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Exercise 50 - Chapter 4

edited August 2018

$$\def\cat#1{{\mathcal{#1}}}$$ $$\def\Cat#1{{\textbf{#1}}}$$ $$\newcommand{\smset}{\Cat{Set}}$$ $$\def\Ob{{\mathrm{Ob}}}$$ $$\newcommand{\cp}{.}$$ $$\def\id{{\mathrm{id}}}$$

Recall from Example 4.47 that $$\cat{V}=(\smset,{1},\times)$$ is a symmetric monoidal category. This means we can apply Definition \ref{def1}. Does the (rough) definition roughly agree with the definition of category given in Definition 3.6? Or is there a subtle difference? $$\label{def1}\tag{4.49}$$ Rough Definition 4.49:

Let $$\cat{V}$$ be a symmetric monoidal category, as in Definition 4.43. To specify a category enriched in $$\cat{V}$$, or a $$\cat{V}$$-category, denoted $$\cat{X}$$,

1. one specifies a collection $$\Ob(\cat{X})$$, elements of which are called objects;
2. for every pair $$x,y\in\Ob(\cat{X})$$, one specifies an object $$\cat{X}(x,y)\in\cat{V}$$, called the hom-object for $$x,y$$;
3. for every $$x\in\Ob(\cat{X})$$, one specifies a morphism $$\id_x\colon I\to\cat{X}(x,x)$$ in $$\cat{V}$$, called the identity element;
4. for each $$x,y,z\in\Ob(\cat{X})$$, one specifies a morphism $$\cp\colon\cat{X}(x,y)\otimes\cat{X}(y,z)\to\cat{X}(x,z)$$, called the composition morphism.

These constituents are required to satisfy the usual associative and unital laws.

Requirement 1. here corresponds to requirement 1. in the definition of a category. The hom-object becomes the set of morphisms. $$\mathrm{id}_x$$ picks out a distinguished element of the set of morphisms from an object to itself, which by the definition of a monoidal category satisfies the left and right unit laws. So, this morphism is the identity morphism at $$x$$. The composition morphism maps $$f\in\mathcal{X}(x,y)$$, $$g\in\mathcal{X}(y,z)$$ to $$\circ(f,g)\in\mathcal{X}(x,z)$$. Because of the symmetric monoidal structure on Set, iterated composites of morphisms are associative, so $$\circ(f,g)$$ can reasonably be called $$g\circ f$$. Note, however, that the laws that the morphisms must obey are only required to hold as isomorphisms, not equalities. So, in general, this does not yield the earlier definition of a category (but it's pretty close).
Comment Source:Requirement 1. here corresponds to requirement 1. in the definition of a category. The hom-object becomes the set of morphisms. \$$\mathrm{id}_x\$$ picks out a distinguished element of the set of morphisms from an object to itself, which by the definition of a monoidal category satisfies the left and right unit laws. So, this morphism is the identity morphism at \$$x\$$. The composition morphism maps \$$f\in\mathcal{X}(x,y)\$$, \$$g\in\mathcal{X}(y,z)\$$ to \$$\circ(f,g)\in\mathcal{X}(x,z)\$$. Because of the symmetric monoidal structure on **Set**, iterated composites of morphisms are associative, so \$$\circ(f,g)\$$ can reasonably be called \$$g\circ f\$$. Note, however, that the laws that the morphisms must obey are only required to hold as isomorphisms, not equalities. So, in general, this does not yield the earlier definition of a category (but it's pretty close).