关于范畴具有局部始对象当且仅当存在余反射子群胚的等价性判定问询

Great question—this is a nice, non-trivial connection between two categorical properties, so let's unpack it step by step, starting with clear definitions to avoid confusion.

Key Definitions #

First, let's recap the critical terms to set the stage:

• Local initial objects: A category $C$ has local initial objects if every slice category $C/X$ (for each object $X \in C$) has an initial object.
  • Important distinction: The word "locally" here refers to slice categories (like in "locally cartesian closed"), not to the size of Hom-sets (the meaning behind "locally small"). This difference is key to avoiding mix-ups!
• Coreflective subcategory: A full subcategory $D \subseteq C$ is coreflective if there exists a functor $G: C \to D$ (called the coreflector) and a natural transformation $\eta_X: G(X) \to X$ such that for any morphism $f: Z \to X$ with $Z \in D$, there's a unique morphism $f': Z \to G(X)$ where $\eta_X \circ f' = f$.
• Groupoid: A category where every morphism is an isomorphism (every arrow has an inverse).

Equivalence Proof #

We'll verify both directions of the "if and only if" statement:

1. Only If: Local Initial Objects ⇒ Coreflective Groupoid Subcategory #

Suppose $C$ has local initial objects. For each $X \in C$, let $\eta_X: G(X) \to X$ be the initial object of the slice category $C/X$.

First, construct our subcategory $D$: define $D$ as the full subcategory of objects $Z \in C$ where $\eta_Z: G(Z) \to Z$ is an isomorphism (intuitively, these are objects where the slice category's initial object is just the identity morphism).

• $D$ is a groupoid: Take any morphism $g: Z \to W$ in $D$. Since $Z, W \in D$, $\eta_Z$ and $\eta_W$ are isomorphisms. By the initiality of $\eta_W$, there's a unique morphism $h: W \to Z$ such that $g \circ h = id_W$. Similarly, the initiality of $\eta_Z$ forces $h \circ g = id_Z$ (since $h \circ g$ must factor uniquely through the identity initial object of $C/Z$). Thus $g$ is invertible, so all morphisms in $D$ are isomorphisms.

• $D$ is coreflective: The functor $G: C \to D$ (mapping each $X$ to $G(X)$, and morphisms $f: X \to Y$ to the unique $G(f): G(X) \to G(Y)$ satisfying $\eta_Y \circ G(f) = f \circ \eta_X$) acts as the coreflector. The natural transformation $\eta_X: G(X) \to X$ satisfies the coreflection condition: for any $f: Z \to X$ with $Z \in D$, the initiality of $\eta_X$ gives a unique $f': Z \to G(X)$ such that $\eta_X \circ f' = f$.

2. If: Coreflective Groupoid Subcategory ⇒ Local Initial Objects #

Suppose $C$ has a coreflective subcategory $D$ (a groupoid) with coreflector $G: C \to D$ and natural transformation $\eta_X: G(X) \to X$.

We claim $\eta_X: G(X) \to X$ is the initial object of $C/X$. For any morphism $f: Y \to X$ in $C$, the coreflection property guarantees a unique morphism $f': Y \to G(X)$ such that $\eta_X \circ f' = f$. This is exactly the defining property of an initial object in $C/X$: every morphism into $X$ factors uniquely through $\eta_X$. Thus every slice category has an initial object, so $C$ has local initial objects.

Edge Case Check #

• Empty category: Vacuously has local initial objects, and its only coreflective subcategory is itself (a groupoid), so the equivalence holds here too.

备注：内容来源于stack exchange，提问作者Geoffrey Trang