# On Proposition $13.27$ of Adamek: full reflective subcategories are limits closed iff they are replete?

In literature there is a great confusion when talking of reflective subcategories. Many authors assume in the definition repleteness. However, I will refer to "The Joy of Cats" and Emily Riehl's book, where repleteness is not assumed.

I will rely heavily on the notations of "The Joy of Cats" (see http://katmat.math.uni-bremen.de/acc/acc.pdf).

Here, the authors say that a full subcategory inclusion $$L: A \to B$$ reflects all limits and colimits existing in $$B$$, but it needs not lift, preserve, nor detect them (according to Example $$13.23(3)$$ and Remark $$13.26$$). Again, in Remark $$13.26$$ they call closed under the formation of limits in $$B$$ any full subcategory $$A$$ of $$B$$ which creates limits. Finally, in the successive Proposition $$13.27$$ the authors prove that a full reflective subcategory $$A$$ of $$B$$ is closed under the formation of limits in $$B$$ if and only if it is replete.

First Question

The implication "replete $$\implies$$ closed under the formation of limits in $$B$$" is quite convincing. In fact, using the reflection of the limit $$L \in B$$ we get an object and a natural source with such an object as its domain. The real problem is in the reverse implication, which the authors dismiss as something obvious. The answer by Marc Olschok on this point is now clear!

Second Question

Riehl's Book - https://math.jhu.edu/~eriehl/context.pdf (page 158)

The author gives the same definition of reflective subcategory as Adamek et al. However in Proposition $$4.5.15$$ she says something which contradicts Adamek's assertion, namely:

If $$D \to C$$ is a reflective subcategory, then:

(i) The inclusion $$D \to C$$ creates all limits that $$C$$ admits;

(ii) $$D$$ has all colimits that $$C$$ admits, formed by applying the reflector to the colimit in $$C$$.

In addition, by Theorems 4.5.2 and 4.5.3 of Riehl, if $$D$$ has limits or colimits, they must be constructed in the way described in Proposition $$4.5.15$$: limits are preserved by the inclusion $$D \to C$$ and colimits of diagrams in $$D$$, regarded as diagrams in $$C$$, are preserved by $$L: C \to D$$.

Suppose Riehl is right. Then, according to her perspective and because of the aformentioned result, every full subcategory is closed under the formation of limits. In other terms, Proposition $$13.27$$ of Adamek does not make sense. Well, my question is the following: who is really right?

• For any object $a$ in $A$, consider the corresponding constant functor $\Delta_a: 1\to B$ from the terminal category $1$. Then any isomorphism $u: b\to a$ should give a limit cone from $b$ to $\Delta_a$. May 28, 2022 at 13:40
• @MarcOlschok So any reflective subcategory is replete? If you read my question, there are several points that in literature are misleading. Some authors states a property, some other disprove it. Can you provide a detailed answer clarifying all my doubts, please? May 28, 2022 at 13:53
• It's a subtle but frankly trivial/pedantic point. If I say that a subcategory is closed under limits, I mean that every diagram in the subcategory that has a limit in the containing category also has a limit in the subcategory that is preserved by the inclusion. I do not insist that literally the same object be in the subcategory – any isomorphic copy will suffice. Some authors differ on this point. May 28, 2022 at 14:04
• @ZhenLin I edited my question. I think now it is clearer. It is possible to have a detailed and clarifying answer, even pedantic, please? May 28, 2022 at 15:22

Let $$B$$ be a full reflective subcategory of $$A$$ (where I will use $$\iota \colon A\to B$$ for the inclusion). Given objects $$a$$ in $$A$$ and $$b$$ in $$B$$ with $$a\cong b$$ then any isomorphism $$u\colon b\to a$$ can be viewed as a limit cone from $$b$$ to the corresponding constant functor $$\Delta_a\colon 1\to B$$ from the terminal category $$1$$.
(1) suppose $$A$$ is closed under limits in $$B$$ in the sense of [AHS]. This means that $$\iota\colon A\to B$$ creates limits as defined in [AHS, Def. 13.17(2)]. Applied to the above, there is a unique cone $$v\colon a'\to a$$ such that $$\iota(a')=b$$, $$\iota(v)=u$$ and moreover $$v\colon a'\to a$$ is a limit cone in $$A$$. Since $$\iota$$ is just the inclusion, this means in particular that $$a'=b$$ and $$v=u$$.
(2) suppose $$A$$ is closed under limits in $$B$$ in the sense of [Riehl, 3.3]. This means that there is some limit cone $$v\colon a'\to a$$ such that $$\iota(v)\colon \iota(a')\to \iota(a)$$ is a limit cone. Again we can drop $$\iota$$ from the notation, but now we only know that $$v\colon a'\to a$$ is a limit cone in $$B$$, so we only know that $$v \cong u$$ in $$B/a$$ and hence $$a'\cong b$$. So we are left with the same situation as before.