Basic concepts of enriched category theory.

*(English)*Zbl 1086.18001Having been a student of the author, and having been a close colleague ever since (particularly during the writing of this book), the reviewer cannot be expected to be objective. Kelly’s writing style and mathematical interests attracted me to category theory. So it is no surprise that I like the book. I have recommended it to my own graduate students and other colleagues; most of them learned a lot from it after some initial efforts, only a few found it too hard.

[The book was reviewed by John Gray for the Bull. Am. Math. Soc. 9, 102–107 (1983); see also G. M. Kelly, Lond. Math. Soc. Lect. Note Ser. 64 (1982; Zbl 0478.18005)]. I believe that review was unduly negative in claiming that it is not a textbook. I believe it is an ideal text for a second course on category theory. Moreover, the need for enriched category theory, especially in homotopy theory, have proved the subject’s importance.

In fact, this reprinting is in response to demand. Obtaining a copy has been difficult. Now, thanks to TAC, it is freely available. Thanks to a remarkable team typing effort, it looks better than the original and has some minor errors corrected.

So what is an enriched category? Recall that a category \(\mathcal{A}\) has objects and, for every pair of objects \(A\) and \(B\), a set \(\mathcal{A}( A,B) \), called a homset, whose elements are called morphisms. Each \(\mathcal{A}( A,A) \) contains a member, called the identity of \(A\), which can be viewed as a function \(j_{A}:\mathbb1\to \mathcal{A}( A,A) \) from the singleton set \(\mathbb1\). There are also functions \(M_{ABC}:\mathcal{A}( B,C) \times \mathcal{A}( A,B) \to \mathcal{A}( A,C) \), called composition. Axioms of associativity and identities can be expressed as three diagrams involving the functions \(j_{A}\) and \(M_{ABC}\).

Now we come to enrichment which, in this context, applies to the homsets. We repeat the definition of category where now the homsets are to be objects of some category \(\mathcal{V}\), not sets at all! Then the functions \(j_{A}\) and \(M_{\mathrm{ABC}}\) must become morphism of \(\mathcal{V}\). However, what should we take instead of the cartesian product \(\times \) of sets and the singleton set \(\mathbb1\)? The relevant answer is an abstract tensor product \(\otimes \) with unit \(I\) satisfying axioms: that is, \(\mathcal{V}\) should be a monoidal category.

This definition of \(\mathcal{V}\)-enriched category was very natural and would have remained rather unremarkable if left at that. What was remarkable was how few further requirements on \(\mathcal{V}\) allowed the development of a lot of category theory to the enriched context. Ingenuity was needed at many points in the development and, in most cases, was supplied by Kelly.

Kelly’s book tells that story. The main characters are limits and Kan extensions with completion processes and limit theories providing the plot. At the time of writing, some of this work was new even for ordinary categories.

The book has some notable omissions. Kelly expresses regret in the introduction at not discussing the theory of monads on enriched categories, or the completeness and cocompleteness of the 2-category of \(\mathcal{V}\)-enriched categories. He also saved his lovely theory of locally presentable enriched categories for a separate publication [see “Structures defined by finite limits in the enriched context. I”, Cah. Topol. Géom. Différ. 23, 3–42 (1982; Zbl 0538.18006)]. Then there is the theory of distributors (also called profunctors, bimodules, or even just modules) which the book avoids by working with functor categories; this approach has its limitations and distributors are natural and inevitable [see F. W. Lawvere, “Metric spaces, generalized logic, and closed categories”, Rend. Sem. Mat. Fis. Milano 43, 135–166 (1974; Zbl 0335.18006); also Repr. Theory Appl. Categ. 2002, 1–37 (2002; Zbl 1078.18501)]. Finally, there is the replacement of the monoidal category \(\mathcal{V}\) as base by a bicategory \(\mathcal{W}\); this theory was developed after Robert Walters’ description of sheaves as enriched categories with a bicategory (built from a space or Grothendieck site) as base [see R. Street, “Cauchy characterization of enriched categories”, Rend. Sem. Mat. Fis. Milano 51, 217–233 (1981; Zbl 0538.18005); also Repr. Theory Appl. Categ. 2004, No. 4, 1–16 (2004; Zbl 1099.18005)].

I very much welcome this reprinting. My well-worn, well-borrowed copy can now stay with me.

[The book was reviewed by John Gray for the Bull. Am. Math. Soc. 9, 102–107 (1983); see also G. M. Kelly, Lond. Math. Soc. Lect. Note Ser. 64 (1982; Zbl 0478.18005)]. I believe that review was unduly negative in claiming that it is not a textbook. I believe it is an ideal text for a second course on category theory. Moreover, the need for enriched category theory, especially in homotopy theory, have proved the subject’s importance.

In fact, this reprinting is in response to demand. Obtaining a copy has been difficult. Now, thanks to TAC, it is freely available. Thanks to a remarkable team typing effort, it looks better than the original and has some minor errors corrected.

So what is an enriched category? Recall that a category \(\mathcal{A}\) has objects and, for every pair of objects \(A\) and \(B\), a set \(\mathcal{A}( A,B) \), called a homset, whose elements are called morphisms. Each \(\mathcal{A}( A,A) \) contains a member, called the identity of \(A\), which can be viewed as a function \(j_{A}:\mathbb1\to \mathcal{A}( A,A) \) from the singleton set \(\mathbb1\). There are also functions \(M_{ABC}:\mathcal{A}( B,C) \times \mathcal{A}( A,B) \to \mathcal{A}( A,C) \), called composition. Axioms of associativity and identities can be expressed as three diagrams involving the functions \(j_{A}\) and \(M_{ABC}\).

Now we come to enrichment which, in this context, applies to the homsets. We repeat the definition of category where now the homsets are to be objects of some category \(\mathcal{V}\), not sets at all! Then the functions \(j_{A}\) and \(M_{\mathrm{ABC}}\) must become morphism of \(\mathcal{V}\). However, what should we take instead of the cartesian product \(\times \) of sets and the singleton set \(\mathbb1\)? The relevant answer is an abstract tensor product \(\otimes \) with unit \(I\) satisfying axioms: that is, \(\mathcal{V}\) should be a monoidal category.

This definition of \(\mathcal{V}\)-enriched category was very natural and would have remained rather unremarkable if left at that. What was remarkable was how few further requirements on \(\mathcal{V}\) allowed the development of a lot of category theory to the enriched context. Ingenuity was needed at many points in the development and, in most cases, was supplied by Kelly.

Kelly’s book tells that story. The main characters are limits and Kan extensions with completion processes and limit theories providing the plot. At the time of writing, some of this work was new even for ordinary categories.

The book has some notable omissions. Kelly expresses regret in the introduction at not discussing the theory of monads on enriched categories, or the completeness and cocompleteness of the 2-category of \(\mathcal{V}\)-enriched categories. He also saved his lovely theory of locally presentable enriched categories for a separate publication [see “Structures defined by finite limits in the enriched context. I”, Cah. Topol. Géom. Différ. 23, 3–42 (1982; Zbl 0538.18006)]. Then there is the theory of distributors (also called profunctors, bimodules, or even just modules) which the book avoids by working with functor categories; this approach has its limitations and distributors are natural and inevitable [see F. W. Lawvere, “Metric spaces, generalized logic, and closed categories”, Rend. Sem. Mat. Fis. Milano 43, 135–166 (1974; Zbl 0335.18006); also Repr. Theory Appl. Categ. 2002, 1–37 (2002; Zbl 1078.18501)]. Finally, there is the replacement of the monoidal category \(\mathcal{V}\) as base by a bicategory \(\mathcal{W}\); this theory was developed after Robert Walters’ description of sheaves as enriched categories with a bicategory (built from a space or Grothendieck site) as base [see R. Street, “Cauchy characterization of enriched categories”, Rend. Sem. Mat. Fis. Milano 51, 217–233 (1981; Zbl 0538.18005); also Repr. Theory Appl. Categ. 2004, No. 4, 1–16 (2004; Zbl 1099.18005)].

I very much welcome this reprinting. My well-worn, well-borrowed copy can now stay with me.

Reviewer: Ross H. Street (North Ryde)

##### MSC:

18-02 | Research exposition (monographs, survey articles) pertaining to category theory |

18D20 | Enriched categories (over closed or monoidal categories) |

18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |