nLab pointed mapping space

Idea

The incarnation of internal homs/mapping spaces in the context of pointed objects.

Definition

Definition

Let $\mathcal{C}$ be a closed monoidal category with finite limits.

For $X, Y \in \mathcal{C}^{\ast}$ two pointed objects in $\mathcal{C}$, their pointed mapping space

$$[X,Y]_* \in \mathcal{C}^{\ast/}$$

(the “object of basepoint-preserving maps”), is the pullback

$$\array{ [X,Y]_* & \overset{}{\longrightarrow} & \ast \\ \downarrow &(pb)& \downarrow \\ [X,Y] & \underset{}{\longrightarrow} & [1,Y] }$$

where the morphism $[X,Y]\to [1,Y]$ is induced from the point $\ast\to X$, and the morphism $\ast\to [\ast,Y]$ is the adjunct to $\ast \otimes \ast \to \ast \to Y$.

Regard $[X,Y]_*$ as a pointed object with basepoint induced by the map $\ast\to [X,Y]$ whose adjunct is $\ast \otimes X \to \ast \to Y$.

Properties

Proposition

Let $\mathcal{C}$ be a closed monoidal category with finite limits and with finite colimits.

For every pointed object $X \in \mathcal{C}^{\ast}$ the operation of forming the pointed mapping space out of $X$, and the operation of forming the smash product with $X$, form a pair of adjoint functors

$$( X \wedge (-) \;\dashv\; [X,-]_\ast ) \;\colon\; \mathcal{C}^{\ast/} \leftrightarrow \mathcal{C}^{\ast/} \,.$$

This makes $\mathcal{C}^{\ast/}$ itself a closed monoidal category, which is symmetric if $\mathcal{C}$ is. The tensor unit is $I_+$ for $I$ the unit for the monoidal structure on $\mathcal{C}$.

(Elmendorf-Mandell 07, lemma 4.20)

Remark

The case when $\mathcal{C}$ is cartesian, or at least semicartesian, is most common in the literature.

Remark

If $\mathcal{C}$ is monoidal but not closed, the same definition of the smash product makes $\mathcal{C}^{\ast/}$ monoidal as long as the tensor product of $\mathcal{C}$ preserves finite colimits in each variable separately.

If not, the smash product can fail to be associative. For instance, the smash product on the ordinary category Top (without any niceness conditions imposed) is not associative.