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Though this might not be what you are expecting, I will explain you "oidification" or horizontal categorification as I understood (Experts are fell free to add or edit as necessary). This is the process that generalizes a "certain type of category with a single object" to "such type of categories with multiple objects". This is done mostly via "enriching" the initial category $\mathcal{C}$ over another monoidal category $\mathcal{K},$ which roughly says homsets (set of arrows two objects) of $\mathcal{C}$ are replaced by objects of $\mathcal{K}.$

Examples include

You can find more details ringoid, and algebroid here. But as far as I know Hopf algebroids and Lie algebroids does not fit into this general definition of algebroids, but still multi-object generalizations of their counterparts.

Though this might not be what you are expecting, I will explain you "oidification" or horizontal categorification as I understood (Experts are fell free to add or edit as necessary). This is the process that generalizes a "certain type of category with a single object" to "such type of categories with multiple objects". This is done mostly via "enriching" the initial category $\mathcal{C}$ over another monoidal category $\mathcal{K},$ which roughly says homsets (set of arrows two objects) of $\mathcal{C}$ are replaced by objects of $\mathcal{K}.$

Examples include

You can find more details ringoid, and algebroid here. But as far as I know Hopf algebroids and Lie algebroids does not fit into this general definition of algebroids, but still multi-object generalizations of their counterparts.

Also, it should worth remind that not every enrichment of a categories consider as an "odification". For example (Lawvere) generalized metric spaces is a category enriched in the monoidal poset category $([0,\infty],\ge),$ where the monoidal product is taken to be addition.

Though this might not be what you are expecting, I will explain you "oidification" or horizontal categorification as I understood (Experts are fell free to add or edit as necessary). This is the process that generalizes a "certain type of category with a single object" to "such type of categories with multiple objects". This is done mostly via "enriching" the initial category $\mathcal{C}$ over another monoidal category $\mathcal{K},$ which roughly says homsets (set of arrows two objects) of $\mathcal{C}$ are replaced by objects of $\mathcal{K}.$

Examples include \begin{array}{|l|l|l| } X & X\text{-oid} & \text{Enrichment}\\ \hline \text{monoid} & \text{Category} & \text{ categories enriched over } \mathbf{Set} \\ \text{Category} & 2\text{-Category} & \text{ categories enriched over } \mathbf{Cat}\\ \text{Group} & \text{Groupoid} \\ \text{Ring} & \text{Ringoid} & \text{category enriched in tensor category } \mathbf{Ab}\\ \text{Quantale} & \text{Quantaloid} & \text{category enriched in suplattices}\\ \text{Algebra} & \text{Algebroid} & \text {category enriched in } \mathbf{Vect} \text{ or } R\mathbf{Mod} \\ C^{\ast}\text{-algebra} & C^{\ast}\text{-category} & \ast\text{-category enriched in } \mathbf{Ban} \end{array} You can find more details ringoid, and algebroid here. But as far as I know Hopf algebroids and Lie algebroids does not fit into this general definition of algebroids, but still multi-object generalizations of their counterparts.

Though this might not be what you are expecting, I will explain you "oidification" or horizontal categorification as I understood (Experts are fell free to add or edit as necessary). This is the process that generalizes a "certain type of category with a single object" to "such type of categories with multiple objects". This is done mostly via "enriching" the initial category $\mathcal{C}$ over another monoidal category $\mathcal{K},$ which roughly says homsets (set of arrows two objects) of $\mathcal{C}$ are replaced by objects of $\mathcal{K}.$

Examples include

You can find more details ringoid, and algebroid here. But as far as I know Hopf algebroids and Lie algebroids does not fit into this general definition of algebroids, but still multi-object generalizations of their counterparts.

Though this might not be what you are expecting, I will explain you "oidification" or horizontal categorification as I understood (Experts are fell free to add or edit as necessary). This is the process that generalizes a "certain type of category with a single object" to "such type of categories with multiple objects". This is done mostly via "enriching" the initial category $\mathcal{C}$ over another monoidal category $\mathcal{K},$ which roughly says homsets (set of arrows two objects) of $\mathcal{C}$ are replaced by objects of $\mathcal{K}.$

Examples include \begin{array}{|l|l|l| } X & X\text{-oid} & \text{Enrichment}\\ \hline \text{monoid} & \text{Category} & \text{ categories enriched over } \mathbf{Set} \\ \text{Category} & 2\text{-Category} & \text{ categories enriched over } \mathbf{Cat}\\ \text{Group} & \text{Groupoid} \\ \text{Ring} & \text{Ringoid} & \text{category enriched in tensor category } \mathbf{Ab}\\ \text{Algebra} & \text{Algebroid} & \text {category enriched in } \mathbf{Vect} \text{ or } R\mathbf{Mod} \\ C^{\ast}\text{-algebra} & C^{\ast}\text{-category} & \ast\text{-category enriched in } \mathbf{Ban} \end{array} You can find more details ringoid, and algebroid here. But as far as I know Hopf algebroids and Lie algebroids does not fit into this general definition of algebroids, but still multi-object generalizations of their counterparts.

Though this might not be what you are expecting, I will explain you "oidification" or horizontal categorification as I understood (Experts are fell free to add or edit as necessary). This is the process that generalizes a "certain type of category with a single object" to "such type of categories with multiple objects". This is done mostly via "enriching" the initial category $\mathcal{C}$ over another monoidal category $\mathcal{K},$ which roughly says homsets (set of arrows two objects) of $\mathcal{C}$ are replaced by objects of $\mathcal{K}.$

Examples include \begin{array}{|l|l|l| } X & X\text{-oid} & \text{Enrichment}\\ \hline \text{monoid} & \text{Category} & \text{ categories enriched over } \mathbf{Set} \\ \text{Category} & 2\text{-Category} & \text{ categories enriched over } \mathbf{Cat}\\ \text{Group} & \text{Groupoid} \\ \text{Ring} & \text{Ringoid} & \text{category enriched in tensor category } \mathbf{Ab}\\ \text{Quantale} & \text{Quantaloid} & \text{category enriched in suplattices}\\ \text{Algebra} & \text{Algebroid} & \text {category enriched in } \mathbf{Vect} \text{ or } R\mathbf{Mod} \\ C^{\ast}\text{-algebra} & C^{\ast}\text{-category} & \ast\text{-category enriched in } \mathbf{Ban} \end{array} You can find more details ringoid, and algebroid here. But as far as I know Hopf algebroids and Lie algebroids does not fit into this general definition of algebroids, but still multi-object generalizations of their counterparts.