Allegories in easy words?

1) What is, in easy words, the definiton of an allegory?

2) And when are allegories useful?

What does it have to do with the category theory and categories?

With the definiton of category, it is easy to have an idea of what is a category, but with allegories I'm totally lost.

An allegory is a special kind of category that has properties like the category $\textbf{Rel}$ of sets and relations. More precisely,

Definition. An allegory is a category $\mathcal{A}$ equipped with a poset structure on each hom set $\mathcal{A} (X, Y)$ and an identity-on-objects functor $(-)^{\circ} : \mathcal{A}^{\textrm{op}} \to \mathcal{A}$ such that

• composition $\mathcal{A} (Y, Z) \times \mathcal{A} (X, Y) \to \mathcal{A} (X, Z)$ is monotonic,

• the map $(-)^{\circ} : \mathcal{A} (X, Y) \to \mathcal{A} (Y, X)$ is monotonic,

• the hom-poset $\mathcal{A}(X, Y)$ has binary meets, and

• for all $\psi : X \to Y$, $\psi : Y \to Z$ and $\chi : X \to Z$ in $\mathcal{A}$, we have the following inequality: $$(\psi \circ \phi) \cap \chi \le (\psi \cap (\chi \circ \phi^{\circ})) \circ \phi$$

Example. The category $\textbf{Rel}$ is an allegory, where $\phi \le \phi'$ means $\phi \subseteq \phi'$ (as subsets of $X \times Y$) and $\phi^{\circ}$ is the relation $Y \to X$ such that $\phi^{\circ} (y, x)$ if and only if $\phi (x, y)$. It is clear that the poset $\textbf{Rel} (X, Y) = \mathscr{P} (X \times Y)$ has binary meets (= intersections) and so we just have to verify the displayed inequality. But, if there are $x, y, z$ such that $\chi (x, z)$, $\phi (x, y)$, and $\psi (y, z)$, then certainly $\phi (x, y)$, $\phi^{\circ} (y, x)$, $\chi (x, z)$, and $\psi (y, z)$, so the inequality is indeed true.

In fact, the same kind of reasoning shows that for any first-order theory $\mathbb{T}$ we can construct an allegory whose objects are predicates over $\mathbb{T}$ (written as $\{ (x_1, \ldots, x_n) : \alpha \}$, where $x_1, \ldots, x_n$ are the free variables of a formula $\alpha$) modulo equivalence under renaming (but not permutation) of variables, and whose arrows $\phi : \{ (x_1, \ldots, x_n) : \alpha \} \to \{ (y_1, \ldots, y_m) : \beta \}$ are those formulae $\phi$ with free variables $x_1, \ldots, x_n, y_1, \ldots, y_m$ such that $\mathbb{T}, \phi \vdash \alpha$ and $\mathbb{T}, \phi \vdash \beta$, modulo provable-equivalence under $\mathbb{T}$. Composition is, of course, the composition of relations: so given $\phi : \{ (x_1, \ldots, x_n) : \alpha \} \to \{ (y_1, \ldots, y_m) : \beta \}$ and $\psi : \{ (y_1, \ldots, y_m) : \beta \} \to \{ (z_1, \ldots, z_l) : \gamma \}$, $\psi \circ \phi$ is the predicate corresponding to $\exists y_1 . \ldots . \exists y_m . \phi \land \psi$. This is called the syntactic allegory of $\mathbb{T}$.

• This Will take a lot time to be assimilated (for me xD ), but sounds clear, meaningfull and a very usefull concept. thanks. – MphLee May 2 '13 at 18:24
• @Zhen Lin you said 'has properties like Rel', but the last property you listed seems really strange and not at all clear why it is natural to request. I mean monotonicity of the other ops seems obviously clear as it connects those operators with the ordering, and there aren't many ways to connect the two. But the last one...just what!?? – Musa Al-hassy Sep 24 '15 at 18:11
• It is rather strange. I don't really know what it means myself. – Zhen Lin Sep 24 '15 at 19:58