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?
Monoidal categories, but not in SET
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category-theory
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2This 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.? – 2012-06-17
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0Hi 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. – 2012-06-17
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1There 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. – 2012-06-17
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2Perhaps you want a monoidal version of http://ncatlab.org/nlab/show/internal+category+in+a+monoidal+category ? – 2012-06-17
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0Hi, Qiaochu Yuan, I think maybe I do. I have visited this notion before. The category enriched over a monoidal category also sounds neat. – 2012-06-17
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0@BenSprott Do you have an answer to your question? If yes, can you post it, so that the question gets removed from the unanswered tab. – 2013-06-08
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0Hi Julian, I'm afraid I cannot answer the question here. Qiaochu's post sounds like it might be an answer, but I would butcher it if I tried to write out an answer based on his suggestion. – 2013-06-09