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Definition

Subcategory

Category Theory · category.tex
Let $\mathcal{C}$ be a category (Definition~Category). A subcategory $\mathcal{D}$ of $\mathcal{C}$ consists of
  1. a collection of objects $$ \operatorname{Ob}(\mathcal{D}) \subseteq \operatorname{Ob}(\mathcal{C}) $$
  2. for each pair of objects $X,Y \in \operatorname{Ob}(\mathcal{D})$, a subset $$ \Hom_{\mathcal{D}}(X,Y) \subseteq \Hom_{\mathcal{C}}(X,Y) $$
such that
  1. identity morphisms from Definition~Category belong to $\mathcal{D}$,
  2. composition in $\mathcal{D}$ is the restriction of composition in $\mathcal{C}$.
Depends on
Dependency Graph
flowchart LR classDef current fill:#6366f1,color:#fff,stroke:#4f46e5 nf2c6bdf5["Category"] na02613f0["Subcategory"]:::current nf2c6bdf5 --> na02613f0 click nf2c6bdf5 "../objects/f2c6bdf5.html" "_self"