Construction of biclosed categories

Reference. Construction of biclosed categories

Brian John Day · 1970 · category-theory · DOI

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@phdthesis{day1970construction,
 title = {Construction of biclosed categories},
 author = {Day, Brian John},
 year = {1970},
 school = {University of New South Wales PhD thesis}
}

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yaml · 7 lines

day1970construction:
  type: thesis
  title: Construction of biclosed categories
  author: Day, Brian John
  date: 1970
  organization: University of New South Wales PhD thesis
  genre: Doctoral dissertation

Download: day1970construction.bib · day1970construction.yml

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2-dimensional Lawvere theories, commutativity, and higher Day convolution perutka_2026arXiv · 2026

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Theorem. Day Convolution is Closed day-closed-structure

Let $𝒱︀$ be a symmetric monoidal closed category that is complete and cocomplete. Let $(𝒞︀, \otimes_{𝒞︀}, 𝐼)$ be a small monoidal $𝒱︀$ -enriched category and $𝐴, 𝐵$ be $𝒱︀$ -enriched presheaves on $𝒞︀$ . Define

(𝐴 ⊸ 𝐵) 𝑐 = \int_{𝑢} 𝐴 𝑢 ⊸_{𝒱︀} 𝐵 (𝑢 \otimes_{𝒞︀} 𝑐) .

Then, for Day convolution $\otimes_{Day}$ ,

𝐴 \otimes_{Day} - ⊣ 𝐴 ⊸_{Day} - .

Proof. Proof that Day Convolution is Closed day-closed-structure-proof

For $𝑋, 𝐵 : {𝒞︀}^{op} \to 𝒱︀$ , the enriched hom in $[{𝒞︀}^{op}, 𝒱︀]$ is given by the end:

\begin{aligned} \int_{𝑐} ((𝐴 \otimes_{Day} 𝑋) 𝑐 ⊸_{𝒱︀} 𝐵 𝑐) & ≅ \int_{𝑐} ((\int^{𝑢, 𝑣} 𝒞︀ (𝑐, 𝑢 \otimes_{𝒞︀} 𝑣) \otimes_{𝒱︀} 𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} 𝐵 𝑐) \\ ≅ \int_{𝑐} \int_{𝑢, 𝑣} ((𝒞︀ (𝑐, 𝑢 \otimes_{𝒞︀} 𝑣) \otimes_{𝒱︀} 𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} 𝐵 𝑐) \\ ≅ \int_{𝑢, 𝑣} ((𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} \int_{𝑐} (𝒞︀ (𝑐, 𝑢 \otimes_{𝒞︀} 𝑣) ⊸_{𝒱︀} 𝐵 𝑐)) \\ ≅ \int_{𝑢, 𝑣} ((𝐴 𝑢 \otimes_{𝒱︀} 𝑋 𝑣) ⊸_{𝒱︀} 𝐵 (𝑢 \otimes_{𝒞︀} 𝑣)) \\ ≅ \int_{𝑣} (𝑋 𝑣 ⊸_{𝒱︀} \int_{𝑢} (𝐴 𝑢 ⊸_{𝒱︀} 𝐵 (𝑢 \otimes_{𝒞︀} 𝑣))) \\ ≅ \int_{𝑣} (𝑋 𝑣 ⊸_{𝒱︀} (𝐴 ⊸ 𝐵) 𝑣) . \end{aligned}

∎

Symmetrically, $(𝐵 ⟜ 𝐴) 𝑐 = \int_{𝑣} 𝐴 𝑣 ⊸_{𝒱︀} 𝐵 (𝑐 \otimes_{𝒞︀} 𝑣)$ and $- \otimes_{Day} 𝐴 ⊣ - ⟜ 𝐴$ , so the enriched presheaf category $[{𝒞︀}^{op}, 𝒱︀]$ is biclosed [1].

Definition. Day Convolution day-convolution

Let $𝒱︀$ be a symmetric monoidal closed category that is complete and cocomplete. Let $(𝒞︀, \otimes_{𝒞︀}, 𝐼)$ be a small monoidal $𝒱︀$ -enriched category. The Day convolution of $𝒱︀$ -enriched presheaves $𝐴, 𝐵 : {𝒞︀}^{op} \to 𝒱︀$ is the enriched presheaf

(𝐴 \otimes_{Day} 𝐵) 𝑐 = \int^{𝑢, 𝑣} 𝒞︀ (𝑐, 𝑢 \otimes_{𝒞︀} 𝑣) \otimes_{𝒱︀} 𝐴 𝑢 \otimes_{𝒱︀} 𝐵 𝑣,

with unit the representable $𝒞︀ (-, 𝐼)$ .

Day convolution makes the enriched presheaf category $[{𝒞︀}^{op}, 𝒱︀]$ a monoidal category, symmetric when $𝒞︀$ is [1]. It is moreover closed.

Under the enriched Yoneda embedding the convolution of representables is representable, $𝒞︀ (-, 𝑥) \otimes_{Day} 𝒞︀ (-, 𝑦) ≅ 𝒞︀ (-, 𝑥 \otimes_{𝒞︀} 𝑦)$ , so Day convolution is the cocontinuous extension of the tensor of $𝒞︀$ .

As a Kan Extension

Equivalently, $𝐴 \otimes_{Day} 𝐵$ is the left Kan extension of $(𝑢, 𝑣) \mapsto 𝐴 𝑢 \otimes_{𝒱︀} 𝐵 𝑣$ along $\otimes_{𝒞︀}^{op} : {𝒞︀}^{op} \times {𝒞︀}^{op} \to {𝒞︀}^{op}$ .

In $𝐒𝐞𝐭$

When $𝒱︀ = 𝐒𝐞𝐭$ , we recover the ordinary Day convolution of presheaves $𝐴, 𝐵 : {𝒞︀}^{op} \to 𝐒𝐞𝐭$ , where the formula simplifies to:

(𝐴 \otimes_{Day} 𝐵) 𝑐 = \int^{𝑢, 𝑣} 𝒞︀ [𝑐, 𝑢 \otimes_{𝒞︀} 𝑣] \times 𝐴 𝑢 \times 𝐵 𝑣 .

Cites 15 works (0 here)

External (15)

On topological quotient maps preserved by pullbacks or products (1970)
On closed categories of functors (1970)
Enriched functor categories (1969)
V-completions by V-monads through the use of Kan extensions (1969)
Adjunction for enriched categories (1969)
Closed categories generated by commutative monads (1969)
Coequalisers in categories of algebras (1969)
Relationship of Spanier's quasi-topological spaces to k-spaces (1968)
On monads in symmetric monoidal closed categories (1968)
Properties of dense and relative adjoint functors (1968)
Introduction to bicategories (1967)
Cohomology in tensored categories (1966)
A generalisation of the functorial calculus (1966)
Autonomous equational categories (1966)
On Kan functor extensions (1966)