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If $F: \mathscr{A} \rightarrow \mathscr{B}$ is an equivalence of categories, and $A,\tilde{A} \in \mathrm{obj}(\mathscr{A})$ such that $\tilde{A} \subseteq A$ in $\mathscr{A}$, does it immediately hold that $F(\tilde{A}) \subseteq F(A)$ in the category $\mathscr{B}$?

Just trying to sanity check here! If I think of the example of certain representations/invariant vector spaces which may be thought of as modules, this seems clear, but I'm struggling to explicitly show this! Though I am tempted to say that this property is not sufficiently `categorical' in the sense of answers such as: Properties preserved under equivalence of categories

Best,

M

mathieu_matheux
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2 Answers2

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$\newcommand{\A}{\mathscr{A}}\newcommand{\B}{\mathscr{B}}$This is true.

Say $h:a\to a’$ is a monic in $\A$ (the standard meaning of $a\subseteq a’$ is that such an $h$ exists); we want to show that $F(h)$ is monic. Let $\eta:GF\cong1_{\A}$ be any natural isomorphism; at least one exists since $F,G$ are pseudoinverse.

Suppose there is a $b\in\B$ and $f,g:b\to F(a)$ with $F(h)\circ f=F(h)\circ g$.

Then: $$\eta_{a’}GF(h)\circ G(f)=\eta_{a’}GF(h)\circ G(g)$$So: $$h\eta_a\circ G(f)=h\eta_a\circ G(g)$$So (since $h\eta_a$ is monic as a composite of monics) we have $G(f)=G(g)$. Because $G$, as an equivalence, must be faithful, we find $f=g$ as desired. Thus $F(h)$ is monic.

Another way to see this: let $h:a\to a’$ be a monic. This is exactly the same thing as saying that the arrows: $$h^\ast:\A(a’’,a)\to\A(a’’,a’)$$Are monic for all $a’’\in\A$ (that is, they are injective).

Fix $b\in\B$. Since $F$ is essentially surjective, there exists some $a’’\in\A$ with $F(a’’)\cong b$. Therefore there is a commutative diagram: $$\require{AMScd}\begin{CD}\B(b,F(a))@>F(h)^\ast>>\B(b,F(a’))\\@V\cong VV@VV\cong V\\\B(F(a’’),F(a’))@>F(h)^\ast>>\B(F(a’’),F(a))\\@V F\,\,\cong VV@VV\cong\,\,FV\\\A(a’’,a)@>>h^\ast>\A(a’’,a’)\end{CD}$$

Where the vertical edges are all isomorphisms. Therefore the top arrow is monic if and only if the bottom one is; $F(h)$ is monic if and only if $h$ is.

FShrike
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  • Thank you for this! Will have to read up on this helpful notion of monics! – mathieu_matheux Jul 11 '23 at 15:48
  • @mathieu_matheux You’re welcome. In a general category the notion $A\subseteq A’$ suggests the concept of “subobject” which is basically the same thing as saying, $A$ ‘injects’ into $A’$ (there is a monic $A\to A’$). In set based categories this is basically just the sub(set, ring, field, …) relation but you have to be careful; e.g. it’s not true in the category of topological spaces that subobjects are the same thing as subspaces – FShrike Jul 11 '23 at 20:21
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Whether an abstract object $A$ is a „subset“ (whatever that may mean) of an abstract object $B$ is an additional relation on the category, which is not preserved under equivalence. Note that a priori the notion $A\subseteq B$ is not defined for arbitrary categories. Maybe you‘re thinking about subobjects?

Nephry
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  • Thank you for the helpful remark and clarification! Yes this would mean subobjects, but I wanted to stress that those are members of the category in their own right. I think the other answer deals with this relation with the more general notion of "monic", which I was not aware of! – mathieu_matheux Jul 11 '23 at 15:42