Re. #24. Just to point out that the structure you are describing is a two-variable adjunction. The internal hom can be phrased in both ways, but I agree with Hurkyl that in practice one is often interested in the setting in which the hom exists only for some objects.

]]>I suspect the motivation for that framing is that it’s often interesting to talk about exponentiable objects even when not every object is exponentiable, which leads to one talking about the situation where *every* object is exponentiable.

I would actually suggest rewriting this definition in the opposite way: define the internal hom pointwise – i.e. for each pair $c_0, c_1$, define an internal hom to be a representing object* for $\hom(c_0 \otimes -, c_1)$ if one exists – and then invoke yoneda to prove that it’s uniquely functoral on the full subcategory where they exist for any global choice of internal homs for each pair. That way it’s more general; you can talk about internal homs even when they don’t exist for every pair. I suppose taking the functor as part of the definition sweeps the “once you make a choice” under the rug, for better or for worse.

*: For the sake of precision, by “representing object” I mean to include both the object $[c_0, c_1]$ and the natural isomorphism $\hom(-, [c_0, c_1]) \cong \hom(c_0 \otimes -, c_1)$ is part of the data.

]]>Given a monoidal category $(\mathcal{C},\otimes)$, the usual “provisional” definition of ‘right internal hom of $(\mathcal{C}, \otimes)$’ is a (unique in the appropriate sense, if it exists) collection indexed over objects $c$ of $\mathcal{C}$ of functors $[c, -] \colon \mathcal{C} \to \mathcal{C}$ along with adjunctions $c \otimes (-) \dashv [c, -]$.

While this agrees with the ultimately desired bifunctorial notion on objects and on morphisms in the second argument, it is not a priori functorial in the first. In particular, the definition of ‘right internal hom of $(\mathcal{C},\otimes)$’ that we ultimately desire is a bifunctor $[-, -] \colon \mathcal{C}^{op} \times \mathcal{C} \to \mathcal{C}$ that satisfies

$Hom_{\mathcal{C}}(c_{0} \otimes c', c_{1}) \simeq Hom_{\mathcal{C}}(c', [c_{0}, c_{1}])$naturally in all three arguments.

Accordingly, Proposition 3.1 establishes that a right internal hom in the former provisional sense can always be (uniquely, in the appropriate sense) upgraded to a right internal hom in the latter bifunctorial sense.

But Yoneda’s lemma implies that the weaker condition that for all pairs of objects $c_{0}$, $c_{1}$ of $\mathcal{C}$, the functor $c' \mapsto Hom_{\mathcal{C}}(c_{0} \otimes c', c_{1})$ is representable suffices to imply the existence of the aformentioned bifunctor.

I guess my question is whether it would it be of any utility to rewrite Definition 2.1 so that ‘a right internal hom of $(\mathcal{C}, \otimes)$’ is defined to be a bifunctor $[-, -] \colon \mathcal{C}^{op} \times \mathcal{C} \to \mathcal{C}$ equipped with a natural (in $c$, $c_{0}$, and $c_{1}$) isomorphism

$\kappa \colon Hom_{\mathcal{C}}(c_{0} \otimes c', c_{1}) \to Hom_{\mathcal{C}}(c', [c_{0}, c_{1}])$and then to prove as propositions

That there is a unique natural transformation from any one such right internal hom to any other such internal hom that is compatible with its Currying isomorphism

Thus that a right internal Hom of $(\mathcal{C}, \otimes)$ is unique if it exists)

That the existence of such a right internal hom is equivalent to the practically weaker condition that for all pairs $(c_{0}, c_{1})$ of objects of $\mathcal{C}$, the functor $c' \mapsto Hom_{\mathcal{C}}(c_{0} \otimes c', c_{1})$ is representable.

In particular, we would be eliding adjunction from the discussion in favor of applying Yoneda’s lemma directly.

]]>Added link to residual, though I’m not sure these pages ought to be distinct.

]]>Fixed typo.

]]>Changed to “functions with domain X”, which seems maximally unambiguous.

]]>I think “function on” is familiar to me, but mostly when it’s something like real- or complex-valued functions. It would be nice to know what “contradictory” meaning linkhyrule5 has in mind here. I think “functions out of $X$” or “functions from $X$” both work for me (both with the implication of arrows pointing away from, not toward, $X$).

]]>What? The only other meaning I can think of is “taking things out of a box” but that doesn’t even make any sense as an interpretation here (“the set of all functions” is not “inside” of $X$ in any sense). And “function out of $X$” is a standard mathematical usage for “function with domain $X$”, more standard in my idiolect than “function on $X$”.

]]>Clarification; ’taking … out of’ has an existing and contradictory meaning in English.

linkhyrule5

]]>I am sorry that I apparently consistently come across as being mysterious, I wish I knew what is causing it. All I am saying is that eventually the material on topological mapping spaces should be split off from the entry on internal homs to a stand-alone entry. It’s a comment on formatting/entry layout. It seems completely obvious to me, which may be the cause of me not communicating it properly.

]]>Possibly I could be of assistance, if I knew what you’d want included.

]]>The topological mapping spaces should have a dedicated article of their own, not inside the much broader entry on internal homs.

]]>This is not doubting you, but what aspects of “mapping space” spill outside this redirect?

]]>just a reminder to myself and anyone who cares:

currently *mapping space* is still just redirecting to *internal hom*. Clearly this is not sustainable. Ought to be improved on…

added a small paragraph on and pointer to *stable splitting of mapping spaces*.

Thanks for catching this. Fixed now.

]]>I think there's a typo in the first equation of proposition 3.3, colimit notation should be changed to the limit notation.

Thanks ]]>

I gave the entry a section *Basic properties* with statement and proof of some of the basic properties:

$X \otimes (-) \dashv [X,-]$ for each $X$ is sufficient to have bifunctor $[-,-]$;

hom adjunction internalizes as $[-,[-,-]] \simeq [-\otimes -,-]$;

$[-,-]$ preserves limits in second argument and sends colimits in first to limits

I also did some editorial edits:

Made explicit that the def as stated is for

*symmetric*monoidal categories.Moved the definition of evaluation map and composition map from “Properties” to “Definition”.

internal hom contained a (trivial) mistake in the sentence

Formally this says that the functor $(-) \times X$ of taking the cartesian product with the set $S$ has a right adjoint given by the construction $[X,-]$.

I mean, it is obvious from this sentence, at least when taken out of context like above, that the $S$ is wrong and should be $X$.

It becomes sort-of-less-wrong in context, i.e. in its former version,

One way to make this precise is to mimic the basic property of a function set $[X,Y] = \{f : X \to Y\}$ of functions between two sets $X$ and $Y$: that is uniquely characterized by the fact that for any other set $S$ the functions $S \to [X,Y]$ are in natural bijection with the functions $S \times X \to Y$ out of the cartesian product of $S$ with $X$. Formally this says that the functor $(-) \times X$ of taking the cartesian product with the set $S$ has a right adjoint given by the construction $[X,-]$.

in view of the “formal variable” $S$, which is presumably the cause for the slight inaccuracy. I therefore simply took out the repeating-what-the-notation-already-says phrase “taking the cartesian product […]” from the statement of the adjunction, but somehow conserved it by giving it a place inside a verbal repetition of the adjunction, in the spirit of the known functors-are-verbs-metaphor.

While I was at it, I made various *stylistic* changes to the paragraph, all of which I think are improvements. I will not run through all of them here. I found fault with “the basic property” and “Formally this says” (I mean, wasn’t the sentence before rather formal already; I changed it to a simple “That is:”)

Todd pointed out at the Cafe that two of the examples at internal hom used different notation from the rest of the page (and, arguably, poorer notation as well), so I changed them to make the page consistent. I also rewrote the section on internal-homs of super vector spaces to make sense in terms of the definition of a graded vector space as an indexed family of vector spaces, which is the category-theoretically sensible one.

]]>I kept adding some things to *internal hom*. It's not done yet, but I need to quit now.

afterwards I realized that Idea-section at *internal hom* was not in good shape. I have now spent some minutes on trying to improve it.

I notice this generally, and it is a bit of a pity: of all entries on the $n$Lab specifically those related to basic category are often poor… and not touched for a long while. This is because we created many of them right when the $n$Lab was very young, and didn’t yet have a more “professional” perspective on writing entries. But it’s still a pity. I wish somebody would give a course on basic category theory and use the occasion to boost the relevant $n$Lab entries to at least the level of decent course material.

]]>have expanded at *internal hom* the first two subsections of the *Examples*-section a little: *In a sheaf topos* and *In smooth spaces*.

Thank you for nice formatted archiving version. I have left a link at the internal hom to this discussion (I think we should nearly always do that when archiving). We need also a canonical reference for this entry (maybe Kelly ??).

]]>