Why are multicategories important?
I'm not biased against them, just curious. Let me clarify:
- Is there any construction that can be done in a multicategory, but not in a (traditional) category with products, using morphisms $X_1\times \dots\times X_n \to Y$?
- If the answer to above is no, are multicategories still convenient because there are additional axioms which in traditional categories would instead be conditions that have to be proven? Or at least do multicategories provide a more convenient notation?
I may be completely off the mark here, since I'm far from being an expert.
Edit: Instead of the categorical product, we can more generally ask the same question using "any" tensor product, assuming we are in a monoidal category. (Thanks to Mariano for the helpful comment!)
The multicategories which arise from a tensor product are usally said to be "representable". In fact, even in these cases it is usally more natural to define first the multicategory structure; the tensor is then characterized by its universal property with respect to multimaps (the associative and coherence properties of tensor follow from that). For a non-representable example, you may consider a full multicategory of a representable one; for instance, take the one-object multicategory which consists of a set and all its endomultimappings (internal operations). For another natural instance, consider (for any category C) the multicategory with the same objects as C and with finite sequences of maps in C with a common target as multimaps; it is representable iff C has finite sums.