Reflective Subcategories, Limit closedness and Repleteness

Question

The classical notion of reflective subcategory of a given category is the following one (taken from Wikipedia):

A full subcategory $\mathcal{A}$ of a category $\mathcal{B}$ is said to be reflective in $\mathcal{B}$ if for each $\mathcal{B}$-object $B$ there exists an $\mathcal{A}$-object ${\displaystyle A_{B}}$ and a $\mathcal{B}$-morphism ${\displaystyle r_{B}\colon B\to A_{B}}$ such that for each $\mathcal{B}$-morphism ${\displaystyle f\colon B\to A}$ to an $\mathcal{A}$-object ${\displaystyle A}$ there exists a unique $\mathcal{A}$-morphism ${\displaystyle {\overline {f}}\colon A_{B}\to A}$ with ${\displaystyle {\overline {f}}\circ r_{B}=f}$.

Now, in the present definition repleteness is NOT required. In Adamek's book "The Joy of Cats" (Definition $4.16$, page $52$), the definition is the same with the exception that subcategories need not to be full.

One of the properties listed in the Wikipedia page is the following:

If $\mathcal{A}$ is a (full) reflective subcategory of $\mathcal{B}$, then the inclusion functor $\mathcal{A} \to \mathcal{B}$ creates all limits that are present in $\mathcal{B}$.

I do not know the proof of such a statement (if possible, can you give me a bibliographic reference, showing the claim in the non-replete case?), however if we work with concrete categories (namely a category $\mathcal{B}$ endowed with a faithful functor $U: \mathcal{B} \to \mathcal{X}$ to a base category $\mathcal{X}$) and the reflections are identity-carried morphisms (namely applying the faithful functor to the reflection we get an identity morphism in the base category), then it may be easily shown that the limits computed in $\mathcal{B}$ actually belong to $\mathcal{A}$!

Adamek, instead, in his remark $13.26$ asserts that

Embeddings $E: \mathcal{A} \to \mathcal{B}$ of full subcategories obviously reflect limits. Hence they lift limits if and only if they create them. A more suggestive term for such full subcategories is that they are closed under the formation of limits (or just limit-closed) in $\mathcal{B}$.

Read moreover the comment to my post

On an (possibly) equivalent way of expressing closure under limits for a full subcategory

If the containing category has all limits and the subcategory is closed under products and equalisers, then the subcategory is closed under limits. If the containing category does not have products and equalisers then the condition may be vacuous.

Only a remark: the previous result is neither stated in Adamek's book nor Wikipedia's page. I did not find it elsewhere, in the non-replete assumption. Can you give me a bibliographic reference, please?

If we put together Wikipedia's page, Adamek's remark and the previous post, one is brought to infer that: "If $\mathcal{A}$ is a reflective subcategory of a complete category $\mathcal{B}$, then $\mathcal{A}$ is limit-closed".

Nevertheless, in the successive Proposition $13.27$, Adamek shows that

A full reflective subcategory $\mathcal{A}$ of $\mathcal{B}$ is limit-closed in $\mathcal{B}$ if and only if $\mathcal{A}$ is isomorphism-closed (i.e. replete).

Something is strange. In fact, if $\mathcal{B}$ is assumed to be complete, the requirement of being limit-closed should hold also for non-isomorphism-closed reflective subcategories.

Therefore, putting all together, I deduce that any reflective subcategory of a complete category is isomorphism-closed. Is it possible? This is very strange that such a fundamental result has been not yet stated elsewhere!

Finally, in Exercise $4D$ Adamek requires to show that the category ${\bf Set}$ has precisely three full, isomorphism-closed, reflective subcategories, and infinitely many reflective subcategories. Here, the point is that Adamek does not assume that a reflective subcategory is full. Among the aforementioned infinitely many reflective subcategories of ${\bf Set}$, does there exist a full not isomorphism-closed reflective subcategory?

Here, in fact, a counterexample:

Reflective subcategories of the category of sets

Answer 1

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".

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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.

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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).