nLab pullback power

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 $E_1\times E_2 \to E_3$, while a pullback power is defined with respect to a functor of the form $E_2^{op} \times E_3\to E_1$ or $E_1^{op}\times E_3 \to E_2$, 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 $\mathcal{C}$ be category with finite limits and let

$$[-,-] \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow \mathcal{C}$$

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

$$g \;\colon\; X \to Y$$

and

$$f \;\colon\; A \to B$$

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

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

into the evident fiber product on the right.

Related concepts

• pullback power axiom

• powering

• Joyal-Tierney calculus

• enriched model category