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.
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:
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.
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
From products and equalizers, we can get arbitrary limits.
Notions of pullback:
pullback, fiber product (limit over a cospan)
lax pullback, comma object (lax limit over a cospan)
(β,1)-pullback, homotopy pullback, ((β,1)-limit over a cospan)
Analogues in dependent type theory:
The terminology wide pullback appears in:
Wide pullbacks are considered under the term fibred product in:
Last revised on February 10, 2024 at 23:24:18. See the history of this page for a list of all contributions to it.