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user57432
user57432

Motivated by this answer, I first tried to conceptualize the concrete reflector as a reflector which is a concrete functor. But unfortunately this is not the case. Then I tried to conceptualize the concrete reflector as a reflector which is a concrete functor that also preserves the information that "the underlying $\mathbf{X}$-objects of the domain and codomain of a $\mathbf{A}$-reflection arrow for a $\mathbf{B}$-object $B$ is same". But frankly, this is just reinterpreting the definition in a different language and hence I am not satisfied with this and think that there must be some deep reason for doingadding this condition.

While searching for a motivation for the reason for introducing (2), I stumbled upon J. Fiadeiro's book Categories for Software Engineering. There it is written that,

Intuitively, for the (co)reflection to be "concrete", i.e. to be consistent with the classification that the underlying functor provides, we would like to remain within the same fibre. That is, we would like that the (co)reflection arrows be identities.

But I didn't understand this comment since I don't even have a vague intuitive idea for a concrete functor (the answers to the question that I linked above only focuses on the big-picture viewpoint of the notion of concrete isomorphism).

Motivated by this answer, I first tried to conceptualize the concrete reflector as a reflector which is a concrete functor. But unfortunately this is not the case. Then I tried to conceptualize the concrete reflector as a reflector which is a concrete functor that also preserves the information that "the underlying $\mathbf{X}$-objects of the domain and codomain of a $\mathbf{A}$-reflection arrow for a $\mathbf{B}$-object $B$ is same". But frankly, this is just reinterpreting the definition in a different language and hence I am not satisfied with this and think that there must be some deep reason for doing this.

Motivated by this answer, I first tried to conceptualize the concrete reflector as a reflector which is a concrete functor. But unfortunately this is not the case. Then I tried to conceptualize the concrete reflector as a reflector which is a concrete functor that also preserves the information that "the underlying $\mathbf{X}$-objects of the domain and codomain of a $\mathbf{A}$-reflection arrow for a $\mathbf{B}$-object $B$ is same". But frankly, this is just reinterpreting the definition in a different language and hence I am not satisfied with this and think that there must be some deep reason for adding this condition.

While searching for a motivation for the reason for introducing (2), I stumbled upon J. Fiadeiro's book Categories for Software Engineering. There it is written that,

Intuitively, for the (co)reflection to be "concrete", i.e. to be consistent with the classification that the underlying functor provides, we would like to remain within the same fibre. That is, we would like that the (co)reflection arrows be identities.

But I didn't understand this comment since I don't even have a vague intuitive idea for a concrete functor (the answers to the question that I linked above only focuses on the big-picture viewpoint of the notion of concrete isomorphism).

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user57432
user57432

Understanding the reason for the particular formulation of the definition of a concrete reflector (as stated in The Joy of Cats)

This question is essentially a followup of this question. But before going into the question let me introduce the relevant definitions as given in The Joy of Cats.

Definition 1. Let $\bf{X}$ be a category. A concrete category over $\bf{X}$ is a pair $({\bf{A}},U)$, where $\bf{A}$ is a category and $U :{\bf{A}} \to X$ is a faithful functor.

Definition 2. If $({\bf{A}},U)$ and $({\bf{B}}, V )$ are concrete categories over $\bf{X}$ , then a concrete functor from $({\bf{A}},U)$ and $({\bf{B}}, V )$ is a functor $F : {\bf{A}}\to {\bf{B}}$ with $U = V \circ F$. We denote such a functor by $F : ({\bf{A}},U)\to ({\bf{B}}, V )$.

Definition 3. If $({\bf{A}},U)$ and $({\bf{B}}, V )$ are concrete categories over $\bf{X}$ , then a concrete functor from $F:({\bf{A}},U)\to ({\bf{B}}, V )$ is said to be a concrete isomorphism iff $F:{\bf{A}}\to{\bf{B}}$ is an isomorphism.

Definition 4. If $({\bf{A}},U)$ and $({\bf{B}}, V )$ are concrete categories over $\bf{X}$ and $E:\mathbf{A}\hookrightarrow\mathbf{B}$ be the inclusion functor. Then $(\mathbf{A},U)$ is called a concrete subcategory of $(\mathbf{B},V)$ if $U=V\circ E$.

Definition 5. Let $({\bf{A}},U)$ and $({\bf{B}}, V )$ are concrete categories over $\bf{X}$ such that $({\bf{A}},U)$ is a concrete subcategory of $({\bf{B}},V)$. Then $({\bf{A}},U)$ is said to be a concretely reflective subcategory of $({\bf{B}},V)$ if

(1) for each $\mathbf{B}$-object $B$ there exists a $\mathbf{B}$-morphism $r:B\to A$ (where $A$ is an $\mathbf{A}$-object) such that for any $\mathbf{A}$-object $A'$ and any $\mathbf{B}$-morphism $f:B\to A'$, there exists an unique $\mathbf{A}$-morphism $g$ such that $g\circ r=f$. These $r$'s are called $\mathbf{A}$-reflection arrows for $B$'s.

(2) for each such $r$ we have $V(B)=V(A)$ and $V(r)=\text{id}_{V(A)}$.

Definition 6. Let $({\bf{A}},U)$ and $({\bf{B}}, V )$ are concrete categories over $\bf{X}$ such that $({\bf{A}},U)$ is a concrete reflective subcategory of $({\bf{B}},V)$. Then the reflector fuctor thus induced is called a concrete reflector.

If $\mathbf{A}$ and $\mathbf{B}$ be two categories and $\mathbf{A}$ is a subcategory of $\mathbf{B}$, then $\mathbf{A}$ is called a reflective subcategory of $\mathbf{B}$ if (1) of the above is satisfied. In this terminology Definition 5 simply says that if $({\bf{A}},U)$ and $({\bf{B}}, V )$ are concrete categories over $\bf{X}$ such that $({\bf{A}},U)$ is a concretely reflective subcategory of $({\bf{B}},V)$ then in particular $\mathbf{A}$ is a reflective subcategory of $\mathbf{B}$.

My Attempt

Motivated by this answer, I first tried to conceptualize the concrete reflector as a reflector which is a concrete functor. But unfortunately this is not the case. Then I tried to conceptualize the concrete reflector as a reflector which is a concrete functor that also preserves the information that "the underlying $\mathbf{X}$-objects of the domain and codomain of a $\mathbf{A}$-reflection arrow for a $\mathbf{B}$-object $B$ is same". But frankly, this is just reinterpreting the definition in a different language and hence I am not satisfied with this and think that there must be some deep reason for doing this.

Question

I am trying to understand the reason for adding (2). What is the motivation for this?