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Given two sets M and N endowed with monoid structure (or, in general, any finite number of monoids, M 1, ..., M k), their Cartesian product M × N, with the binary operation and identity element defined on corresponding coordinates, called the direct product, is also a monoid (respectively, M 1 × ⋅⋅⋅ × M k). [5] Fix a monoid M.
A monoid object in the category of monoids (with the direct product of monoids) is just a commutative monoid. This follows easily from the Eckmann–Hilton argument. A monoid object in the category of complete join-semilattices Sup (with the monoidal structure induced by the Cartesian product) is a unital quantale.
Any category with finite products can be regarded as monoidal with the product as the monoidal product and the terminal object as the unit. Such a category is sometimes called a cartesian monoidal category. For example: Set, the category of sets with the Cartesian product, any particular one-element set serving as the unit.
The coproduct in the category of sets is simply the disjoint union with the maps i j being the inclusion maps.Unlike direct products, coproducts in other categories are not all obviously based on the notion for sets, because unions don't behave well with respect to preserving operations (e.g. the union of two groups need not be a group), and so coproducts in different categories can be ...
A Florida woman who allegedly snatched a three-year-old boy from his fenced-in yard and ran off down the street last week told the cops she shouldn’t be arrested because she “gave it back ...
In the category of graphs, the product is the tensor product of graphs. In the category of relations, the product is given by the disjoint union. (This may come as a bit of a surprise given that the category of sets is a subcategory of the category of relations.) In the category of algebraic varieties, the product is given by the Segre embedding.
After a childhood behind the Iron Curtain, Angela Merkel served as Chancellor of a united Germany for 16 years, becoming the most powerful woman in the world while dealing with its most powerful men.
The last two months, I think they’re playing as good football as anybody in the country.” Texas quarterback Quinn Ewers “It’s the stuff you think about as a kid, for sure.