The n-Category Café

1. Background and motivation

The starting point was Chris, Andre and Nesta’s recent axiomatic characterisation of the category $\mathbf{Con}$ of all Hilbert spaces and linear contractions, which in turn depends on Chris and Andre’s earlier axiomatic characterisation of the category $\mathbf{Hilb}$ of all Hilbert spaces and all bounded linear maps. The nice thing about these characterisations is that they do not refer to analytic notions such as norms, continuity, (metric) completeness, or the real or complex numbers. Instead, the axioms are about simple category-theoretic structures and properties.

The fundamental structure is that of a dagger — an involutive identity-on-objects contravariant endofunctor $(-)^\dagger$. The dagger encodes adjoints of linear maps. Following the “way of the dagger” philosophy, all of the other axioms involve some kind of compatibility condition with the dagger. For instance, rather than asking merely for the existence of equalisers, we ask for the existence of equalisers that are dagger monic, that is, equalisers $m$ such that $m^\dagger m = 1$; in $\mathbf{Hilb}$ and $\mathbf{Con}$, the dagger monomorphisms are precisely the isometries.

Of course, a natural question to ask is where do the analytic properties come from? In the original article, the heavy lifting is done by Solèr’s Theorem (See also Prestel’s account). This theorem gives conditions under which a hermitian space — a kind of generalised Hilbert space — over an involutive field is actually a Hilbert space over the field $\mathbb{R}$ or $\mathbb{C}$. Much of the initial part of the original article is spent constructing such a hermitian space over the scalar field; the fact that the scalar field is $\mathbb{R}$ or $\mathbb{C}$ then magically pops out. The proof of Solèr’s theorem is rather unenlightening, using a series of obscure “tricks” to show that the self-adjoint scalars form a Dedekind-complete Archimedean ordered field; that they are the real numbers then follows by the classical characterisation. A more satisfying explanation of why the scalars are the real or complex numbers would describe explicitly how to construct something like limits of sequences, directly from the axioms in a category-theoretic manner.

Another limitation of the Solèr approach is that Solèr’s theorem may only be applied to infinite-dimensional spaces. In many applications of Hilbert spaces, such as quantum computing, we only care about the finite-dimensional spaces. To characterise categories of finite-dimensional Hilbert spaces, a different proof strategy is required.

2. Infima from directed colimits

The only axiom for $\mathbf{Con}$ of infinitary nature is the one asserting that all directed diagrams have a colimit. As completeness of metric spaces is an infinitary condition, the completeness of the scalar field and the spaces associated to each object must be encoded in this axiom. This is confirmed by the following explicit construction of such colimits in $\mathbf{Con}$, which features infima of decreasing sequences in $\mathbb{R}_+ = \{x \in \mathbb{R}\mid x \geq 0\}$, as well as completion of inner-product spaces with respect to their norm.

For simplicity, consider the directed diagram in $\mathbf{Con}$ generated by the sequence

of objects and morphisms; diagrams of this shape are called sequential whilst diagrams of the opposite shape are called cosequential. As each of the $f_n$ are contractions, for each $k \in \mathbb{N}$ and each $x \in X_k$, the sequence

of positive reals is decreasing, so it has both an infimum and a limit, and these coincide. Let $\sim$ be the binary relation on the set $\bigcup_{n = 1}^\infty X_n$ defined, for all $j, k \in \mathbb{N}$, each $x \in X_j$ and each $x' \in X_k$, by $x \sim x'$ if

Then $\big(\bigcup_{n = 1}^\infty X_n\big)\big/{\sim}$ inherits the structure of an inner-product space from each of the $X_k$. In particular, its norm is defined, for each $k \in \mathbb{N}$ and each $x \in X_k$, by the equation

where $[x]$ denotes the equivalence class of $x$. The completion $X$ of the inner-product space $\big(\bigcup_{n = 1}^\infty X_n\big)\big/{\sim}$, together with the maps $X_j \to X$ that send each element to its equivalence class, from a colimit cocone on the diagram.

Our main idea to bypass Solèr’s theorem is to turn this construction around. Positive scalars are now defined to be the “norms” of elements of objects. A partial order $\leqslant$ on the positive scalars is defined so that the scalar corresponding to one element is at most the scalar corresponding to another exactly when there is a “contraction” mapping the first element to the second. Infima of decreasing sequences and suprema of bounded increasing sequences may then be recovered from the colimits of the associated sequential and cosequential diagrams of “contractions”.

3. Characterising the positive reals

The key ingredient that allowed us to proceed with this approach was our discovery of a beautiful and concise article by Ralph DeMarr from the 60s. It gives a variant of the classical characterisation of the real numbers that (1) only assumes a partial order rather than a total one, and (2) replaces the assumptions of Archimedianness and Dedekind-completeness by monotone sequential completeness. In operator algebra, a partial order is called monotone sequentially complete (or monotone $\sigma$-complete) if every bounded increasing sequence has a supremum. For a partially ordered field, this is equivalent to asking that every decreasing sequence of positive elements has an infimum.

At this point, whilst it is not so relevant for the rest of this blog post, I feel compelled to highlight DeMarr’s clever use of the humble geometric series to prove totality of the order. The main idea is that, for each $u \geq 0$, either $u \leq 1$ or $u \geq 1$ depending on whether or not the increasing sequence $s_n = 1 + u + u^2 + \dots + u^n$ has a supremum. In turn, this depends on whether or not the infimum of the decreasing sequence $1/s_n$, which always exists, is zero.

The challenge now is that DeMarr’s theorem applies only to partially ordered fields, whilst the construction of infima of positive scalars described above is with respect to a partial order that is defined only for the partially ordered semifield of positive scalars. Resolving this challenge turned out to be much trickier than we first thought.

Our initial approach was to adapt DeMarr’s proof to similarly characterise $\mathbb{R}_+$ among partially ordered semifields, and then show that the partially ordered semifield of positive scalars satisfies this new characterisation. To guide us, we had Tobias Fritz’s recent characterisation of $\mathbb{R}_+$ among partially ordered strict semifields (see Theorem 4.5) in terms of Dedekind completeness and a multiplicative variant of Archimedeanness. Our goal was to use ideas from DeMarr’s work to replace these classical assumptions with some condition about the existence of suprema or infima of monotone sequences. In the absence of additive inverses, such suprema and infima are not necessarily compatible with addition. As compatibility with addition was used in several steps of DeMarr’s approach, it was clear that we would need to incorporate it into our assumptions.

Multiplicative inversion allows us to pass between considering suprema of bounded increasing sequences and infima of decreasing sequences, modulo being careful about zero, which is not invertible. With this in mind, it is not hard to show that a partially ordered strict semifield has suprema of bounded increasing sequences if and only if it has infima of decreasing sequences. On the other hand, whilst compatibility of such suprema with addition implies compatibility of such infima with addition (this is not so obvious in the absence of additive inverses), the converse is not true.

Indeed, consider the subset $\mathbb{S} = \Big\{\Big(\begin{smallmatrix} 0 \\ 0 \end{smallmatrix}\Big)\Big\} \cup (0, \infty) \times (0, \infty)$ of $\mathbb{R} \times \mathbb{R}$. It is a partially ordered strict semifield with $0 = \Big(\begin{smallmatrix} 0 \\ 0 \end{smallmatrix}\Big)$, $1 = \Big(\begin{smallmatrix} 1 \\ 1 \end{smallmatrix}\Big)$, pointwise addition and multiplication, and $\Big(\begin{smallmatrix} x \\ y \end{smallmatrix}\Big) \leqslant \Big(\begin{smallmatrix} u \\ v \end{smallmatrix}\Big)$ exactly when $x \leqslant y$ and $y \leqslant v$. It is also monotone sequentially complete, and suprema are compatible with addition. However, as

infima are not compatible with addition. The issue is decreasing sequences whose infimum is zero, because zero is not multiplicatively invertible.

Taking this into account, and cleverly adapting DeMarr’s proof to avoid using additive inverses, yields the following result, which is called Proposition 48 in our article.

Proposition 1. A partially ordered strict semifield is isomorphic to $\mathbb{R}_+$ if and only if it is monotone sequentially complete, infima are compatible with addition, and $1 + 1 \neq 1$.

Assuming that the axioms for $\mathbf{Con}$ also imply that infima of positive scalars are compatible with addition, it follows from these axioms and Proposition 1 that the semifield of positive scalars is isomorphic to $\mathbb{R}_+$. It is then purely a matter of algebra to show that the field of all scalars is isomorphic to $\mathbb{R}$ or $\mathbb{C}$. Unlike the proof of this fact via Solèr’s theorem, our new proof is informative, explaining how infima of decreasing sequences of positive scalars arise from sequential colimits.

4. Completeness axioms

This is, however, not the end of the story. You see, aside from the issue of showing that infima of positive scalars are compatible with addition (which we will address shortly), the category $\mathbf{FCon}$ of finite-dimensional Hilbert spaces and linear contractions does not even have all sequential colimits, let alone all directed ones. For example, the sequential diagram

whose colimit in $\mathbf{Con}$ is the infinite dimensional Hilbert space $\ell_2(\mathbb{N})$ of square-summable sequences, does not have a colimit in $\mathbf{FCon}$. The search for an appropriate completeness axiom for $\mathbf{FCon}$ is a delicate balancing act. We must ask that enough directed colimits exist that the scalar field is complete, but not so many that the category also necessarily has infinite-dimensional objects.

Both of the following candidates for the infinitary axiom seemed likely to strike this balance.

Axiom A. Every sequential diagram of epimorphisms has a colimit.

Axiom B. Every cosequential diagram with a cone of epimorphisms has a limit.

Indeed, both hold in $\mathbf{FCon}$ and $\mathbf{Con}$, and both are sufficient to prove that the positive scalars are monotone sequentially complete. Axiom A is a categorification of the requirement that every decreasing sequence of positive scalars has an infimum. Axiom B is a categorification of the requirement that every bounded increasing sequence of positive scalars has a supremum. Given that Axiom A is simpler, and, being about infima rather than suprema, is better matched with Proposition 1, our initial focus was on Axiom A.

Unsuccessful in our attempts to derive compatibility of addition with infima using only Axiom A and the other axioms for $\mathbf{Con}$, we decided to allow ourselves one additional conservative assumption: that each functor $X \oplus -$ preserves colimits of sequential diagrams of epimorphisms. By the middle of last year, we had almost finished a draft of our article based on this approach. Unfortunately, at this point, I noticed a subtle error in our compatibility proof, which took several months to resolve. In the end, we assumed the following more-complicated variant of Axiom A, which, in our article, is called Axiom 9’.

Axiom A’. Every sequential diagram of epimorphisms has a colimit, and, for each natural transformation of such diagrams whose components are dagger monic, the induced morphism between the colimits is also dagger monic.

5. Finite dimensionality

From here, we thought that it would be easy sailing to the end. All that remained, really, was to show that the inner-product space associated to each object is finite dimensional, and Andre had already sketched out to us how this might work.

An object in a dagger category is called dagger finite if every dagger monic endomorphism on that object is an isomorphism. The origin of this notion is operator algebra, although it quite similar to the notion of Dedekind finiteness from set theory.

An object of $\mathbf{Con}$ is dagger finite if and only if it is finite dimensional. The idea is that every infinite-dimensional Hilbert space $X$ contains a copy of the space $\ell_2(\mathbb{N})$ of square-summable sequences. The direct sum of the canonical right shift map on this subspace with the identity map on its orthogonal complement is a dagger monic endomorphism on $X$ that is not an isomorphism. Andre adapted this proof to the abstract axiomatic setting, using sequential colimits to construct abstract analogues of the copy of $\ell_2(\mathbb{N})$ and its right shift map.

Unfortunately, the colimits required to make this proof work are of sequential diagrams of dagger monomorphisms, whilst the axiom that we had assumed was about sequential diagrams of epimorphisms. The equivalence between the category of dagger subobjects of a fixed object and dagger quotients of that object, given by taking dagger kernels and dagger cokernels, presented a possible workaround. For any object $X$, through this equivalence, the category of dagger subobjects of $X$ has sequential colimits. If $X$ is dagger infinite, then we may construct a dagger subobject of $X$ corresponding to $\ell_2(\mathbb{N})$ using such a sequential colimit. Unfortunately, as the abstract analogue of right shift map is not a morphism in this category of dagger subobjects, we cannot construct it using the universal property of this sequential colimit.

Ultimately, our approach using Axiom A’ and Proposition 1 was abandoned, and the parts of it that did work were moved to an appendix.

6. Accounting for the field embedding

Axiom B, which is about suprema rather than infima, is not well matched with Proposition 1. If we assume Axiom B, then we need to somehow exclude the partially ordered strict semifields, like $\mathbb{S}$, that have badly behaved decreasing sequences with infimum zero. What should have been obvious in hindsight is that, whilst our semifield of interest — the positive scalars — embeds in a field, the problematic semifield $\mathbb{S}$ does not. If it did, then

so either $\Big(\begin{smallmatrix} 2 \\ 1 \end{smallmatrix}\Big) = \Big(\begin{smallmatrix} 1 \\ 1 \end{smallmatrix}\Big)$ or $\Big(\begin{smallmatrix} 2 \\ 1 \end{smallmatrix}\Big) = \Big(\begin{smallmatrix} 2 \\ 2 \end{smallmatrix}\Big)$, which is a contradiction.

This trick of forming a quadratic equation to show that two elements are equal actually forms the basis of the following result, called Lemma 35 in the article.

Lemma 2. In a partially ordered strict semifield that is monotone sequentially complete and embeds in a field, if suprema are compatible with addition, then $\inf(a + u^n) = a$ for all non-zero $a$ and all $u \lt 1$.

Noting that $\inf u^n = 0$ when $u \lt 1$, we see that this extra assumption of a field embedding allowed us to deduce that addition is compatible with at least one general class of decreasing sequences with infimum zero.

We actually knew all along that we could partially order the field of self-adjoint scalars by $a \preccurlyeq b$ if and only if $b - a$ is a positive scalar. To apply DeMarr’s theorem to the self-adjoint scalars, this partial order must be monotone sequentially complete. As discussed earlier, we also knew how to show that the “contraction” partial order $\leqslant$ on the positive scalars is montone sequentially complete. The difficulty is that, a priori, the partial order $\leqslant$ merely refines the partial order $\preccurlyeq$. When we started this project, we had no idea how to lift monotone sequential completeness from $\leqslant$ to $\preccurlyeq$, so we quickly dismissed this idea. However, now armed with a better understanding of infima and suprema in partially ordered semifields, and Lemma 2 in particular, we could finally make headway.

With a few more tricks, including a clever use of completing the square, we arrived at the following proposition, called Proposition 36 in our article.

Proposition 3. Let $C$ be an involutive field with a partially ordered strict subsemifield $(P, \leqslant)$ whose elements are all self-adjoint and include $a^\dagger a$ for all $a \in C$. If $P$ is monotone sequentially complete and its suprema are compatible with addition, then there is an isomorphism of $C$ with $\mathbb{R}$ or $\mathbb{C}$ that maps $P$ onto $\mathbb{R}_+$.

All that remains, is to show that suprema are compatible with addition, and that the inner-product space associated to each object is finite dimensional. For compatibility, unlike when we assumed Axiom A, it is actually enough that each functor $X \oplus -$ preserve limits of cosequential diagrams with a cone of epimorphisms. Actually, very recently, I stumbled on a new characterisation of dagger biproducts and a few more tricks that enabled us to finally prove this fact from the other axioms. For finite-dimensionality, Andre’s proof already essentially assumed the dual of Axiom B, and so works unchanged.

7. Conclusion

It’s quite a miracle that everything worked out so well, and without concession of ugly assumptions. Our faith that the scalars being $\mathbb{R}$ or $\mathbb{C}$ should be provable in a manner that works equally well with the axioms for $\mathbf{Con}$ as for $\mathbf{FCon}$ certainly guided us in the right direction. However, there were many points at which we almost forfeited this goal to accept a subpar final result.

Unfortunately, in published mathematics, incentives for clarity, conciseness, and positive results leave little space to tell stories of how ideas and results came to be, even when these stories are interesting and insightful. I hope that for our article, this blog post conveys at least some part of the emotional rollercoaster, the twists and turns and failed attempts, that got us to the end.