Pullbacks and pushouts as limits and colimits of cospans and spans

Depends on

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

Used by

• A monotone functor between poset categories preserves every monomorphism but need not preserve pullbacks Counterexample
• Base change of classical varieties when the pullback exists Definition
• Direct and inverse image of a subobject Definition
• Pullback and pushout of an extension Definition
• Slice categories, composition, and pullback along a morphism Definition
• Subobject classifier Definition
• The one-step generator extension functor Definition
• A pullback in Top is the fibre product with the subspace topology inherited from the product Example
• A weighted limit computing a kernel pair Example
• Pullbacks in Set are fibre products and pushouts are quotients of tagged disjoint unions Example
• The twisted arrow category of the walking arrow is a cospan Example
• FALSE: every weighted limit is the ordinary limit of the diagram it weights False statement
• A pullback of a monomorphism is a monomorphism, and a pushout of an epimorphism is an epimorphism Lemma
• Direct and inverse image satisfy Beck–Chevalley for pullback squares of sets Lemma
• A cartesian square over an epimorphism is also cocartesian Theorem
• A locally cartesian closed category has pullbacks, and with a terminal object it has all finite limits Theorem
• A square is cartesian exactly when a short sequence is exact Theorem
• Epimorphy is detected by members Theorem
• Exactness is detected by members Theorem
• Finite, nonempty finite, and connected finite (co)limit criteria in terms of products, equalizers, pullbacks, terminal objects, and their duals Theorem
• Member equivalence is transitive Theorem
• Pullback and pushout pasting, with cancellation of the square adjacent to the outer edge Theorem
• Schanuel's lemma in an abelian category Theorem
• The connecting morphism exists and is unique Theorem
• The meet of two subobjects is their pullback Theorem

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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