The Stacks project

Definition 6.15.1. A type of algebraic structure is given by a category $\mathcal{C}$ and a functor $F : \mathcal{C} \to \textit{Sets}$ with the following properties

  1. $F$ is faithful,

  2. $\mathcal{C}$ has limits and $F$ commutes with limits,

  3. $\mathcal{C}$ has filtered colimits and $F$ commutes with them, and

  4. $F$ reflects isomorphisms.

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