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Assume I have simplicial sets $X:\Delta^{op}\rightarrow Set$ and $Y:\Delta^{op}\rightarrow Set$, then I can form the simplicial set

$\operatorname{Hom}(X,Y)_n := \operatorname{Hom}_{sSet}(\Delta^n\times X,Y)$ where $\Delta^n := \operatorname{Hom}_{\Delta}(\cdot,[n])$. Then I have the ‘adjunction relation’:

$\operatorname{Hom}(Z,\operatorname{Hom}(X,Y)) \simeq \operatorname{Hom}(Z\times X,Y)$

Can I do the same thing for simplicial topological spaces $X,Y:\Delta^{op}\rightarrow \operatorname{Top}$?

Are there references on that or is that trivially the same thing? What would be $\Delta^n$ and what topologies do you take on the $\operatorname{Hom}(X,Y)_n$?

In particular, I would like to have the same adjunction relation: $\operatorname{Hom}(Z,\operatorname{Hom}(X,Y)) \simeq \operatorname{Hom}(Z\times X,Y)$

but I can’t figure out the details of how to construct the simplicial topological space $\operatorname{Hom}(X,Y)$.

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- Can it happen that the image of a functor is not a category?

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- The categories Set and Ens
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- Theorems implied by Yoneda's lemma?
- Example of an additive functor admitting no right derived functor

Here is some general nonsense that will do what you want. Suppose $\mathcal{C}$ is a locally small cartesian-closed category with limits and colimits for all small diagrams (so, for example, $\textbf{CGHaus}$ or $\textbf{Set}$) and suppose $\mathcal{D}$ is a small category. Then the functor category $[\mathcal{D}, \mathcal{C}]$ will be a cartesian-closed category with limits and colimits for all small diagrams as well. The second part is a standard fact, so I will only explain why $[\mathcal{D}, \mathcal{C}]$ is cartesian-closed.

Let $Y$ and $Z$ be objects in $[\mathcal{D}, \mathcal{C}]$, i.e. functors $\mathcal{D} \to \mathcal{C}$. We define a new object $Z^Y$ using the following formula:

$$Z^Y (t) = \int_{d : \mathcal{D}} {Z (d)}^{\mathcal{D}(t, d) \otimes Y (d)}$$

The integral notation refers to ends, and $\otimes$ is defined by the adjunction below:

$$\mathcal{C}(A \otimes C’, C) \cong \textbf{Set}(A, \mathcal{C}(C’, C))$$

The set of natural transformations can also be computed by an end:

$$[\mathcal{D}, \mathcal{C}](X, W) \cong \int_{t : \mathcal{D}} \mathcal{C} (X (t), W (t))$$

Moreover, $\mathcal{C}(C, -)$ preserves ends, so we have

$$[\mathcal{D}, \mathcal{C}] \left( X, Z^Y \right) \cong \int_{t : \mathcal{D}} \int_{d : \mathcal{D}} \mathcal{C} \left( X (t), {Z (d)}^{\mathcal{D}(t, d) \otimes Y (d)} \right)$$

and by the definition of exponential objects in $\mathcal{C}$,

$$[\mathcal{D}, \mathcal{C}] \left( X, Z^Y \right) \cong \int_{t : \mathcal{D}} \int_{d : \mathcal{D}} \mathcal{C} \left( X (t) \times (\mathcal{D}(t, d) \otimes Y (d)), Z (d) \right)$$

and since $C \times (-) : \mathcal{C} \to \mathcal{C}$ has a right adjoint, it preserves $\otimes$, i.e.

$$X (t) \times (\mathcal{D}(t, d) \otimes Y (d)) \cong \mathcal{D}(d, t) \otimes (X (t) \times Y (d))$$

and so, using the definition of $\otimes$, we obtain

\begin{align}

[\mathcal{D}, \mathcal{C}] \left( X, Z^Y \right)

& \cong \int_{t : \mathcal{D}} \int_{d : \mathcal{D}} \textbf{Set} \left( \mathcal{D}(t, d), \mathcal{C} \left( X (t) \times Y (d), Z (d) \right) \right) \\

& \cong \int_{d : \mathcal{D}} \int_{t : \mathcal{D}} \textbf{Set} \left( \mathcal{D}(t, d), \mathcal{C} \left( X (t) \times Y (d), Z (d) \right) \right)

\end{align}

where the second isomorphism is the interchange theorem for ends; but the end version of Yoneda lemma says

$$\int_{t : \mathcal{D}} \textbf{Set} \left( \mathcal{D}(t, d), F t \right) \cong F d$$

for any functor $F : \mathcal{D}^\textrm{op} \to \textbf{Set}$, hence,

$$[\mathcal{D}, \mathcal{C}] \left( X, Z^Y \right) \cong \int_{d : \mathcal{D}} \mathcal{C} \left( X (d) \times Y (d), Z (d) \right) \cong [\mathcal{D}, \mathcal{C}] (X \times Y, Z)$$

and this isomorphism is natural in $X$, $Y$, and $Z$.

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