nLab
pullback power

Context

Category theory

Concepts

Universal constructions

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Theorems

Extensions

Applications

Contents

Idea

The formal dual to the pushout product (frequently considered in the context of enriched model category theory) does not have a widely established name, but plausibly deserves to be called the pullback powering operation. Note that it sometimes called pullback hom.

More precisely, a pushout product is defined with respect to a functor of the form E 1×E 2E 3E_1\times E_2 \to E_3, while a pullback power is defined with respect to a functor of the form E 2 op×E 3E 1E_2^{op} \times E_3\to E_1 or E 1 op×E 3E 2E_1^{op}\times E_3 \to E_2, of the sort that would be the right adjoints in a two-variable adjunction.

Pullback powers and pushout products are related to factorization systems by the Joyal-Tierney calculus.

Definition

Let 𝒞\mathcal{C} be category with finite limits and let

[,]:𝒞 op×𝒞𝒞 [-,-] \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow \mathcal{C}

a functor (out of the product category of the opposite category of 𝒞\mathcal{C} with 𝒞\mathcal{C} itself). Then for

g:XY g \;\colon\; X \to Y

and

f:AB f \;\colon\; A \to B

two morphisms in 𝒞\mathcal{C}, their pullback powering g fg^f is the morphism

[B,X][i,p][A,X]× [A,Y][B,Y] [B,X] \stackrel{[i , p]}{\to} [A,X] \times_{[A,Y]} [B,Y]

into the evident fiber product on the right.

Last revised on July 7, 2018 at 09:23:51. See the history of this page for a list of all contributions to it.