Quantaloid

In mathematics, a quantaloid is a category enriched over the category Sup of complete lattices with supremum-preserving maps. In other words, for any objects a and b the Hom object between them is not just a set but a complete lattice, in such a way that composition of morphisms preserves all joins:

$(\underset{i}{\bigvee }{f}_i)\circ (\underset{j}{\bigvee }{g}_j)=\underset{i,j}{\bigvee }(f_i\circ g_j)$

The endomorphism lattice $\mathrm{H}\mathrm{o}\mathrm{m}(X,X)$ of any object $X$ in a quantaloid is a quantale, whence the name.

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