nLab
wide pullback

Contents

Definition

A wide pullback in a category π’ž\mathcal{C} is a product (of arbitrary cardinality) in a slice category π’žβ†“C\mathcal{C} \downarrow C. In terms of π’ž\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 coterminal morphisms f i:A i→Cf_i\colon A_i \to C is an object PP equipped with projection p i:P→A ip_i\colon P\to A_i such that f ip if_i p_i is independent of ii, 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

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

Proposition

A category CC with all wide pullbacks and a terminal object 11 is complete. If CC is complete and F:C→DF: C \to D preserves wide pullbacks and the terminal object, then it preserves all limits.

Proof

To build up arbitrary products ∏ i∈Ic i\prod_{i \in I} c_i in CC, take the wide pullback of the family c iβ†’1c_i \to 1. Then to build equalizers of diagrams f,g:cβ†’β†’df, g: c \stackrel{\to}{\to} d, construct the pullback of the diagram

d ↓δ c β†’βŸ¨f,g⟩ dΓ—d\array{ & & d \\ & & \downarrow \delta \\ c & \underset{\langle f, g \rangle}{\to} & d \times d }

From products and equalizers, we can get arbitrary limits.

References

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

Revised on January 4, 2013 21:16:50 by Mike Shulman (108.225.239.218)