Mentioned the result from MacLane that allows to conclude that the natural isomorphism is natural in three variables from just an adjunction for each object.

]]>deleting the pointer to

- François Foltz, Christian Lair, G. M. Kelly,
*Algebraic categories with few monoidal biclosed structures or none*, Journal of Pure and Applied Algebra**17**2 (1980) 171-177 [pdf, doi:10.1016/0022-4049(80)90082-1]

here and moving it to *symmetric closed monoidal category*

Added the example that Cat has precisely two closed monoidal structures, as shown in:

- François Foltz, Christian Lair, and G. M. Kelly,
*Algebraic categories with few monoidal biclosed structures or none*, Journal of Pure and Applied Algebra**17**2 (1980), 171-177. (pdf, doi:10.1016/0022-4049(80)90082-1)

Fix reference to unbound variable.

Tuplanolla

]]>touched wording and formatting of the first few examples (here)

and added mentioning of the previously missing examples of presheaves, vector spaces, and chain complexes

]]>Okay, good. Thanks for fixing the typo.

(Incidentally, that entry *closed monoidal category* deserves the attention of an energetic editor: So far it does not do much beyond re-stating the definition of internal homs, three times in a row :-/)

Thanks for the improvement! I’m glad to help. I fixed a easy typo in your edit \maspto to \mapsto.

]]>What you suggest does not quite type-check, and I think the statement in the entry was correct. But I have now edited the pragraph a little to harmonize the variable names, which may make it easier to spot what the left- and the right adjoints are doing:

…such that for each object $Y \,\in\, \mathcal{C}$

the functor

$(-) \otimes Y \;\colon\; C \longrightarrow C$of forming the tensor product with $Y$,

has a right adjoint functor

$[Y,-] \;\colon\; C \longrightarrow C$forming the internal-hom out of $Y$

in that for all triples of objects $X,\, Y,\, Z$ there is a natural hom-isomorphism of the following form:

$Hom_{\mathcal{C}}(X \otimes Y,\, Z) \;\simeq\; Hom_{\mathcal{C}}\big(X, [Y,Z] \big)$]]>

Should we distinguish the definition we give as a defining a right closed monoidal category?

The Wikipedia article //en.wikipedia.org/wiki/Closed_monoidal_category (https) mentions the left version under the first definition.

added pointer to Borceux.

Does Borceux ever mention compact closure? Or else, what would be a good textbook reference for compact closed categories?

]]>Added lctvs with the inductive tensor product as an example

]]>I have added the statement that in a closed monoidal category the tensor/hom adjunction isos always internalize. (here)

]]>Let’s continue at affine space.

]]>@Mike: I was wondering the same thing.

@David: It may not have been said in exactly those terms, but similar points have been made elsewhere (e.g., an affine space is the same as a torsor of its space of translations). And that kind of point has some validity and force, surely, but not enough (for me, anyway) to override the opposing feeling that affine spaces over a field should form a variety, should form a complete/cocomplete category, etc., and that we teach our students that the solution space to Ax = b may be empty, etc. etc. This might be another instance of the nice object / nice category dichotomy. (Speaking only for myself, I’m not very comfortable at present with the nLab’s leaning toward the “nice object” side of the debate in affine space.)

In the end whichever notion of affine space we choose probably comes down to context and expediency, but I’d be happier to see the *category* of inhabited affine spaces arise organically from satisfying categorical considerations. Along lines vaguely analogous to Tom Leinster’s Café post on how he came to love the nerve construction, which among other things lays out organic categorical reasons for choosing the simplex category over the augmented simplex category. There is a vague sense within the current thread (which I suppose should go elsewhere than “closed monoidal category”) that the structure of the slice construction $Vect/k$, which arises by applying the coreflection of monoidal categories into semicartesian monoidal categories to $(Vect, \otimes)$, might provide a clue to this, maybe via this observation about $Vect/k$ also being a collage construction.

If you think of an affine space as a (locally trivial) principal V-bundle over a point, then it should be inhabited (maybe this point has been made already elsewhere).

]]>Ah, I see. Cute!

I wonder if one might regard this as another argument against admitting the empty affine space.

]]>Yes, they are almost disjoint, except that you do have maps $\phi: (V, 0) \to (W, g)$ from a “vector space” to an “affine space” whenever $\phi \circ g = 0$.

I think what we have in fact is $Vect/k$ realized as a collage of vector spaces with inhabited affine spaces (I’ll denote that category by $Aff_+$). It’s the collage induced by the bimodule $B: Vect^{op} \times Aff_+ \to Set$ where $B(V, A) = Vect(V, T(A))$; here $T$ is the functor which takes an affine space $A$ to its vector space of translations $T(A)$, which we can also express as the composite

$Aff_+ \to Vect/k \stackrel{\ker}{\to} Vect/0 \simeq Vect$where $\ker$ is of course the pullback along $0 \to k$. I guess you’d just call that (some version of) the cograph of $T$.

]]>@Todd: Cute. And the objects that aren’t inhabited are exactly those in the image of $Vect$, right? So $Vect\cup Aff$ is all of $Vect/k$, and $Vect\cap Aff$ is $0\to k$, the zero vector space and the empty affine space. There are no maps from an inhabited affine space to a vector space, and there is only the zero map from a vector space to an affine space, so the two are almost disjoint.

]]>By the way, Coecke and Lal call semicartesian monoidal categories *causal categories*.

Thanks! Something I found a little amusing yesterday was to consider the specific case $Vect/k$ where $k$ is the ground field for $Vect$. Both $Vect$ and $Aff$ (affine spaces over $k$, including the empty one) embed fully in this category (the latter as a *monoidal* subcategory). For $Vect$ the embedding is $V \mapsto (V, 0: V \to k)$.

The embedding of $Aff$ takes a little more time to spell out. It’s $A \mapsto (1 \sqcup A, \pi)$ where $\sqcup$ is the coproduct in affine spaces (akin to a simplicial join), $1$ is the terminal affine space, and $\pi$ is the composite of $1 \sqcup !: 1 \sqcup A \to 1 \sqcup 1$ with a natural identification $\mu: 1 \sqcup 1 \cong k$. Both $1 \sqcup !$ and $\mu$ which are morphisms of $Aff$ may be regarded as morphisms of $1 \downarrow Aff \simeq Vect$ (pointed affine spaces are vector spaces) if we let the first inclusion $i_0: 1 \to 1 \sqcup 1$ be the pointing of $1 \sqcup 1$ and $0: 1 \to k$ the pointing of $k$ and define $\mu$ by $\mu \circ i_0 = 0$, $\mu \circ i_1 = 1$ (the element $1 \in k$). (So $\mu$ is like two ends of a meter stick used to set up coordinates on the line $k$.)

“Most” objects $(V, f: V \to k)$ of $Vect/k$ are “inhabited” in the sense that the projection to the terminal is regular epic; this means $f: V \to k$ is epic. For such objects, a morphism $(V, f) \to (W, g)$ determines and is uniquely determined by the affine map $f^{-1}(1) \to g^{-1}(1)$ between the fibers over $1 \in k$, and thus we identify the full subcategory of inhabited objects of $Vect/k$ with the category of inhabited affine spaces.

]]>I kind of doubt that the notion of semicartesian monoidal category is commonly enough used for anything to be “standard”. But the observation about slices is nice!

]]>The usual examples do have a symmetric monoidal product though. It made me pause to wonder whether the nLab definition is the standard one.

**Edit:** I’ll retract my previous sentence; I’ll bet it’s quite standard. In fact for any monoidal category $M$, the slice $M/I$ over the unit is semicartesian, and that should be a universal example. I added some words to that effect (which should be checked for accuracy).

Surely not.

]]>