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We normally present the theory of categories in SET, that is, we define a category as a set of objects and a set of morphisms. If we do not present categories in SET, how do we present the abstract structure of a monoidal category?

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This question is not really precise. Do you know the notion of an enriched category? Are you looking for enriched monoidal categories? Have you looked at the standard references by Kelly, Street, etc.? –  Martin Brandenburg Jun 17 '12 at 17:32
    
Hi Martin, The only monoidal structure I want to capture is the kind of structure we find in the wikipedia entry on "Monoidal Category". I want to present a category where I can tensor objects and then tensor morphisms to map products to products. I think the real problem is that it will be hard to present a category without sets, but that is the challenge. –  Ben Sprott Jun 17 '12 at 17:52
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There are certain general principles which can be followed to lift definitions from the non-enriched case to the enriched case. (It is in fact very easy to describe categories enriched over any monoidal category.) I am sure it is possible in principle to describe a monoidal enriched category. –  Zhen Lin Jun 17 '12 at 21:40
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Perhaps you want a monoidal version of ncatlab.org/nlab/show/internal+category+in+a+monoidal+category ? –  Qiaochu Yuan Jun 17 '12 at 22:58
    
Hi, Qiaochu Yuan, I think maybe I do. I have visited this notion before. The category enriched over a monoidal category also sounds neat. –  Ben Sprott Jun 17 '12 at 23:33
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