Subcategory

definition example

If $\mathcal{C}$ is a Category, a subcategory $\mathcal{D}$ consists of a subset ${\mathcal{D}}_0\subseteq {\mathcal{C}}_0$ of objects and, for each $x,y\in {\mathcal{D}}_0$, a subset ${\mathrm{Hom}}_{\mathcal{D}}(x,y)\subseteq {\mathrm{Hom}}_{\mathcal{C}}(x,y)$ such that all identities are in $\mathcal{D}$ and composites of morphisms in $\mathcal{D}$ are in $\mathcal{D}$. $\mathcal{D}$ is wide if ${\mathcal{D}}_0={\mathcal{C}}_0$ and full if ${\mathrm{Hom}}_{\mathcal{D}}(x,y)={\mathrm{Hom}}_{\mathcal{C}}(x,y)$ for all $x,y\in {\mathcal{D}}_0$; the only wide full subcategory is $\mathcal{C}$ itself.

Sources: Kittenlab Lecture 5; DaoFP §9.5 (full subcategory on a weakly terminal set); 7 Sketches Example 3.74 (abelian groups in groups); CTfS Example 4.1.1.7, Definition 4.6.3.1, Example 4.6.3.2, Remark 4.6.3.3, Example 4.6.3.4

Subtlety (Kittenlab). $\mathsf{Preorder}$ is a full subcategory of $\mathbf{Cat}$ — but a preorder-as-(set, relation) is not literally a thin category; there is only an injective Functor into $\mathbf{Cat}$. So the categorical notion of subobject is generalized: a subcategory of $\mathcal{C}$ is any category with an injective (faithful, injective-on-objects) functor into $\mathcal{C}$, and “we should not distinguish between isomorphic objects”. Everyone then follows convention and ignores the pedantry. Compare Subobject and Monomorphism.

Full subcategories (CTfS Definition 4.6.3.1): given a set $X$ of objects of $\mathcal{C}$, keep all morphisms between them. CTfS’s list: finite sets in sets, the sets $\underline{n}$ in finite sets, the linear orders $[n]$ in finite linear orders ($\Delta \subseteq \mathbf{FLin}$), groups in monoids, monoids in categories, sets in graphs (as discrete graphs), partial and linear orders in preorders, discrete and indiscrete categories in $\mathbf{Cat}$. A subcategory is just an injective-on-objects-and-arrows functor, full iff the functor is full (Full and Faithful Functor); the full subcategory on $X$ is even a fiber product $\mathcal{C}{\times }_{\mathrm{Ind}(\mathrm{Ob}\mathcal{C})}\mathrm{Ind}(X)$ of categories (CTfS Example 4.6.3.4). Being full is a real choice: for preorders with all joins, the full subcategory of $\mathbf{Preord}$ and the (non-full) subcategory of join-preserving maps are both reasonable (CTfS Example 4.1.1.7).

Examples. $\mathbf{FinSet}$ $\subseteq \mathbf{Set}$ (full); $\mathbf{Preord}$ $\subseteq \mathbf{Cat}$ (full); $\mathbf{Ab}\subseteq \mathbf{Grp}$ (full, with a left adjoint — a reflective subcategory); injections form a wide non-full subcategory of $\mathbf{Set}$; the Yoneda Embedding exhibits $\mathcal{C}$ as a full subcategory of presheaves (representables).

Lean

#check CategoryTheory.FullSubcategory      -- FullSubcategory (Z : C → Prop)
#check CategoryTheory.InducedCategory      -- induced category along a function
#check CategoryTheory.Functor.Faithful
#check CategoryTheory.Functor.Full