Category Theory/Subcategories

Definition (subcategory):

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

Definition (full):

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

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

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

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

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

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

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

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