yields falsehood when preceded by his own quotation
![tweet from Olivia Nuzzi on April 24, 2019: "Nothing but respect for MY president, @bariweiss" followed by a link to a Vanity Fair article about her. Bari Weiss replies: "Olivia! [heart eyes emojis]"](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:judtbidg6b2ye3jzverecw3g/bafkreidwo7666ejnzflryhzdfrde7xsiqooj55tnn34h3ymcvro6gtjnge)
![What's the "dual" of a set $A$ with some operations $A^n \to A$ for some $n$?
1. **Comodels**: Of course, the dual of $A \times A$ must be $A + A$, so an $X$-ary function $A^X \to A$ has dual $$A \to X \cdot A$$ where $X \cdot A$ is just the $X$-fold coproduct $A + \dots + A$.
2. **Co-operations**: Of course, the dual of $A^2 = [2, A]$ must be $[A, 2] = 2^A$, so an $X$-ary function $A^X \to A$ has dual $$1^A \to X^A$$ which is, equivalently, an element of $X^A = [A, X]$.
3. **Co-structure of a signature:** For each $n < \omega$, let $F_n$ be the set of function symbols of arity $n$ in the signature $\Sigma$. Then a $\Sigma$-structure is a function $F_n \times A^n \to A$ for each $n < \omega$. Dually, a co-structure of $\Sigma$ should be a function $$A \to F_n \times A^n$$ for each $n < \omega$.
4. **Polynomial coalgebra**: A function $A \times A \to A$ is an algebra of the polynomial endofunctor $p: A \mapsto A^2$, so the answer should be a coalgebra of $p$, that is, a set $A$ and a function $A \to A \times A$. In general, we consider coalgebras of polynomial endofunctors, i.e. a set $A$ and a function $$A \to \sum_n X_n \cdot A^n$$](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:judtbidg6b2ye3jzverecw3g/bafkreidqsbqjxm5sxhl4byhcohbv3znavdihwqwz44uolhle6jguzazcom)
yields falsehood when preceded by his own quotation