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added to closed monoidal category a proof that the pointwise tensor product on a functor category with complete codomain is closed.
Discussion elsewhere suggested that readers found the bi-closed aspect at closed monoidal category not clear enough, or not highlighted properly to be recognizable.
Therefore I have now edited slightly, trying to make the point clearer. Also added a few more hyperlinks to the existing text.
And added pointers to Lambek’s original articles to the References-section, together with a pointer to the history of the subject over at linear type theory.
As a symmetric closed monoidal category, a cartesian closed category $(C, \times, 1)$ has the properties that (1) the unit $1$ is the terminal object and (2) the tensor product $\times$ distributes over finite coproducts.
Is there a name for symmetric closed monoidal categories that satisfy properties (1) and (2)?
(2) is redundant since in a symmetric monoidal closed category, it is automatic that the tensor product distributes over all colimits.
As for (1): is it true that I’ve heard “affine monoidal”? (Terminology based on the example of affine spaces = vector spaces but forgetting the origin.) I’ll have to check later.
Edit: Oh, the nLab calls it a semicartesian monoidal category, so you could call your thing a semicartesian closed category I guess.
Thanks Todd, that was helpful. I added to semicartesian monoidal category the motivating example $([0,\infty], \ge, + , 0)$ of the extended real numbers.
Surely not.
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).
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!
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.
By the way, Coecke and Lal call semicartesian monoidal categories causal categories.
@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.
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 $T$.
Ah, I see. Cute!
I wonder if one might regard this as another argument against admitting the empty affine space.
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).
@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.
Let’s continue at affine space.
I have added the statement that in a closed monoidal category the tensor/hom adjunction isos always internalize. (here)
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