$ as above. So for instance the next three would be

A \times C \times B \stackrel{(Id_A \circ p_1) \times (f \circ p_1) \times ( Id_C\circ p_2) \times (Id_B \circ p_3)}{\to} A \times C \times C \times B

and

A \times C \times B \stackrel{(Id_A \circ p_1) \times (Id_C \times Id_C \circ p_2) \times (Id_B \circ p_3)}{\to} A \times C \times C \times B

and

A \times C \times B \stackrel{(Id_A \circ p_1) \times (Id_C \circ p_2) \times ( g \circ p_3) \times (Id_B \circ p_3)}{\to} A \times C \times C \times B

This way, unless I am making a mistake, the ordinary limit over the diagram

A \times B \stackrel{\to}{\to} A \times C \times B \stackrel{\to}{\stackrel{\to}{\to}} A \times C \times C \times B

would still be $A \times_C B$ , but the homotopy limit would pick up the right first degree correction for the homotopy fiber product.

Revision on May 18, 2009 at 14:32:19 by Urs Schreiber See the history of this page for a list of all contributions to it.