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I have a way of constructing (something like) a 2-category from a category with products $\mathcal{C}$.

My question: Is this a correct construction, and if so, does it have a name?

Define $\mathcal{C'}$ to consist of one object. The 1-morphims are objects of $\mathcal{C}$, the identity is the terminal object of $\mathcal{C}$, and $A \circ B$ is $A \times B$. The 2-morphisms $A \to B$ are morphisms $A \to B$ from $\mathcal{C}$. Vertical composition is composition in $\mathcal{C}$ and the horizontal composition of $f$ and $g$ is given by $f \times g$.

Notes:

  1. I said "something like" a 2-category because it seems like composition will be defined up to isomorphism, rather than equality.

  2. Perhaps this would work with any monoidal category, rather than just a category with products.

  3. I want this construction so I can "reverse" the usual relationship between monoid object and monad, and say that a monoid in $\mathcal{C}$ is given by a monad in $\mathcal{C'}$.

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    $\begingroup$ Take a look at the bottom of page three here: math.berkeley.edu/~aaron/livetex/highercats.pdf Also the bottom of page 5 is illuminating. $\endgroup$ Oct 20 '13 at 2:26
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    $\begingroup$ This is a correct construction (though by "terminal" object, you probably mean unit object) and should work for any monoidal structure. I'm not sure if there's a name for it, but higher algebra people think about this kind of construction often. It shows you can trade off between category number and number of compatible multiplications (composition and monoidal structure give 2 multiplications). For instance, an (oo,n)-category with only n-morphisms is the same thing as an E_n algebra. $\endgroup$ Oct 20 '13 at 3:00
  • $\begingroup$ @BabyDragon: Ah it is a correspondance in both directions! That makes sense. I didn't even consider that. $\endgroup$
    – Tom Ellis
    Oct 20 '13 at 11:40
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Yes, and more generally this works for any monoidal category. In fact a monoidal category is precisely a (weak) 2-category with one object, in the same way that a monoid is precisely a category with one object. (Weak 2-categories are the "correct" notion of 2-category; what you have in mind is a strict 2-category.)

The story continues: recall that a monoid in the category of monoids is a commutative monoid by the Eckmann-Hilton argument, and then things stabilize; taking monoid objects doesn't give you anything new. One can ask what a monoid in the category of monoidal categories is, and this turns out to be a braided monoidal category. Then a monoid in the category of braided monoidal categories is a symmetric monoidal category.

More generally, see the periodic table of higher categories.

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