# Category Theory/Subcategories

Definition (subcategory):

Let ${\displaystyle {\mathcal {C}}}$ be a category. Then a subcategory ${\displaystyle {\mathcal {D}}=(\operatorname {Ob} ({\mathcal {D}}),\operatorname {Mor} ({\mathcal {D}}))}$ of ${\displaystyle {\mathcal {C}}}$ is a category such that ${\displaystyle \operatorname {Ob} ({\mathcal {D}})\subseteq \operatorname {Ob} ({\mathcal {C}})}$ and ${\displaystyle \operatorname {Mor} ({\mathcal {D}})\subseteq \operatorname {Mor} ({\mathcal {C}})}$.

Definition (full):

A subcategory ${\displaystyle {\mathcal {D}}}$ of a category ${\displaystyle {\mathcal {C}}}$ is called full iff for all ${\displaystyle a,b\in {\mathcal {D}}}$, we have

${\displaystyle \operatorname {Hom} _{\mathcal {D}}(a,b)=\operatorname {Hom} _{\mathcal {C}}(a,b)}$.

Proposition (limits are preserved when restricting to a full subcategory):

Let ${\displaystyle {\mathcal {C}}}$ be a category, let ${\displaystyle J:{\mathcal {J}}\to {\mathcal {C}}}$ be a diagram in ${\displaystyle {\mathcal {C}}}$, and let ${\displaystyle {\mathcal {D}}}$ be a full subcategory of ${\displaystyle {\mathcal {C}}}$. Suppose that ${\displaystyle (L,(\phi _{\alpha })_{\alpha \in A})}$ is a limit over ${\displaystyle J}$ in ${\displaystyle {\mathcal {C}}}$ such that ${\displaystyle L}$ and all targets of the ${\displaystyle \phi _{\alpha }}$ are in ${\displaystyle {\mathcal {D}}}$. Then ${\displaystyle (L,(\phi _{\alpha })_{\alpha \in A})}$ is a limit over ${\displaystyle J}$ in ${\displaystyle {\mathcal {D}}}$.

Proof: Certainly, the underlying cone of ${\displaystyle (L,(\phi _{\alpha })_{\alpha \in A})}$ is contained within ${\displaystyle {\mathcal {D}}}$, because the subcategory is full. Now let another cone ${\displaystyle (Q,(\psi _{\alpha })_{\alpha \in A})}$ in ${\displaystyle {\mathcal {D}}}$ over the diagram ${\displaystyle J}$ (which, analogously, is a diagram in ${\displaystyle {\mathcal {D}}}$) be given. By the universal property of ${\displaystyle (L,(\phi _{\alpha })_{\alpha \in A})}$ in ${\displaystyle {\mathcal {C}}}$, there exists a unique morphism ${\displaystyle f:Q\to L}$ which satisfies ${\displaystyle \psi _{\alpha }=\phi _{\alpha }\circ f}$ for all ${\displaystyle \alpha \in A}$. Since ${\displaystyle {\mathcal {D}}}$ is full, ${\displaystyle f}$ is in ${\displaystyle {\mathcal {D}}}$. ${\displaystyle \Box }$

Analogously, we have:

Proposition (colimits are preserved when restricting to a full subcategory):

Let ${\displaystyle {\mathcal {C}}}$ be a category, let ${\displaystyle J:{\mathcal {J}}\to {\mathcal {C}}}$ be a diagram in ${\displaystyle {\mathcal {C}}}$, and let ${\displaystyle {\mathcal {D}}}$ be a full subcategory of ${\displaystyle {\mathcal {C}}}$. Suppose that ${\displaystyle (C,(\phi _{\alpha })_{\alpha \in A})}$ is a colimit over ${\displaystyle J}$ in ${\displaystyle {\mathcal {C}}}$ such that ${\displaystyle C}$ and all domains of the ${\displaystyle \phi _{\alpha }}$ are in ${\displaystyle {\mathcal {D}}}$. Then ${\displaystyle (C,(\phi _{\alpha })_{\alpha \in A})}$ is a colimit over ${\displaystyle J}$ in ${\displaystyle {\mathcal {D}}}$.

Proof: This follows from its "dual" proposition, reversing all arrows in its statement and proof except the direction of the diagram functor. ${\displaystyle \Box }$