wide pullback in nLab
Context
Limits and colimits
1-Categorical
2-Categorical
(∞,1)-Categorical
Model-categorical
Contents
Definition
A wide pullback or wide fiber product or wide fibre product in a category 𝒞\mathcal{C} is a product (of arbitrary cardinality) in a slice category 𝒞↓C\mathcal{C} \downarrow C. In terms of 𝒞\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:A i→Cf_i\colon A_i \to C (a wide cospan) 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
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A category has wide pullbacks (of all small cardinalities) if and only if it has (binary) pullbacks and cofiltered limits.
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The saturation of the class of wide pullbacks is the class of limits over categories CC whose fundamental groupoid Π 1(C)\Pi_1(C) is trivial.
On the other hand, together with a terminal object, wide pullbacks generate 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 i→1c_i \to 1. Then to build equalizers of diagrams f,g:c⇉df, g\colon c \rightrightarrows d, construct the pullback of the diagram
d ↓δ c →⟨f,g⟩ d×d\array{ & & d \\ & & \downarrow \delta \\ c & \underset{\langle f, g \rangle}{\to} & d \times d }
From products and equalizers, we can get arbitrary limits.
Notions of pullback:
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pullback, fiber product (limit over a cospan)
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lax pullback, comma object (lax limit over a cospan)
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(∞,1)-pullback, homotopy pullback, ((∞,1)-limit over a cospan)
Analogues in dependent type theory:
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
Last revised on February 10, 2024 at 23:24:18. See the history of this page for a list of all contributions to it.