Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: theory: category theory

Topic: From *-autonomous to compact closed

Tobias Fritz (Apr 17 2023 at 07:30):

The nlab page on $*$-autonomous categories states in Remark 2.4 that if a $*$-autonomous category has a natural isomorphism $(A \otimes B)^* \cong A^* \otimes B^*$, then it is compact closed. The proof of this is not obvious, so I wonder if anyone knows of a reference where it's spelled out?

I do have a proof sketch (so I'm not asking how to prove it), but it would be nice to have something to refer to. I've been looking at a few papers on $*$-autonomous categories without seeing it mentioned.

Context: I'm dealing with a symmetric monoidal category which appears to be compact closed, but proving that directly seems tedious. But fortunately, proving $*$-autonomy as well as $(A \otimes B)^* \cong A^* \otimes B^*$ is something that I should be able to do.

JS PL (he/him) (Apr 17 2023 at 07:34):

So I looked into this also and my conclusion was that a proof has never been published in a journal paper

JS PL (he/him) (Apr 17 2023 at 07:35):

The result is mentioned in passing here: https://core.ac.uk/download/pdf/82696829.pdf on page 2

JS PL (he/him) (Apr 17 2023 at 07:38):

Tobias Fritz said:

I do have a proof sketch (so I'm not asking how to prove it)

I do have photos of whiteboards where I prove this using string diagrams. If anyone is interested. And I agree, the proof is definitely not obvious

Tobias Fritz (Apr 17 2023 at 07:47):

Thank you! Yes, I'd still be interested in seeing a clean string-diagrammatic proof :smiling_face:

JS PL (he/him) (Apr 17 2023 at 08:21):

Sure!

JS PL (he/him) (Apr 17 2023 at 08:22):

First, I had to prove that a symmetric monoidal closed category is compact closed if the canonical map
$\kappa: (A \multimap I) \otimes A \to A \multimap A$ is an isomorphism.

JS PL (he/him) (Apr 17 2023 at 08:23):

This is stated in this paper HERE but without proof. So here it is in string diagrams:

JS PL (he/him) (Apr 17 2023 at 08:23):

IMG_0028.JPG

JS PL (he/him) (Apr 17 2023 at 08:25):

((Chris Heunen pointed out to me that I may have twisted things the wrong way, depending on your conventions))

JS PL (he/him) (Apr 17 2023 at 08:26):

Now to prove that star autonomous category is compact closed if the canonical map $m_{A,B}: A^\ast \otimes B^\ast \to (A \otimes B)^\ast$ is an isomorphism.

JS PL (he/him) (Apr 17 2023 at 08:27):

Now star autonomous categories are closed. So we need only show that $\kappa$ is an isomorphism

JS PL (he/him) (Apr 17 2023 at 08:29):

It turns out that $\kappa$ is equal to the composite of isomorphisms, one of which is $m^{-1}$, so your done!

JS PL (he/him) (Apr 17 2023 at 08:29):

IMG_0032.JPG

JS PL (he/him) (Apr 17 2023 at 08:31):

So it's not a direct proof. But I recall trying to prove a direct proof was not very fun.

JS PL (he/him) (Apr 17 2023 at 08:32):

You can combine these two to get a direct proof. But this is what I have

JS PL (he/him) (Apr 17 2023 at 08:33):

Tobias Fritz said:

clean string-diagrammatic proof :smiling_face:

Well hopefully that's clean enough to follow/read. (I took those pictures back in 2021) Hope this helps!
Happy to write up a cleaner version (not on a whiteboard!)

Cole Comfort (Apr 17 2023 at 10:10):

JS PL (he/him) said:

Tobias Fritz said:

clean string-diagrammatic proof :smiling_face:

Well hopefully that's clean enough to follow/read. (I took those pictures back in 2021) Hope this helps!
Happy to write up a cleaner version (not on a whiteboard!)

If you do end up typesetting this, you should post it to the nab!

dusko (Apr 17 2023 at 21:27):

Tobias Fritz said:

The nlab page on $*$-autonomous categories states in Remark 2.4 that if a $*$-autonomous category has a natural isomorphism $(A \otimes B)^* \cong A^* \otimes B^*$, then it is compact closed. The proof of this is not obvious, so I wonder if anyone knows of a reference where it's spelled out?

I do have a proof sketch (so I'm not asking how to prove it), but it would be nice to have something to refer to. I've been looking at a few papers on $*$-autonomous categories without seeing it mentioned.

Context: I'm dealing with a symmetric monoidal category which appears to be compact closed, but proving that directly seems tedious. But fortunately, proving $*$-autonomy as well as $(A \otimes B)^* \cong A^* \otimes B^*$ is something that I should be able to do.

i am doing my best to not be an old fart who complains how the world was better when they were young. but star-autonomous categories emerged as the refinement of compact categories when the involutive duality does not preserve the tensor, but creates another one. so the fact that a star-autonomous category where the duality preserves the tensor is compact was a throwaway remark in most early papers, and should be in barr's book where he defined the star-autonomy. it was the stepping stone into the constructions of interaction categories in my paper with samson abramsky from the 90s, and was a stepping stone into 2-dimensional star-autonomy here: https://arxiv.org/abs/1006.1011.

the earliest example of star-autonomy was the category of pairs of vector spaces $\langle V\supseteq U\rangle$ with $\langle V\supseteq U\rangle^\ast = \langle V^\ast\supseteq U^\bot\rangle$, in yuri manin's original framework for non-commutative geometry from the early 80s. lawvere was always saying that george mackey's 1942 thesis was the original source for the chu construction. there the duality on the infinite-dimensional spaces induces star-autonomy, and reduces to compactness for finite-dimensional spaces.

another "obvious" place where star-autonomy emerges from compactness and collapses back to it were girard's coherence spaces that led him to linear logic. the category of relations is, of course, the epitome of compactness, and restricting the relations by coherences splits the relational tensor into two. that was i think also in my paper with samson.

while the general proof that star autonomy where the two tensors boil down to one should be in barr, and should be something like 2 steps of bijections on hom-sets, i am sure that everyone will prefer the string diagrammatic proof. barr's book was still typed on a typewriter by the departmental secretaries...

Mike Shulman (Apr 18 2023 at 01:23):

Also perhaps worth noting is that while being compact is a mere property of a symmetric monoidal category, being $\ast$-autonomous is structure, and a given smc can "be $\ast$-autonomous" in more than one way. As shown in this answer if an smc has one $\ast$-autonomous structure, then the groupoid of $\ast$-autonomous structures is a torsor over the [[Picard 2-group]].

JS PL (he/him) (Apr 18 2023 at 02:15):

A simple example of a symmetric monoidal category with multiple star-autonomous structure is the poster $(\mathbb{R}, \leq)$ where $x \otimes y = x + y$. Any $r \in \mathbb{R}$ can be a dualizing object and the resulting par (which I will write as $\oplus$) is $x \oplus y = x + y -r$. Also note that $(\mathbb{R}, \leq)$ is compact closed!

JS PL (he/him) (Apr 18 2023 at 02:18):

While compact closed is a property, a compact closed category can be a star-autonomous category in another way by choosing another dualizing object. Compactness comes from specifically choosing the monoidal unit as the dualizing object plus some extra identities holding.

Mike Shulman (Apr 18 2023 at 04:21):

The same construction works for any partially ordered group, or indeed any 2-group.

Tobias Fritz (Apr 18 2023 at 05:40):

dusko said:

i am doing my best to not be an old fart who complains how the world was better when they were young. but star-autonomous categories emerged as the refinement of compact categories when the involutive duality does not preserve the tensor, but creates another one. so the fact that a star-autonomous category where the duality preserves the tensor is compact was a throwaway remark in most early papers, and should be in barr's book where he defined the star-autonomy.

That sounds very reasonable. But I just don't see the statement in Barr's book, which is of course one of the places where I had looked before asking here. So if you still think that it is in there, I would appreciate a page number. I haven't read the whole book, and I don't want to: all I need is a reference for the simple fact discussed in this thread, and I do not work with $*$-autonomous categories otherwise.

while the general proof [..] should be something like 2 steps of bijections on hom-sets

That's also what I've been thinking. I will try to work it out in detail and post it here if it works, as that would be simpler than @JS PL (he/him)'s proof.

Tobias Fritz (Apr 18 2023 at 06:59):

I've finally found a reference! It's the implication (a)=>(b) of Theorem 1.3 in Duality, trace and transfer by Dold and Puppe:
image.png

They apparently didn't know about $*$-autonomous categories (which were introduced a few years prior to that), and their statement has slightly weaker assumptions in that they start with a natural isomorphism $\mathscr{C}(A \otimes B, I) \cong \mathscr{C}(A, B^*)$ only, but they also require the induced morphisms $A^* \otimes B^* \to (A \otimes B)^*$ to be isomorphisms. The proof isn't fully spelled out there either, but it's good enough as a reference for my purposes.

Tobias Fritz (Apr 18 2023 at 07:01):

Perhaps someone more familiar with $*$-autonomous categories than I am can double-check that this is the right statement, and if it is I would add this to the nLab page.

dusko (Apr 18 2023 at 08:16):

Mike Shulman said:

Also perhaps worth noting is that while being compact is a mere property of a symmetric monoidal category, being $\ast$-autonomous is structure, and a given smc can "be $\ast$-autonomous" in more than one way. As shown in this answer if an smc has one $\ast$-autonomous structure, then the groupoid of $\ast$-autonomous structures is a torsor over the [[Picard 2-group]].

are we really considering two isomorphic star-autonomous structures as different structures?

there is also a large groupoid of singletons in the category of sets. i can use each of them as a different unit of the compact structure and there will be a groupoid of compactness structures of $Rel$. and there is a large groupoid of right adjoints to the diagonal $Set\to Set\times Set$, and each of them will give us a different tensor for $Rel$.

the correspondence with picard group is of course interesting, but it might be useful to refine the property/structure terminology and keep the term "property" as for "structure that is unique up to unique isomorphism"?

aha, but the star-autonomous structures have nontrivial groups of isomorphisms?

so then we should follow klein and say that star-autonomy is not a property but an invariant of the category? (and leave structure for, say, the group structure: "the set of 8 elements has 5 group structures". and each of them has 8! invariant versions. otherwise the set of 8 elements will have 5*8! group structures.

JS PL (he/him) (Apr 18 2023 at 10:32):

Tobias Fritz said:

but they also require the induced morphisms $A^* \otimes B^* \to (A \otimes B)^*$ to be isomorphisms.

Ah good find! However its not exactly the statement that for every objects $A$ and $B$ that $A^* \otimes B^* \to (A \otimes B)^*$ is an isomorphism (with $A$ and $B$ being unrelated). For the theorem they are only assuming that $A^* \otimes A^{**} \to (A^* \otimes A)^*$ is an isomorphism (with $A^\ast$ being the strong dual of $A$). But then if this is true for all objects $A$, then you are compact closed per their theorem.

Tobias Fritz (Apr 18 2023 at 11:38):

JS PL (he/him) said:

Ah good find! However its not exactly the statement that for every objects $A$ and $B$ that $A^* \otimes B^* \to (A \otimes B)^*$ is an isomorphism (with $A$ and $B$ being unrelated). For the theorem they are only assuming that $A^* \otimes A^{**} \to (A^* \otimes A)^*$ is an isomorphism (with $A^\ast$ being the strong dual of $A$).

Right, but isn't that even better, as they have weaker assumptions?

Mike Shulman (Apr 18 2023 at 16:00):

dusko said:

are we really considering two isomorphic star-autonomous structures as different structures?

No, we aren't. Choosing non-isomorphic dualizing objects produces non-isomorphic $\ast$-autonomous structures.

dusko (Apr 18 2023 at 20:29):

aha very nice. so collapsing the two tensors to one removes the freedom of choosing the dualizing object by reducing the two tensor units to one. very nice.

do we know whether the dualizing object gets nailed down in any way when it carries the mix-morphism to the tensor unit (and induces the morphism from one tensor to the other).

is it easy to find... ah i was going to ask of genuinely different examples. but seely was talking about his shift monoids, which are genuinely different star-autonomous posets, and i never updated the intuition accordingly. well, i guess it's never too late to be embarassed and pay attention :)

thank you mike!

JS PL (he/him) (Apr 18 2023 at 21:38):

Tobias Fritz said:

JS PL (he/him) said:

Ah good find! However its not exactly the statement that for every objects $A$ and $B$ that $A^* \otimes B^* \to (A \otimes B)^*$ is an isomorphism (with $A$ and $B$ being unrelated). For the theorem they are only assuming that $A^* \otimes A^{**} \to (A^* \otimes A)^*$ is an isomorphism (with $A^\ast$ being the strong dual of $A$).

Right, but isn't that even better, as they have weaker assumptions?

Oh yes yes! I was just pointing out it wasn't exactly the same assumptions you originally stated.
But yah absolutely, the weaker assumptions the better of course.

Mike Shulman (Apr 19 2023 at 04:59):

dusko said:

do we know whether the dualizing object gets nailed down in any way when it carries the mix-morphism to the tensor unit (and induces the morphism from one tensor to the other).

I don't understand this question, can you explain it in more detail?

JS PL (he/him) (Apr 19 2023 at 20:56):

He's talking about mixed star autonomous categories. Which is a star autonomous category where the dualizing object $\perp$ has map to the monoidal unit $\perp \to I$. This mix map induces a canonical map from tensor to par: $A \otimes B \to A \oplus B$

JS PL (he/him) (Apr 19 2023 at 20:56):

The question was if a dualizing object with this mixed map was unique.

JS PL (he/him) (Apr 19 2023 at 20:57):

The answer is no: dusko answered:

JS PL (he/him) (Apr 19 2023 at 20:57):

dusko said:

is it easy to find... ah i was going to ask of genuinely different examples. but seely was talking about his shift monoids, which are genuinely different star-autonomous posets, and i never updated the intuition accordingly. well, i guess it's never too late to be embarassed and pay attention :)

thank you mike!

JS PL (he/him) (Apr 19 2023 at 21:01):

See: https://www.math.mcgill.ca/rags/linear/wdc-fix.pdf after Prop 5.2

Mike Shulman (Apr 19 2023 at 21:48):

Ah, I see! Thanks. When I read "A carries B to C" it sounds like A is a function whose value at B is equal to C, but here it meant that A is equipped with a morphism B from A to C.

Tim Campion (Apr 24 2023 at 14:21):

What's an example of a $\ast$-autonomous category which is not compact closed, but where the dualizing object is the unit?

Mike Shulman (Apr 24 2023 at 15:33):

${\rm Chu}(C,I)$ where $(C,\otimes,I)$ is any closed symmetric monoidal category with pullbacks.

Mike Shulman (Apr 24 2023 at 15:33):

For example, $\rm Set\times Set^{op} = Chu(Set,1)$.

Damiano Mazza (Apr 24 2023 at 18:04):

In fact, most concrete, non-degenerate models of linear logic live in categories like that (coherence spaces, hypercoherence spaces, probabilistic coherence spaces, totality spaces, finiteness spaces...). To linear logicians, this is the fact that nearly all "naturally occurring" models of linear logic validate the "empty sequent" rule (a rule allowing you to derive the empty sequent :big_smile:).

Mike Shulman (Apr 24 2023 at 18:06):

You mean, most of the models in categories that were cooked up in order to model linear logic? All Chu constructions model linear logic, but they don't have this property unless you choose the Chu-dualizing object to be the unit in the original monoidal category.

Damiano Mazza (Apr 24 2023 at 18:07):

Yes, sorry, I guess "naturally occurring" is unfair as a characterization of these models.

Mike Shulman (Apr 24 2023 at 18:07):

Well, one person's "naturally occurring" is another person's "cooked up". (-:

Mike Shulman (Apr 24 2023 at 18:08):

Maybe I should have said something like "the models in categories that linear logicians tend to focus on".

Damiano Mazza (Apr 24 2023 at 18:11):

Yes, that's definitely a better formulation of what I meant!

Damiano Mazza (Apr 24 2023 at 18:20):

In fact, it's even less general than that: if I am not mistaken, there are several categories of games which model linear logic and in which $1\neq\bot$, and those are definitely models on which linear logicians have focused a lot. So, mine was just an extreme overstatement, period :upside_down: (But the main fact remains true: Tim's question has a lot of very well-known examples in the "world" of linear logic).

Jean-Baptiste Vienney (Apr 25 2023 at 00:21):

Some $*$-autonomous categories such that $1 = \bot$ which are naturally occuring in functional analysis are the models of differential linear logic built from locally convex spaces, in this case $1 = \bot = \mathbb{R}$ or $\mathbb{C}$. I don't know exactly how to define precisely any of them but some people are expert in this kind of things (like Marie Kerjean).

dusko (Apr 26 2023 at 20:12):

Mike Shulman said:

Well, one person's "naturally occurring" is another person's "cooked up". (-:

while this is certainly true in general, the "neokantian" slant of the categorical terminology is a record of the hope that there should be something objective behind concepts. the naturality requirement was hoped to capture some evidence that an operation is not cooked up. but natural. in many cases it did capture that evidence. in their respective theses, girard and reynolds were trying to characterize uniform constructions. reynold called them polymorphic. they both stayed with category theory because the naturality requirement expressed the concept that they were looking for. their starting positions were quite skeptical.

now girard's coherence spaces were definitely not cooked up. just like manin explained how *-autonomy arises geometrically, by carrying subspaces annihilated by observations, girard explained it by coherences, required to capture indistinguishability (like partial equivalences) as dual to distinguishability (a.k.a. apartness). this leads to dropping the transitivity from partial equivalences. where equivalences induce partitions, coherences induce cliques... (to some people, winskel's closely related event structures provided more concrete intuitions of this kind.)

but while we do know how to characterize coherence spaces as a universal construction over $(Rel,\times,1)$, we do not seem to know (or at least i don't know) how to characterize the compact category $(Rel,\times,1)$ itself as a universal construction. $(Rel,+,0)$ is a clone, and obviously the free category with special biproducts (special = the diagonal followed by the codiagonal is identity). is there in the meantime a universal/natural characterization of $(Rel,\times,1)$? ((i know that aurelio carboni was trying to characterize it as a suitable cartesian bicategory arising from functorial semantics...))

without a characterization, for all we know $(Rel,\times,1)$ itself could be cooked up by a compact conspiracy.

Cole Comfort (Apr 26 2023 at 20:23):

dusko said:

Mike Shulman said:

Well, one person's "naturally occurring" is another person's "cooked up". (-:

while this is certainly true in general, the "neokantian" slant of the categorical terminology is a record of the hope that there should be something objective behind concepts. the naturality requirement was hoped to capture some evidence that an operation is not cooked up. but natural. in many cases it did capture that evidence. in their respective theses, girard and reynolds were trying to characterize uniform constructions. reynold called them polymorphic. they both stayed with category theory because the naturality requirement expressed the concept that they were looking for. their starting positions were quite skeptical.

now girard's coherence spaces were definitely not cooked up. just like manin explained how *-autonomy arises geometrically, by carrying subspaces annihilated by observations, girard explained it by coherences, required to capture indistinguishability (like partial equivalences) as dual to distinguishability (a.k.a. apartness). this leads to dropping the transitivity from partial equivalences. where equivalences induce partitions, coherences induce cliques... (to some people, winskel's closely related event structures provided more concrete intuitions of this kind.)

but while we do know how to characterize coherence spaces as a universal construction over $(Rel,\times,1)$, we do not seem to know (or at least i don't know) how to characterize the compact category $(Rel,\times,1)$ itself as a universal construction. $(Rel,+,0)$ is a clone, and obviously the free category with special biproducts (special = the diagonal followed by the codiagonal is identity). is there in the meantime a universal/natural characterization of $(Rel,\times,1)$? ((i know that aurelio carboni was trying to characterize it as a suitable cartesian bicategory arising from functorial semantics...))

without a characterization, for all we know $(Rel,\times,1)$ itself could be cooked up by a compact conspiracy.

Entertaining this tangent: under the cartesian product the category of relations of finite sets is a multicoloured prop, coloured by all primes. I gave a presentation for the full subcategory of spans generated by 2; a presentation of the corresponding category of relations following by imposing a single additional equation making the "triangle gate" idempotent:
https://arxiv.org/pdf/2004.05287.pdf

The question for other bases remains open! An even weirder question is to give a premonoidal presentation for (full subcategories of) corelations/cospans under the cartesian product, because it is not even monoidal.

dusko (Apr 27 2023 at 05:40):

Cole Comfort said:

dusko said:

Mike Shulman said:

Well, one person's "naturally occurring" is another person's "cooked up". (-:

while this is certainly true in general, the "neokantian" slant of the categorical terminology is a record of the hope that there should be something objective behind concepts. the naturality requirement was hoped to capture some evidence that an operation is not cooked up. but natural. in many cases it did capture that evidence. in their respective theses, girard and reynolds were trying to characterize uniform constructions. reynold called them polymorphic. they both stayed with category theory because the naturality requirement expressed the concept that they were looking for. their starting positions were quite skeptical.

now girard's coherence spaces were definitely not cooked up. just like manin explained how *-autonomy arises geometrically, by carrying subspaces annihilated by observations, girard explained it by coherences, required to capture indistinguishability (like partial equivalences) as dual to distinguishability (a.k.a. apartness). this leads to dropping the transitivity from partial equivalences. where equivalences induce partitions, coherences induce cliques... (to some people, winskel's closely related event structures provided more concrete intuitions of this kind.)

but while we do know how to characterize coherence spaces as a universal construction over $(Rel,\times,1)$, we do not seem to know (or at least i don't know) how to characterize the compact category $(Rel,\times,1)$ itself as a universal construction. $(Rel,+,0)$ is a clone, and obviously the free category with special biproducts (special = the diagonal followed by the codiagonal is identity). is there in the meantime a universal/natural characterization of $(Rel,\times,1)$? ((i know that aurelio carboni was trying to characterize it as a suitable cartesian bicategory arising from functorial semantics...))

without a characterization, for all we know $(Rel,\times,1)$ itself could be cooked up by a compact conspiracy.

Entertaining this tangent: under the cartesian product the category of relations of finite sets is a multicoloured prop, coloured by all primes. I gave a presentation for the full subcategory of spans generated by 2; a presentation of the corresponding category of relations following by imposing a single additional equation making the "triangle gate" idempotent:
https://arxiv.org/pdf/2004.05287.pdf

The question for other bases remains open! An even weirder question is to give a premonoidal presentation for (full subcategories of) corelations/cospans under the cartesian product, because it is not even monoidal.

adding the triangle gate is fine. but the same question can be asked for $(Span,+,0)$, which is simply the free category with coproduts, and $(Span,\times, 1)$... ah but this time the exponentiation does take us from one to the other, which does not work for $Rel$. do we know a universal property of $(Span,\times, 1)$? if we could go back to $(Span,+,0)$, we would have a "Stone-Tannaka" duality. is there a *-autonomous thread of ZX-calculus?...

sorry, i got tired today so i keep coming back here to ask pie in the sky questions. thanks for sharing the paper :)

John Baez (Apr 27 2023 at 05:46):

In what sense is (Span, +, 0) the free category with coproducts? I'm pretty sure (FinSet, +, 0) is the free category with finite coproducts on one object, namely the 1-element set. I imagine that (Set, +, 0) is the free category with small coproducts on one object.

Alexander Gietelink Oldenziel (May 04 2023 at 15:50):

How can you characterize coherence spaes as a universal construction over Rel?

Do you have a reference for the 'Manin explained how *-autonomy arises geometrically'. Much apprecaited ! =)

I think I sort of know what coherence spaces are. It's a reflexive graph where the relation is supposed to model compatibility. There is a natural dual construction that sends this relation to incompatibility. Is thit shwat you mean by indistinguishability and distinguishability?
I'm not sure what you mean by 'dropping the transivity from partial equivalences' - could you explicate that? :)

Todd Trimble (Dec 22 2023 at 20:42):

This is a terribly old thread, but I'm going through some old unread posts, now that the semester is over. From the last comment (by @Alexander Gietelink Oldenziel )

Do you have a reference for the 'Manin explained how *-autonomy arises geometrically'.

Todd Trimble (Dec 22 2023 at 20:42):

I think the allusion is to quadratic algebras as forming a star-autonomous category, which Manin posed as a useful framework for discussing quantum groups and quantum geometry, although he didn't use the term "star-autonomous" or the like. I expect that when he was writing his article (here), he was unaware of the pre-existing categorical literature available then (the publication year is 1987, the same year as Girard's seminal article on Linear Logic).

Todd Trimble (Dec 22 2023 at 20:43):

A quadratic algebra over a field $k$ is a graded algebra $A$ that is generated (as an algebra) by the space of degree 1 homogeneous elements $V = A_1$, a finite-dimensional vector space, modulo a space $R = R_A \subseteq V \otimes V$ of degree 2 homogeneous elements. Thus $A \cong T(V)/I$ where $T(V)$ is the tensor algebra and $I$ is the two-sided algebra ideal generated by the inclusion $R \subseteq V \otimes V \hookrightarrow T(V)$ of $R$ in $T(V)$. Such a graded algebra is determined by and uniquely determines such a pair $(V = A_1, R_A \subseteq V \otimes V)$, and a graded algebra morphism $A \to B$ is tantamount to a linear map $f: V \to W$ such that $(f \otimes f)(R_A) \subseteq R_B$.

Todd Trimble (Dec 22 2023 at 20:43):

Examples of quadratic algebras include standard presentations of tensor algebras, symmetric algebras, and exterior algebras. It's pretty clear that this is a graded notion. There's also a filtered notion of quadratic algebra, where the homogeneity requirement $R \subseteq V \otimes V$ is relaxed to allow inhomogeneous degree 2 terms, i.e., $R \subseteq k \oplus V \oplus V^{\otimes 2}$ in the tensor algebra, and this allows you to include more examples like universal enveloping algebras of finite-dimensional Lie algebras, and also Clifford algebras. But below I'll stick with the graded notion.

Todd Trimble (Dec 22 2023 at 20:43):

As for star-autonomy: the dual of a quadratic algebra $A = (V, R)$, which Manin denotes as $A^!$, is

$A^! = (V^\ast, R^\perp)$

where $V^\ast = \hom(V, k)$ is the dual space and $R^\perp \subseteq V^\ast \otimes V^\ast \cong (V \otimes V)^\ast$ is the orthogonal complement, consisting of those functionals $\phi: V \otimes V \to k$ such that $\phi(R) = 0$. The multiplicative tensor $\otimes$ is defined by

$(V, R) \otimes (W, S) = (V \otimes W, R \otimes S \subseteq (V \otimes V) \otimes (W \otimes W) \cong (V \otimes W) \otimes (V \otimes W))$

where the isomorphism in both cases is middle-four interchange. The $\otimes$-unit is $(k, k \otimes k)$, which presents the algebra of dual numbers $k[\varepsilon]/(\varepsilon^2)$.

Todd Trimble (Dec 22 2023 at 20:43):

Letting $\wp$ denote the dual tensor, defined by $A \wp B = (A^! \otimes B^!)^!$, the proof of the adjunction

$\mathrm{QAlg}(A \otimes B, C) \cong \mathrm{QAlg}(A, B^! \wp C)$

can be found in the nLab here.