+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Limits and colimits +-- {: .hide} [[!include infinity-limits - contents]] =-- =-- =-- \tableofcontents ## 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 [[over category|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 category|small]] cardinalities) if and only if it has (binary) [[pullbacks]] and [[cofiltered limits]]. * The [[saturated class of limits|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: +-- {: .num_prop #WidePbToComplete} ###### Proposition A [[category]] $C$ with all [[wide pullbacks]] and a [[terminal object]] $1$ is [[complete category|complete]]. If $C$ is complete and $F\colon C \to D$ preserves wide pullbacks and the terminal object, then it preserves all limits. =-- +-- {: .proof} ######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]] [[!include notions of pullback -- contents]] **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](http://math.ca/10.4153/CJM-1990-038-6) [[!redirects wide pullback]] [[!redirects wide pullbacks]] [[!redirects wide fiber product]] [[!redirects wide fiber products]] [[!redirects wide fibre product]] [[!redirects wide fibre products]] [[!redirects wide pullback type]] [[!redirects wide pullback types]] [[!redirects wide pushout]] [[!redirects wide pushouts]]