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Tensor and cotensor in a V-category
Definition
Let be a -category and assume is right closed, so that it is enriched in itself (A closed monoidal category is enriched in itself).
For and :
- a cotensor of by is an object written together with isomorphisms in natural in ;
- a tensor of by is an object written together with isomorphisms in natural in .
These are the one-object-indexed weighted limit and weighted colimit cases of Set-weighted limits and colimits after replacing set-valued weights by -valued ones and ordinary hom-sets by enriched hom-objects.
In the special case , cotensors are the powers and tensors are the copowers.
Depends on
Used by
- A bijection on underlying hom-sets need not exhibit a cotensor Counterexample
- A cotensor in Set is a power Example
- A V-category is tensored exactly when each covariant hom has a left enriched adjoint Theorem
- Conical weights are a proper special case of enriched weights Theorem
- Enriched adjoint functor theorem for cotensored categories Theorem
- Enriched completeness is cotensors plus small conical limits Theorem
- When a category is tensored, every limit in it is a conical enriched limit Theorem
Dependency tree · two levels
17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- G. M. Kelly, Basic Concepts of Enriched Category Theory, equations (3.42) and (3.43) (standard reference, not scraped)
- Emily Riehl, Categorical Homotopy Theory, Section 3.7 (standard reference, not scraped)