Monoidal Categories, Enriched, and the Closed¶
Job: develop monoidal categories (tensor / unit / coherence), closed / internal Hom / CCC, enriched categories over \(\mathcal{V}\), and \(k\)-linear / abelian — with \(\mathbf{Vec}_k\) the monoidal closed / \(k\)-linear enriched as the worked example.
Core¶
The monoidal category¶
The monoidal category \(\langle\mathcal{C},\otimes,I\rangle\) (the tensor \(\otimes:\mathcal{C}\times\mathcal{C}\to\mathcal{C}\), the unit \(I\), the associator \(\alpha:(X\otimes Y)\otimes Z\to X\otimes(Y\otimes Z)\), the unitors \(\lambda:I\otimes X\to X\), \(\rho:X\otimes I\to X\), the Mac Lane coherence).
| Monoidal / symmetric / braided | Description |
|---|---|
| Monoidal \(\langle\mathcal{C},\otimes,I\rangle\) | the tensor \(\otimes\), the unit \(I\), the associator / unitors (the Mac Lane coherence: all diagrams commute) |
| Symmetric | the \(\sigma_{X,Y}:X\otimes Y\to Y\otimes X\) (the symmetry, \(\sigma^2=1\)) |
| Braided | the \(B_{X,Y}:X\otimes Y\to Y\otimes X\) (the braid, \(B^2\ne1\), the topology) |
| Mac Lane coherence | all coherence diagrams commute (the Mac Lane theorem, the pentagon + triangle) |
The closed / the internal Hom¶
| Closed / internal Hom / CCC | Description |
|---|---|
| Closed monoidal | \([X,Y]\) the internal Hom, the \(\otimes\dashv[-,Y]\) (the adjunction \(\mathcal{C}(Z\otimes X,Y)\cong\mathcal{C}(Z,[X,Y])\)) |
| Cartesian closed (CCC) | the \(\times\) the tensor, \([X,Y]=Y^X\) the exponential, the \(\mathbf{Set}\), the homotopy type theory |
The enriched category¶
| Enriched / \(\mathcal{V}\)-Cat | Description |
|---|---|
| Enriched \(\mathcal{C}\) over \(\mathcal{V}\) | the \(\mathcal{C}(X,Y)\in\mathcal{V}\) (the Hom in \(\mathcal{V}\), the composition \(\circ:\mathcal{C}(Y,Z)\otimes\mathcal{C}(X,Y)\to\mathcal{C}(X,Z)\) in \(\mathcal{V}\)) |
| \(\mathcal{V}\)-Cat | the enriched categories (the \(\mathcal{V}\)-Cat the category of \(\mathcal{V}\)-enriched categories, the higher categories) |
The \(k\)-linear / the abelian¶
| \(k\)-linear / abelian / derived | Description |
|---|---|
| \(k\)-linear | \(\mathcal{C}(X,Y)\in k\text{-}\mathbf{Vec}\) (the \(k\)-linear, the representation theory) |
| Abelian / derived | the abelian the \(k\)-linear + the additive + the kernels/cokernels, the homological algebra |
See it¶
\(\mathbf{Vec}_k\) the monoidal closed and the \(k\)-linear enriched. The \(\mathbf{Vec}_k\) the monoidal (\(\otimes_k\), \(k\) the unit), the closed (\([V,W]=\mathrm{Hom}_k(V,W)\), the \(\otimes\dashv\mathrm{Hom}\): \(\mathrm{Hom}_k(U\otimes V,W)\cong\mathrm{Hom}_k(U,\mathrm{Hom}_k(V,W))\)). The \(k\)-linear: the \(\mathcal{C}(X,Y)\in\mathbf{Vec}_k\) (the \(k\)-linear, the representation theory the enriched). The key point: the \(\mathbf{Vec}_k\) the monoidal closed, the \(k\)-linear the enriched.
Check¶
Verify that \(\mathbf{Vec}_k\) is closed monoidal: \(\mathrm{Hom}_k(U\otimes V,W)\cong\mathrm{Hom}_k(U,\mathrm{Hom}_k(V,W))\).
The tensor-hom adjunction: \(\mathrm{Hom}_k(U\otimes_k V,W)\cong\mathrm{Hom}_k(U,\mathrm{Hom}_k(V,W))\) (the \(k\)-linear maps \(U\otimes V\to W\) the bilinear \(U\times V\to W\) the linear \(U\to\mathrm{Hom}(V,W)\), the universal property of the tensor product). So \([V,W]=\mathrm{Hom}_k(V,W)\) the internal Hom, the \(\otimes\dashv[-,W]\), the \(\mathbf{Vec}_k\) the closed monoidal. The unit \(I=k\) (the \(k\) the one-dimensional, the \(V\otimes k\cong V\)), the associator the canonical (\(k\) the commutative, the \(\otimes_k\) the symmetric). So the \(\mathbf{Vec}_k\) the closed symmetric monoidal.
Explain the enriched category over \(\mathcal{V}\), and the \(k\)-linear as the enriched over \(\mathbf{Vec}_k\).
The enriched category \(\mathcal{C}\) over \(\mathcal{V}\): the \(\mathcal{C}(X,Y)\in\mathcal{V}\) (the Hom-object in \(\mathcal{V}\), not the set), the composition \(\circ:\mathcal{C}(Y,Z)\otimes\mathcal{C}(X,Y)\to\mathcal{C}(X,Z)\) (the morphism in \(\mathcal{V}\)), the identity \(j_X:I\to\mathcal{C}(X,X)\) (the unit). The \(k\)-linear: the \(\mathcal{C}\) enriched over \(\mathbf{Vec}_k\) (the \(\mathcal{C}(X,Y)\in\mathbf{Vec}_k\) the \(k\)-vector space, the composition \(k\)-bilinear, the identity \(k\)-linear) — the \(k\)-linear category the \(\mathbf{Vec}_k\)-enriched. The representation theory: the \(k\)-linear the natural (the \(\mathrm{Hom}\) the \(k\)-linear, the enriched the \(\mathbf{Vec}_k\)).
Pitfalls¶
- The Mac Lane coherence (all coherence diagrams commute — the pentagon + triangle, the Mac Lane theorem) — the monoidal the coherent, not the strict (the strictification the equivalence, the non-strict the general).
- The symmetric vs the braided (the \(\sigma^2=1\) the symmetric, the \(B^2\ne1\) the braided — the topology the braid) — the two the different.
- The closed (\([X,Y]\) the internal Hom, \(\otimes\dashv[-,Y]\)) — not every monoidal is closed (the closed the extra structure).
- The enriched (\(\mathcal{C}(X,Y)\in\mathcal{V}\), not the set) — the enriched the generalization, the ordinary the \(\mathbf{Set}\)-enriched.
Refs¶
- Mac Lane, Categories for the Working Mathematician (Ch. VII: monoidal); the coherence theorem
- Kelly, Basic Concepts of Enriched Category Theory (1982); the \(\mathcal{V}\)-Cat
- Etingof–Gelband–Nikshych–Ostrik, Tensor Categories (the braided / symmetric, the representation theory)
Next¶
CT index · Foundations · Monads & Algebras · Accessible & Presentable · Categorical Logic & Topoi · Higher categories · Representation theory · Homological algebra