A wide pullback 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 coterminal morphisms $f_i\colon A_i \to C$ 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)
Last revised on December 2, 2020 at 12:34:47. See the history of this page for a list of all contributions to it.