DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-13

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Pullbacks and pushouts as limits and colimits of cospans and spans

Definition

For a cospan $X \overset{f}{\to} Z \overset{g}{\leftarrow} Y$ , a pullback is its limit (Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties). It consists of an object $X \times_{Z} Y$ and morphisms $p : X \times_{Z} Y \to X$ , $q : X \times_{Z} Y \to Y$ with $f p = g q$ , such that every $a : W \to X$ , $b : W \to Y$ satisfying $f a = g b$ admit a unique $u : W \to X \times_{Z} Y$ with $p u = a$ and $q u = b$ .

For a span $X \overset{f}{\leftarrow} Z \overset{g}{\to} Y$ , a pushout is its colimit. It consists of $X ⨿_{Z} Y$ and morphisms $i : X \to X ⨿_{Z} Y$ , $j : Y \to X ⨿_{Z} Y$ with $i f = j g$ , such that every compatible pair $a : X \to W$ , $b : Y \to W$ has a unique $u : X ⨿_{Z} Y \to W$ satisfying $u i = a$ and $u j = b$ .

Depends on

Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 8 results over 5 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

E. Riehl, Category Theory in Context, Definition 3.1.23 (standard reference, not scraped)