A functor between two monoids seen as 1 object categories is essentially a monoid-homomorphism. What is the equivalent construction for vector spaces and linear maps?
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$\begingroup$ Hi @ManuelAraújo, thank you for your comment! I've looked at the links, but I'm still unclear on how the construction would work... could you please help me clarify what would the category and functor be in the case, say, if I want to categorify alinear map $f:\mathbb{R}^n\to\mathbb{R}^n$ $\endgroup$– rickCommented May 9 at 19:24
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$\begingroup$ Sorry I think I misunderstood your question. I deleted my comment. $\endgroup$– Manuel AraújoCommented May 9 at 19:46
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1$\begingroup$ Well, a vector space is just a group with a scalar multiplication by the underlying field, and a group is just a category with one object where every morphism is invertible, so a natural (trivial) answer would be a one object groupoid together with a scalar multiplication by the underlying field. But there may be more interesting ‘categorical’ answers for the real/complex numbers; we can define a ‘field of scalars’ in any monoidal category if i remember correctly from categorical QM, but I’ll have to check my references when I get home tonight. $\endgroup$– Alec RheaCommented May 9 at 19:58
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$\begingroup$ @AlecRhea thank you! I think is going in the right direction. So in this case the multiplication with scalars would be given by an enriched structure of the groupoid? And we would require the functor to respect this enriched structure $\endgroup$– rickCommented May 9 at 20:47
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1$\begingroup$ Hi, @rick! I think a possible way to answer this question is via enriched categories and presheaves. If you take $A$ to be a monoid and write $\mathbf{B}A$ for the one-object category associated to it, then a presheaf on $\mathbf{B}A$, i.e. a functor $\mathbf{B}A^{\mathsf{op}}\to\mathsf{Sets}$, is precisely the data of a left $A$-set. $\endgroup$– EmilyCommented May 10 at 0:18
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