nLab wide pullback (Rev #15)

**limits and colimits** ## 1-Categorical * limit and colimit * limits and colimits by example * commutativity of limits and colimits * small limit * filtered colimit * directed colimit * sequential colimit * sifted colimit * connected limit, wide pullback * preserved limit, reflected limit, created limit * product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum * finite limit * exact functor * Kan extension * Yoneda extension * weighted limit * end and coend * fibered limit ## 2-Categorical * 2-limit * inserter * isoinserter * equifier * inverter * PIE-limit * 2-pullback, comma object ## (∞,1)-Categorical * (∞,1)-limit * (∞,1)-pullback * fiber sequence ### Model-categorical * homotopy Kan extension * homotopy limit * homotopy product * homotopy equalizer * homotopy fiber * mapping cone * homotopy pullback * homotopy totalization * homotopy end * homotopy colimit * homotopy coproduct * homotopy coequalizer * homotopy cofiber * mapping cocone * homotopy pushout * homotopy realization * homotopy coend

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Definition

A fibre product or fiber product is a product in a slice category $\mathcal{C} \downarrow C$ . The fibre product of two morphisms $f: A \to C$ , $g: B \to C$ is the same as their pullback; accordingly, a fiber product more than two morphisms $f_i : A_i \to C$ is often called a wide pullback.

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

\array{ A \times_C B &\to& B \\ \downarrow && \downarrow^g \\ A &\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.

You're English, David, so how come you don't like ‘fibre’? —Toby

I just thought it was policy here. That’s what the HowTo page says anyway. —David

That's only for page titles. (I know, because I wrote it!) Actually, now that we have a usable system of redirects, even that may not be so important. The English Wikipedia does fine with a policy only of including redirects for alternate spellings. —Toby

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.

Chris: Thanks, Urs. I’ll continue thinking about this. While I was at the coffee shop, I realized that to form $A \times_C B$ in a homotopically meaningful way, we should somehow resolve $B$ . So if $\mathcal{D}$ is our category and $\mathcal{D}_C$ is the category of objects over $C \in \mathcal{D}$ , then we have an adjoint pair of functors $R:= C\times? : \mathcal{D} \rightarrow \mathcal{D}_C$ and $L:=\mathcal{D}_C \rightarrow \mathcal{D}$ forgetting the map to $C$ . (Strange to me, but it seems that forgetting here is the left adjoint.) Then we can apply the monad $R L$ to $B \rightarrow C$ to get a cobar resolution of $B$ , take the product with $A$ , and so get the cosimplicial object $A \times C \times B, A \times C \times C \times B,...$ .

Perhaps this is the wrong section to discuss this, and when we figure it out, maybe we should make a section on fiber products of derived spaces.

Revision on July 4, 2010 at 13:45:44 by Todd Trimble See the history of this page for a list of all contributions to it.

nLab wide pullback (Rev #15)

Contents

Definition

Examples