Functor

Exact functor

Let 𝖢,𝖣 be finitely (co)complete categories. A functor 𝐹 :𝖢 →𝖣 is left exact iff it commutes with finite limits, cat i.e.

𝐹lim⟵⁡𝒟=lim⟵⁡𝐹𝒟

for any finite diagram 𝒟 :𝖩 →𝖢; and right exact iff it commutes with finite colimits, i,e,

𝐹lim⟶⁡𝒟=lim⟶⁡𝐹𝒟

for any finite diagram 𝒟 :𝖩 →𝖢. A functor which is both left and right exact is simply called an exact functor. A more general notion is Continuity and cocontinuity of functors.

The concept originated in the context of Exact functor on abelian categories.

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