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

Depends on

• Constant diagrams, cones, cocones, and their morphisms
• Initial object, terminal object, and zero object

Used by

• A category satisfying the explicit SAFT intersection hypotheses is cocomplete Corollary
• A colimit of a set-valued functor is the set of connected components of its category of elements Corollary
• Absolute colimits Definition
• Equalizers and coequalizers as limits and colimits of a parallel pair Definition
• Filtered categories and filtered colimits Definition
• Finite, small, and large limits and colimits; complete and cocomplete categories Definition
• Lim one obstruction to completeness Definition
• Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors Definition
• Products and coproducts as limits and colimits of discrete diagrams, including their existence-and-uniqueness equations Definition
• Pullbacks and pushouts as limits and colimits of cospans and spans Definition
• Strong convergence of a spectral sequence Definition
• Fubini checked by hand on a product of two walking arrows Example
• The colimit of an increasing chain of sets is its union Example
• The twisted arrow category of the walking arrow is a cospan Example
• FALSE: every functor preserves the ends that exist in its domain False statement
• FALSE: every weighted limit is the ordinary limit of the diagram it weights False statement
• FALSE: under this page's convention a coend is the colimit of the same twisted-arrow diagram whose limit is the end False statement
• A comma-category projection strictly creates the limits preserved by the functor Lemma
• Countable sequence groups and tail filtrations Lemma
• Stable homotopy colimits are independent of a cofinal tail Lemma
• The constant sheaf is the sheaf of locally constant functions Lemma
• The legs of a limiting cone are jointly monic, and the legs of a colimiting cocone are jointly epic Lemma
• Wide pullbacks compute intersections of supplied set-indexed subobject representatives independently of the representatives Lemma
• A limiting cone for a diagram is exactly a colimiting cocone for the formally dual diagram in the opposite category Proposition
• Limits of empty diagrams are terminal objects, and colimits of empty diagrams are initial objects Proposition
• The end of a functor made mute in its contravariant variable is the ordinary limit of that functor Proposition
• A complete locally small category with a jointly weakly initial set has an initial object, without class-indexed choice Theorem
• A complete locally small category with a small coseparating set and intersections of all subobject collections has an initial object Theorem
• A pointwise Kan extension along a fully faithful functor genuinely extends the original functor Theorem
• A power by a set is the product of that many copies and a copower is the coproduct Theorem
• A reflective inclusion creates every ambient limit in the ordinary isomorphism-invariant sense Theorem
• A reflective subcategory has every ambient colimit, obtained by reflecting an ambient colimit Theorem
• A weighted limit is an ordinary limit over the category of elements of the weight, and a weighted colimit an ordinary colimit over it Theorem
• An end is a limit over the twisted arrow category, and a coend is a colimit over its opposite Theorem
• An end is the equalizer of two products, and a coend the coequalizer of two coproducts Theorem
• Any two limits, or any two colimits, of one diagram are uniquely isomorphic compatibly with their structure maps Theorem
• Assuming Choice, precomposition with a final functor does not change colimits, and precomposition with an initial functor does not change limits Theorem
• Chosen limits and colimits are adjoint to the diagonal functor Theorem
• Chosen limits and colimits of a fixed small shape assemble into limit and colimit functors Theorem
• Comma-category limit and colimit formulae compute Kan extensions Theorem

…and 13 more results.

Dependency tree · two levels

7 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, Definitions 3.1.6 and 3.1.11 (standard reference, not scraped)