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Exercise 50 - Chapter 4

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

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.

Comments

  • 1.

    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).
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