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Consider constructs as concrete categories (over $\mathbf S\mathbf e\mathbf t$):

$1)$ $\mathbf M\mathbf C\mathbf P\mathbf o\mathbf s$ (complete lattices and meet-preserving maps),

$2)$ the full subconstruct of $\mathbf A\mathbf l\mathbf g(\mathcal{P})$ consisting of those $\mathcal{P}-$algebras $(X,h)$ that satisfy the following two conditions:

a) $h(\{x\})=x$ for each $x\in X$,
b) $h(\cup\mathcal{A})=h(\{h(A)|A\in\mathcal{A})$ for each $\mathcal{A}\subseteq\mathcal{P}X$.

$\mathcal{P}:\mathbf S\mathbf e\mathbf t\rightarrow\mathbf S\mathbf e\mathbf t$ is covariant power-set functor defined by:
$\mathcal{P}(f:A\rightarrow B)=(\mathcal{P}f):\mathcal{P}A\rightarrow\mathcal{P}B$
where $\mathcal{P}A$ is the power-set of $A$ and for each $X\subseteq A$, $\mathcal{P} f(X)$ is the image $f[X]$ of $X$ under $f$.

How one needs to establish concrete isomorphism between such constructs?

A. G
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  • What is a construct? What is $\mathcal{P}$? – Qiaochu Yuan Oct 04 '17 at 22:30
  • @QiaochuYuan, construct is a concrete category over $\mathbf S\mathbf e\mathbf t$ and $\mathcal{P}$ is the covariant power-set functor. I’ll add information about it. – A. G Oct 04 '17 at 22:35
  • I'm not sure it can really be called a duplicate, but this answer could be useful : https://math.stackexchange.com/a/279717/245577 – Arnaud D. Oct 05 '17 at 08:17

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Given a complete lattice $M$, let $h:\mathcal P(M)\to M$ be the meet operation, i.e. $h(U):=\displaystyle\bigwedge_{u\in U}u$

Conversely, given a set $X$ and an $h:\mathcal P(X)\to X$ satisfying the two conditions, we can define the complete meet operation to be $h$.
More specifically, for any two elements $x,y\in X$ define $x\land y:=h(\{x,y\})$.
It's commutative per nature, idempotent by (a), and associative by (b), hence defines at least a meet-semilattice structure. And, one can also deduce that $h(U)$ is the greatest common lower bound of any subset $U\subseteq X$ in the induced partial order ($x\le y \overset{def}\iff x\land y=x$).

Berci
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