nLab wide pullback

Definition

A wide pullback or wide fiber product or wide fibre product in a category $\mathcal{C}$ is a product (of arbitrary cardinality) in a slice category $\mathcal{C} \downarrow C$. In terms of $\mathcal{C}$, this can be expressed as a limit over a category obtained from a discrete category by adjoining a terminal object.

Yet more explicitly, the wide pullback of a family of morphisms $f_i\colon A_i \to C$ (a wide cospan) is an object $P$ equipped with projection $p_i\colon P\to A_i$ such that $f_i p_i$ is independent of $i$, and which is universal with this property.

Binary wide pullbacks are the same as ordinary pullbacks, a.k.a. fiber products.

Of course, a wide pushout is a wide pullback in the opposite category.

Properties

• A category has wide pullbacks (of all small cardinalities) if and only if it has (binary) pullbacks and cofiltered limits.

• The saturation of the class of wide pullbacks is the class of limits over categories $C$ whose fundamental groupoid $\Pi_1(C)$ is trivial.

On the other hand, together with a terminal object, wide pullbacks generate all limits:

Proposition

A category $C$ with all wide pullbacks and a terminal object $1$ is complete. If $C$ is complete and $F\colon C \to D$ preserves wide pullbacks and the terminal object, then it preserves all limits.

Proof

To build up arbitrary products $\prod_{i \in I} c_i$ in $C$, take the wide pullback of the family $c_i \to 1$. Then to build equalizers of diagrams $f, g\colon c \rightrightarrows d$, construct the pullback of the diagram

$$\array{ & & d \\ & & \downarrow \delta \\ c & \underset{\langle f, g \rangle}{\to} & d \times d }$$

From products and equalizers, we can get arbitrary limits.

Related concepts

• connected limit

Notions of pullback:

• pullback, fiber product (limit over a cospan)

• wide pullback

• lax pullback, comma object (lax limit over a cospan)

• (∞,1)-pullback, homotopy pullback, ((∞,1)-limit over a cospan)

• base change, context extension

• pullback bundle

• contravariant functor

  • pullback in cohomology, d-invariant

  • pullback of differential forms

  • pullback of a distribution

Analogues in dependent type theory:

• wide pullback type

• wide pushout type

References

The terminology wide pullback appears in:

• Paul Taylor, Quantitative domains, groupoids and linear logic, Category Theory and Computer Science: Manchester, UK, September 5–8, 1989 Proceedings. Springer Berlin Heidelberg, 1989.

Wide pullbacks are considered under the term fibred product in:

• Robert Paré, Simply connected limits. Can. J. Math., Vol. XLH, No. 4, 1990, pp. 731-746, CMS