Reading through Joy of cats. I am stuck at the reflective sub-categories. I can't understand how a reflective sub-category be not full (I couldn't understand the last two examples of the section for the same reason). Suppose, a function $f: A \rightarrow A' \notin Hom_X(A, A')$ where $X$ is a subcategory of $Y$ and $A, A' \in X$ then for $B \in Y$ every morphism $g: B \rightarrow A'$ has to have a reflection, lets call it $h$ now $f \circ h \in Y$ but if $f \notin X$ $A$ would't be a reflection now right?
How can a reflective subcategory be non full
category-theory
Related Solutions
What's happening is that the Joy of Cats' notion of being limit-closed is a strict version of what other people mean by limit-closed. This is because their notion of being limit-closed is that the embedding functor uniquely lift/create limits, but their version of creating and lifting limits are the strict versions of these notions (https://ncatlab.org/nlab/show/created+limit#strictness and https://ncatlab.org/nlab/show/lifted+limit).
In fact, ignoring the first sentence of the proof of Proposition 13.27 and replacing the last sentence with an appeal to full embeddings reflecting limits (Example 13.23(3)), yields a proof that a full reflective subcategory is always limit-closed in the non-strict sense, i.e. that the embedding functor lifts limits (non-strictly). Adding back in the last sentence then shows that the subcategory is additionally strictly limit-closed if is isomorphism-closed. The first sentence asserts the converse, but follows from observation in the last section of the chapter: "Creation and reflection of isomorphisms".
In detail, Definition 13.17 has a functor lifting limits (uniquely) of a certain diagram meaning that every limiting cone of the image of the diagram is equal to the image of a (unique) limiting cone of the diagram. The non-strict version is that every limiting cone over the image of a diagram is isomorphic to the image of a limiting cone over the diagram.
The non-strict version of lifting in particular requires that the functor detect limits (as in Definition 13.22), i.e. if the image of the diagram has a limiting cone, then the diagram has a limiting cone. Morever, in that case the limiting cone of the diagram is isomorphic to any other limiting cone, whence the limiting cone of the image of the diagram being isomorphic to the image of a limiting cone over the diagram is equivalent to the image of any limiting cone being a limiting cone.
In other words, the non-strict version of a functor lifting limits can be expressed in the terminology of the Joy of Cats as the functor detecting and preserving limits.
Now a functor creates limits (in the sense of Joy of Cats) if it lifts limits (strictly and uniquely) and reflects them (this is the part labeled as "obvious" in the proof of Proposition 13.25 because it is merely a rewording of the definition). The non-strict version of creating limits is then obtained by using the non-strict version of lifting limits. Thus using the Joy of Cats terminology, the non-strict version of creating can be phrased as detecting, preserveing, and reflecting.
With the above terminology, Remark 13.26 survives: embeddings of full subcategories reflect limits, hence lift limits (strictly or not) if they creat them (strictly or not).
The proof of Proposition 13.27 with the first sentence removed and the last sentence changed to an appeal to reflecting of limits then starts with a a limiting cone of an image of a diagram and shows it isomorphic to an image of a limiting cone of the diagram. In other words, it exactly shows that embeddings of full reflective subcategories lift and hence create limits in the non-strict sense, i.e. that the categories are limit-closed in the non-strict sense (which is how other people use the term limit-closed).
Next, putting the last sentence back in shows that if the subcategory is isomorphism-closed, then the original limiting cone is equal to the image of a limiting cone, i.e. that limits are lifted and created in the strict sense.
The converse, that being strict limit-closed implies being isomorpism-closed, follows from that any isomorphism $Y\cong X$ is the limit of the diagram consisting of the single object $X$ (as in the discussion in the following "Creation and reflection of isomorphisms" section).
Such limits always exist, whence isomorphisms are always detected. Moreover, since the condition of being an isomorphism is equational, such limits are also preserved. In other words, isomorphisms, as limits of such diagrams, are always non-striclty lifted.
The condition that isomorphisms, considered as limits of one-object diagrams, be strictly (uniquely) lifted by a functor is then exactly the notion of the functor being (uniquely) transportable (Definition 5.28). Hence the converse follows from the fact that for an embedding of a category, being uniquely transportable means being closed under isomorphisms.
In fact, (unique) transportability is exactly the condition that converts the non-strict notions of liting (and creating) limits to the strict ones.
Indeed, if a functor is (uniquely) transportable, then any limits it lifts, it does so strictly (and uniquely). Indeed, if a limiting cone over the image of a diagram is isomorphic to the image of a limiting cone, then it is itself the image of a (unique) limiting cone obtained by applying unique transportability to the isomorphism (this is what the last sentence of the proof of Poroposition 13.27 does).
Sure, for instance take $\mathbf{C}\subset \mathbf{Set}$ the full subcategory consisting of the cardinal sets. In general, if $\mathbf{C}\subset \mathbf{D}$ is a reflexive subcategory, then any skeleton of $\mathbf{C}$ is also reflexive in $\mathbf{D}$.
Best Answer
Let me try to understand what you said with your $X,Y,A,A',B$, and then I'll give an example to see what's happening.
You take $B\in Y$, and a reflection of it in $X$, say $h: B\to A$. If $f: A\to A'$ is not in $X$, you get $f\circ h : B\to A'$. You say "this must factor through some $A\to A'$ which is in $X$; and since you chose $f$ not to be in $X$, this seems impossible.
But the point is that there will be some $f' : A\to A'$ such that $f'\circ h = f\circ h$, with $f'\in X$. And this $f'$ will be unique in $X$.
An easy example of a non full reflective subcategory is, given any category $C$ with products, the diagonal $\Delta: C\to C^2$.
We may clearly see it as a subcategory: the subcategory on objets $(A,A)$ and morphisms $(f,f)$ between those. In general, it's not full : there will, in general, be objects $A,B$ with two distinct morphisms $f,g:A\to B$ (if $C$ is not a poset, you're guaranteed to find such $A,B$), and so $(f,g) : (A,A)\to (B,B)$ is not in the image of $C$.
A left adjoint to this inclusion is given by the coproduct $(A,B)\mapsto A\coprod B$ (and if you want to really view it as a subcategory, it's $(A,B)\mapsto (A\coprod B, A\coprod B)$)
It's clear that this is not a full reflective subcategory, as if it were, then the reflector applied to $(A,A)$ would be just $(A,A)$ : here it's $(A\coprod A, A\coprod A)$ which is in general wildly different.
Now in my example, what's happening with your question : take $f,g: A\to B$ that are different, so we get $(f,g) : (A,A)\to (B,B)$ which is not in $\Delta(C)$, and suppose $(A,A)$ is the reflection of some $(E,F)$, that is $A= E\coprod F$.
Then $f,g$ are determined by $f_0,g_0: E\to B$ and $f_1,g_1: F\to B$, and the map $(E,F)\to (B,B)$ is given by $(f_0,g_1)$. But then its reflection is $([f_0,g_1],[f_0,g_1]) : (A,A)\to (B,B)$, which is different from what we started with, i.e. $(f,g) = ([f_0,f_1],[g_0,g_1])$ .
So we see that we do get a different f$'$ which will be in $X$, and which will be the only map in $X$ to satisfy $f'\circ h = f\circ h$.
(where for maps $h : E\to D, k:F\to D$, I let $[h,k]: E\coprod F\to D$ denote the uniquely determiend map)