fibred category theory

Concepts

• Grothendieck fibration

• indexed category

• Grothendieck construction

• fibered limit

Theorems

• fibred Yoneda lemma?
• iterated fibrations?

Generalizations

• fibration in a 2-category

Applications

• categorical semantics of dependent type theory

limits and colimits

1-Categorical

• limit and colimit

  • limits and colimits by example

  • commutativity of limits and colimits

  • small limit

  • filtered colimit

    • directed colimit

      • sequential colimit

  • sifted colimit

  • connected limit, wide pullback

  • preserved limit, reflected limit, created limit

  • product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum

  • finite limit

    • exact functor

• Kan extension

  • Yoneda extension

• weighted limit

• end and coend

• fibered limit

2-Categorical

• 2-limit

  • inserter

  • isoinserter

  • equifier

  • inverter

  • PIE-limit

• 2-pullback, comma object

(∞,1)-Categorical

• (∞,1)-limit

  • (∞,1)-pullback

    • fiber sequence

Model-categorical

• homotopy Kan extension

• homotopy limit

  • homotopy product

  • homotopy equalizer

  • homotopy fiber

  • mapping cone

  • homotopy pullback

  • homotopy totalization

  • homotopy end

• homotopy colimit

  • homotopy coproduct

  • homotopy coequalizer

  • homotopy cofiber

  • mapping cocone

  • homotopy pushout

  • homotopy realization

  • homotopy coend