Building on the notion of normed category as suggested by Lawvere, we introduce notions of Cauchy convergence and cocompleteness for such categories that differ from proposals in previous works. Key to our approach is to treat them consequentially as categories enriched in the monoidal-closed category of normed sets, i.e., of sets which come with a norm function. Our notions largely lead to the anticipated outcomes when considering individual metric spaces or normed groups as small normed categories (in fact, groupoids), but they can be challenging when trying to establish them for large categories, such as those of semi-normed or normed vector spaces and all linear maps as morphisms, not just because norms of vectors need to be allowed to have value $\infty$ in order to guarantee the existence of colimits of (sufficiently many) infinite sequences.
These categories, along with categories of generalized metric spaces,  are the key example categories discussed in detail in this paper.

Working with a general commutative quantale $\mathcal V$ as a value recipient for norms, rather than only with Lawvere's quantale $\mathcal R_+$ of the extended real half-line, we observe that the categorically atypical, and perhaps even irritating, structure gap between objects and morphisms in the example categories is already present in the underlying normed category of the enriching category of $\mathcal V$-normed sets. To show that this normed category and, in fact, all presheaf categories over it, are Cauchy cocomplete, we assume  the quantale $\mathcal V$ to satisfy a couple of light alternative extra properties which, however, are satisfied in all instances of interest to us.
Of utmost importance to the general theory is the fact that our notion of Cauchy convergence is subsumed by the notion of weighted colimit of enriched category theory. With this theory and, in particular, with results of Albert, Kelly and Schmitt, we are able to prove that all $\mathcal V$-normed categories have correct-size Cauchy cocompletions, for $\mathcal V$ satisfying our light alternative assumptions.

We also show that our notions are suitable to prove a Banach Fixed Point Theorem for contractive endofunctors of Cauchy cocomplete normed categories.

Mathematics Subject Classification: 18A35, 18D20; 18F75, 46M99, 54E35.

Keywords: quantale, normed set, normed group, normed category, normed functor, Cauchy sequence, normed colimit, weighted colimit, Cauchy cocomplete category, Cauchy cocompletion, (Lawvere) metric space, (semi-)normed vector space, Banach Fixed Point Theorem.

§ INTRODUCTION

A category $\mathbb X$ is normed if it comes with a function which assigns to every morphism $f:x\to y$ a value $|f|\in[0,\infty]$, such that

$$0\geq |1_x|\qquad\text{and}\qquad|f|+|g|\geq|g\cdot f|$$

for all morphisms $g:y\to z$. Hence, this paper adopts Lawvere's [30] original notion of normed category, as a category enriched in a certain monoidal-closed category. Mentioned by him only rather covertly as a remark in [30], the notion was worked out in full generality soon afterwards by Betti and Galuzzi  [7]. Some authors call such categories weighted (see [16, 12, 35]) rather than normed. We avoid this change of the original terminology, mostly for being able to distinguish between our notion of normed colimit and the established notion of weighted colimit of enriched category theory when we prove the non-trivial fact that the former notion is subsumed by the latter. We do not aim at special categorical environments, like that of triangulated categories (see [34] for an overview), and we refrain from imposing any further a priori conditions on the notion as some papers do (such as [29, 24]) but treat or discuss these as special add-on properties of Lawvere's fundamental notion.

Given Lawvere's  minimalistic and logic-oriented approach, it cannot surprise that examples of normed categories abound.
First of all, individual mathematical objects may often be seen as small normed categories. We mention here only the most obvious example: for every metric space $(X,d)$ one forms the (indiscrete) category $\mathrm iX$ with object set $X$ in which, for any $x,y\in X$, there is precisely one morphism $x\to y$, denoted by $(x,y)$, and its norm is given by the metric: $|(x,y)|=d(x,y)$. In fact, the small normed categories in which all hom-sets are singletons correspond precisely to the generalized metric spaces $(X,d:X\times X\to [0,\infty])$, required only to satisfy the point and triangle inequalities

$$0\geq d(x,x)\qquad\text{and}\qquad  d(x,y)+d(y,z)\geq d(x,z)$$

for all $x,y,z\in X$, with this atypical way of stating them (following [30, 31]) to be explained shortly.

Many of the large, and sufficiently interesting, normed categories have objects with some metric structure which, however, is hardly, or not at all, respected by the morphisms. But the metric structure of the objects may then be used to specify classes of well-behaved morphisms. For instance, let $\mathbb X$ be the (large) category $\mathsf{NVec}_{\infty}$ whose objects are all normed real vector spaces in the usual sense, except that we allow norms to assume the value $\infty$ (see further below for some justification), along with an adjustment of the real arithmetic for this value, and whose morphisms are all linear maps (i.e., the linear $\infty$-Lipschitz maps). From a standard categorical perspective, forming this category appears to be  highly questionable since it makes two objects $X$ and $Y$ isomorphic as soon as they are algebraically isomorphic, regardless of their norms. (In fact, with a choice of a basis for every space granted, $\mathbb X$ becomes equivalent to the category of all real vector spaces and their linear maps.) Nevertheless, $\mathbb X$ has a raison d' $\hat{e}$tre when regarded as a normed category, as it may allow us to investigate morphisms of interest within the same category, such as the (uniformly) continuous maps, or the maps with a given Lipschitz value $\geq 1$. Indeed, for a linear map $f:X\to Y$, with $|\!|\cdot|\!|$ denoting the given norms of vectors in $X$ and $Y$ (which, as already observed in [29], may also be considered as categorical norms when $X$ and $Y$ are treated as one-object categories under addition), writing $\log^{\circ}\alpha:=\max\{0,\log \alpha\}$ when $\alpha >0$ one simply considers

$$|f|=\sup_{x\neq 0}\,\log^{\circ}(\frac{\nv fx\nv}{\nv x\nv})\;,$$

so that $|f|$ becomes minimal in $[0,\infty]$ with respect to the natural order and the property that

holds for all $x\in X$. This makes $e^{|f|}$ the Lipschitz value $\mathrm L(f)$ of the map $f$ whenever $\mathrm L(f)\geq 1$ under a natural extension of the real arithmetic to $\infty$), so that the condition $|f|<\infty$ characterizes $f$ as bounded (or, equivalently, as (uniformly) continuous), while $|f|=0$ describes it as non-expanding, or 1-Lipschitz.

A seemingly trivial situation arises when we consider a normed category $\mathbb X$ in which norms are allowed to take as values only $0$ or $\infty$. Interpreting $0$ as $\top$ (true) and $\infty$ as $\bot$ (false), and rewriting the relation $\geq$ as the implication $\Rightarrow$ and the operation $+$ as the logical connective $\wedge$, then the norm conditions determining the class $\mathcal S$ of all morphisms $f$ that  satisfy $|f|=\top$ or, equivalently, $\top\Rightarrow |f|$, may be recorded equivalently as

$$1_x\in\mathcal S\qquad\text{and}\qquad f\in\mathcal S\;\wedge\;g\in\mathcal S\Longrightarrow g\cdot f\in\mathcal S\;.$$

In other words, such $\mathsf 2$-valued normed categories $\mathbb X$ are just categories that come equipped with a distinguished wide subcategory $\mathcal S$ (i.e., a subcategory with the same class of objects as $\mathbb X$). The wide-subcategory conditions as stated above explain why it is logically preferable to use the natural $\geq$ of the reals as the relevant order, a perspective that gets justified further by Lawvere's enriched category-theoretical view of metrics and norms which we adopt in this paper.

Just as for metric spaces, the concept of Cauchy convergence should be fundamental in the study of normed categories. But what is it? And once defined, what does completeness mean? Do there exist completions, and are there protagonistic normed categories in this context, like the presheaf categories in the completion theory of ordinary categories?
In this paper we try to give answers to these questions and test them in examples. Taking seriously the enriched categorical perspective that is already present in [30, 7, 31], our answers differ from those presented in other papers, such as [29, 34, 24]. Also, instigated not just by the third type of examples above, we increase the potential range of applications by allowing norms to take values in an arbitrary (commutative and unital) quantale, i.e., in a complete lattice $(\V,\leq)$ which, in addition, has a commutative monoid structure $(\V,\otimes,\kk)$, such that $\otimes$ distributes in each variable over arbitrary joins. Hence, we consider $\V$-normed categories where $(\V,\leq,\otimes,\kk)$ will, amongst others, take on the role of the Lawvere quantale $\mathcal R_+=([0,\infty],\geq,+,0)$, or of the Boolean quantale $\mathsf 2=(\{\top,\bot\},\Rightarrow,\wedge,\top)$ used in the examples above.

Seeing the study of $\V$-normed categories as embedded into enriched category theory, we must take seriously the enriching monoidal-closed category $\Setss{\V}$ whose objects are mere sets equipped with a $\V$-valued function; morphisms are maps which keep or increase the $\V$-value of elements. Hence, $\V$-normed categories are categories enriched in $\Setss{\V}$. Despite its simplicity, $\Setss{\V}$ has an unexpected feature: the internal hom of objects $A$ and $B$ is given by all mappings $A\to B$, not just by the ($\Setss{\V}$)-morphisms $A\to B$. Hence, being central to the understanding of the theory, right from start we have to consider the somewhat strange $\V$-normed category induced by the internal hom of $\Setss{\V}$. This leads us to the $\V$-normed category
of $\V$-normed sets with arbitrary maps as their morphisms. It contains the ordinary category $\Setss{\V}$ as a non-full subcategory. As the recipient category for the presheaves over any given $\V$-normed category $\ncatX$, it is key in the study of any kind of completions of $\ncatX$.

For this reason, in Sections 2-4 we take time to present the fundamentals of $\V$-normed category theory in detail. Alongside many examples of such small and large categories, we note that the category $\Catss{\V}$ of all small $\V$-normed categories and $\V$-normed functors is topological [1, 22] over $\Setss{\V}$, which gives the recipe for the construction of  limits and colimits of $\V$-normed categories. We show that it is symmetric monoidal closed [26, 33], as well as locally presentable [15, 3]. Other than $\nSetss{\V}$, we also introduce the $\V$-normed categories $\V\text{-}\mathsf{Lip}$ and $\V\text{-}\mathsf{Dist}$, both having as their objects small $\V$-categories, i.e., Lawvere metric spaces when $\V=\mathcal R_+$, whilst their morphisms are respectively arbitrary maps and $\V$-distributors. The former category facilitates the study of norms for categories of metric spaces, and the norm of the latter category naturally leads to non-symmetrized Hausdorff metrics (as considered in [30] and studied in [4, 36]).

In Sections 5 and 6, introducing the key notions of Cauchy sequence and of normed convergence of a sequence in a $\V$-normed category, we tighten the corresponding definitions as proposed by Kubiś [29] in the context of $\V=\mathcal R_+$, in such a way that, unlike in Kubiś's work, normed colimits become unique and conform with the enriched setting. This allows us to prove in Sections 11-13 the central general theorems of this paper, under two very weak alternative assumptions on the ambient quantale $\V$, the status of which we discuss further in the appended Section 15. Hence, first we show that the normed category of $(\nSetss{V})$-valued presheaves of a $\V$-normed category $\ncatX$ is Cauchy cocomplete[The term, or its dual, is not to be confused with Cauchy completeness in the sense of idempotent completeness; see Corollary <ref> and the footnote there.], i.e., that all of its Cauchy sequences have a normed colimit. Then we exhibit the normed colimits as weighted colimits in the sense of enriched category theory, and finally we invoke the machinery developed by Albert, Kelly and Schmitt [5, 28] to show the existence of a Cauchy cocompletion of $\ncatX$ belonging to the same universe as the given normed category $\ncatX$. Unlike these results, it is easy to see that our notions lead to known concepts and outcomes when applied to individual (Lawvere) metric spaces seen as small normed categories; see in particular [9, 39, 23, 21]. We leave to future work the question whether the methods used in these and other papers may be generalized to produce a more direct construction of the Cauchy cocompletion of a normed category than the one presented here.

Our discussion of norms for categories of normed vector spaces and of metric spaces does not require any advanced categorical tools and can be read independently from the abstract completion theory. We therefore present it earlier (in Sections 8 and 9), even though matters are not as straightforward as one may have hoped. Consider the sequence of normed vector spaces

$$\xymatrix{\mathbb R=\mathbb R_1\ar[r] & \mathbb R_{\frac{1}{2}}\ar[r] & \mathbb R_{\frac{1}{3}}\ar[r] & ...\ar[r] & \mathrm{colim}_n\;\mathbb R_{\frac{1}{n}}=\mathbb R_0\,,\\

where $\mathbb R_c$ is the $1$-dimensional vector space of real numbers normed by $\nv 1\nv=c$ for a constant $c>0$. The connecting identity maps being strictly contractive, it seems reasonable to work in a categorical context which gives as the colimit the space $\mathbb R_0$, and thus to admit the case $c=0$, i.e., to allow norms of non-zero vectors to be $0$, rather than to force the sequence to collapse in the null space in order to obtain a normed vector space. Hence, just like for Lawvere metric spaces, we do not insist on separation and consider semi-normed vector spaces, rather than just normed spaces. Moreover, to have enough limits and colimits, just as it is the case for Lawvere metrics, we should allow norms to have value $\infty$. Here is a simple example of an inverse sequence, again involving only $1$-dimensional normed spaces and strict contractions given by identity maps; in the same notation as above, it reads as

$$\xymatrix{\mathbb R=\mathbb R_1 & \mathbb R_{2}\ar[l] & \mathbb R_{3}\ar[l] & ...\ar[l] & \mathrm{lim}_n\;\mathbb R_{n}=\mathbb R_{\infty}\ar[l]\,.\\

To be able to work in a satisfactory categorical environment, we therefore consider in Section 8 the normed category $\mathsf{SNVec}_{\infty}$ whose objects are semi-normed vector spaces with norms permitted to attain the value $\infty$, and whose morphisms are just linear maps. Consequently, we had to extend (and restrict) not only the real addition and subtraction, but also the multiplication and division to $[0,\infty]$ and, in order to have the category normed, do the same for the exponential function and the logarithm. This works well when one extends inverse operations systematically by adjunction, rather than by ad hoc conventions. We consider the proof showing that, by traditional analytic epsilon arguments, the normed category $\mathsf{SNVec}_{\infty}$ is Cauchy cocomplete as this paper's main result in the realm of concrete normed categories. By contrast, the category $\mathsf{NVec}_{\infty}$ fails to be Cauchy cocomplete.

Expanding on previous work (see in particular [29, 35]), in Section 9 we establish the purely metric version of the vector space result, by proving that the normed category $\MET_{\infty}$ of all Lawvere metric spaces and arbitrary maps is Cauchy cocomplete. We do so by first showing that  the previously used epsilon argumentation works well also in the more general setting of the category $\V\text{-}\mathsf{Lip}$, under reasonable (but no longer mild) conditions on the quantale $\V$ which align with the methods used in Flagg's pioneering work [13, 14] and, more recently, in [20]. After a change of base, which trades  the quantale $\mathcal R_+$ for its multiplicative counterpart, the Cauchy cocompleteness of $\MET_{\infty}$ follows.
Finally, improving a result by Kubiś [29], in Section 14 we present a relatively easily established Banach Fixed Point Theorem for a contractive endofunctor of a Cauchy cocomplete normed category which replicates the classical theorem in the case of a complete metric space, considered as a small normed category.

Acknowledgements: We thank Javier Gutiérrez-García for his assistance in finding an example of a quantale helping to  distinguish the conditions used in Section 11; see Section 15 for details. We also thank Paolo Perrone for providing access to the paper [7], as well as Marino Gran who made possible our joint stay at Louvain-la-Neuve prior to CT 2023 in June 2023. Following the third-named author's talks given in April/May of 2022, joint work on this paper was initiated at that occasion, with some key results presented by the second-named author in a talk given in October 2023.

§ $\V$-NORMED SETS

Throughout this paper $\V=(\V,\leq,\otimes,\kk)$ is a unital and commutative quantale, that is: $(\V,\leq)$ is a complete lattice and $(\V,\otimes,\kk)$ is a commutative monoid with neutral element $\kk$, such that, for all $v\in\V$, the map $-\otimes v:\V\to \V$ preserves arbitrary suprema:

$$(\bigvee_{i\in I}u_i)\otimes v=\bigvee_{i\in I}(u_i\otimes v);$$

in particular, $\bot\otimes v=\bot$ for the bottom element $\bot$ of $\V$. In the (small and thin) symmetric monoidal-closed category $\V$, for all $v,w\in\V$ one has the internal hom, $[v,w]$, characterized by

$$u\leq[v,w]\iff u\otimes v\leq w$$

for all $u\in \V$. The standard quantales considered in this paper are the Boolean quantale, $\two=\{\bot,\top\}$ with $\otimes=\wedge$ and $\kk=\top$, and the Lawvere quantale, $\mathcal R_+=([0,\infty],\geq,+,0)$, ordered by the natural $\geq$-order of the extended real half-line. In $\mathcal R_+$, the internal hom is computed as
$[v,w]=\max\{0,w-v\},\; [v,\infty]=\infty,\;[\infty,w]=[\infty,\infty]=0$ for all $v,w<\infty$, and in $\mathsf 2$ it is given by the implication: $[v,w]=(v\Rightarrow w)$.

A $\V$-normed set is a set $A$ that comes with a function $|\text{-}|_A:A\to\V$, and a $\V$-normed map $(A,|\text{-}|_A)\to(B,|\text{-}|_B)$ is a mapping $f:A\to B$ satisfying $|a|_A\leq|fa|_B$ for all $a\in A$:

A[rr]^f[rd]_|-|_A^  ≤        B[ld]^|-|_B

Henceforth, we usually drop the subscripts.
This defines the category $\Setss{\V}\;.$

This category is simply the formal coproduct completion $\mathrm{Fam}\V$ of the category $(\V,\leq)$. Applying the
$\mathrm{Fam}$-construction to the unique functor $\V\to\mathsf 1$ of $\V$ to the terminal quantale  one obtains the forgetful functor \(\Setss{\V}\to\SET\), which is topological [1]; that is: given a family of any size of mappings $f_i:A\to B_i\;(i\in I)$ with a fixed set $A$ and all $B_i$ $\V$-normed, then there is an “initial” $\V$-norm on $A$, namely

$$|a|=\bigwedge_{i\in I}|f_ia|.$$

Equivalently: given any family of mappings $g_i:A_i\to B\;(i\in I)$ from $\V$-normed sets $A_i$ to a given set $B$, then there is a “final” $\V$-norm on $B$ that is described by

$$|b|=\bigvee_{i\in I}\bigvee_{a\in g_i^{-1}b}|a|.$$

Consequently, \(\Setss{\V}\) is complete and cocomplete. Moreover, the forgetful functor has a left adjoint, putting on every set the discrete $\V$-norm  with constant value $\bot$, as well as a right adjoint, putting on every set the indiscrete $\V$-norm with constant value $\top$. In particular, \(\Setss{\V}\to\SET\) is represented by the discrete singleton $\V$-normed set $\mathrm E_{\bot}$, i.e., by $\{*\}$ with $|*|=\bot$.

More importantly, one has:

The category \(\Setss{\V}\) is symmetric monoidal-closed.

For $\V$-normed sets $A$ and $B$, their tensor product $A\otimes B$ is carried by the cartesian product $A\times B$, normed by $|(a,b)|=|a|\otimes|b|$ in $\V$, and the tensor-neutral set $\mathrm E_{\mathrm k}$ is the set $\{*\}$ normed by $|{*}|=\kk$.

To describe the internal hom-object, denoted by $[A,B]$, we first observe that its elements are necessarily described by $\V$-normed maps $\mathrm E_{\bot}\to[A,B]$, which must correspond to $\V$-normed maps $\mathrm E_{\bot}\otimes A\to B$. But these correspond precisely to arbitrary $\SET$-maps $A\to B$, since $\mathrm E_{\bot}\otimes A$ puts just the discrete structure on the set $A$. Consequently, $[A,B]$ has carrier set $\SET(A,B)$, i.e., the set of all mappings $\varphi: A\to B$, with their norm  defined by

$$|\varphi|=\bigwedge_{a\in A}[|a|,|\varphi a|].$$

This turns out to be the largest structure (in the order induced pointwise by $\V$) making the evaluation map $[A,B]\otimes A\to B$ $\V$-normed; i.e., $|\varphi|$ is maximal with the property

$$|\varphi|\otimes |a|\leq|\varphi a|$$
for all $a\in A$.

(1) We note that, for $\varphi \in[A,B]$, one has $\kk\leq|\varphi|$ precisely when $|a|\leq|\varphi a|$ for all $a\in A$, that is, when $\varphi: A\to B$ is a $\V$-normed map. Hence, $|\varphi|$ is to be seen as the “degree” to which the arbitrary map $\varphi$ is a morphism of \(\Setss{\V}\).

(2) When we consider the lattice $\V$ as a small thin category, the functor $\mathsf 1\to \SET$ of the terminal category $\mathsf 1$ pointing to the terminal object $\{*\}$ of $\SET$ “lifts” to the functor $\mathrm i: \V\to\ \Setss{\V}$, which assigns to every $v\in\V$ the set $\mathrm E_v=\{*\}$, normed by $|*|=v$. It has a left adjoint, $\mathrm s$, which assigns to every object $A$ its “sum”, or “supremum” $\mathrm sA=\bigvee_{a\in A}|a|$, also regarded as its “optimal value” [35]. In the commutative diagram

$\xymatrix{\V\ar[d]\ar[rr]^{\mathrm i} &&<EMAIL_ADDRESS>s}_{\top}\\
\mathsf 1\ar[rr] &&<EMAIL_ADDRESS>
all arrows are monoidal homomorphisms and they, except possibly for $\mathrm s$, preserve also the internal homs.

(3) As a left adjoint, the functor $\mathrm s$ preserves all colimits, and it also preserves products if (and only if) the lattice $\V$ is completely distributive.

(4) Other than the forgetful functor $\Setss{\V}\to\SET$ as in (2), one may, for every $v\in \V$, consider more generally the functor $P_v:\Setss{\V}\to\SET$ which assigns to a $\V$-normed set $A$ the set $\{a\in A\mid v\leq|a|\}$. It has a left adjoint which puts the $\V$-norm with constant value $v$ onto every set, and it is represented by the $\V$-normed set $\mathrm E_v$ as defined in (2). The set of objects

$$\{\mathrm E_v\mid v\in\V\}$$

distinguishes itself as being a strong generator of the category $\Setss{\V}$. Indeed, for any $\V$-normed set $B$, the family of all morphisms $\mathrm E_v\to B$ with some $v\in \V$ is not only jointly epic, but in fact strongly so, since $B$ carries the final structure (as described above) with respect to this family.

Before exploiting the strong-generator property of the set $\{\mathrm E_v\mid v\in\V\}$, we show that every individual member of this set has an important property (as introduced in [15], see also [3]):

For the least regular cardinal $\lambda$ larger than the size of the set \(\V\), and for every element \(v\in\V\), the $\V$-normed set $\mathrm E_v$ is \(\lambda\)-presentable in \(\Setss{\V}\), that is: the representable functor $P_v:\Setss{\V}\to\SET$ preserves $\lambda$-directed colimits.

For a $\lambda$-directed ordered set $I$ (so that any subset of size $<\lambda$ has an upper bound in $I$), we consider an $I$-indexed diagram $(f_{i,j}:A_i\to A_j)_{i\leq j}$ with colimit cocone  $(g_i:A_i\to B)_{i\in I}$ in $\Setss{\V}$. We must show that every morphism $b:\mathrm E_v\to B$ in $\Setss{\V}$  has an essentially unique factorization through some $g_i$. But such morphism is described by an element $b\in B$ with $v\leq |b|$, and since $B$ carries the final structure with respect to the colimit cocone, we have $|b|=\bigvee C_b$ with

C_b={|a||∃i∈I: a∈g_i^-1b}⊆.

For every \(u\in C_b\) we choose an index \(i_u\in I\) and an element $a_u\in g_{i_u}^{-1}b$ with  \(|{a_u}|=u\). Since the terminal object $\{*\}$ is \(\lambda\)-presentable in \(\SET\) and \(I\) is \(\lambda\)-directed, there are \(j\in I\) and \(a\in A_j\) so that, for all \(u\in  C_b\), one has \(i_u\le j\) and \(f_{i_u,j}(a_u)=a\). Therefore \(g_{j}(a)=b\) and, for all \(u\in C_b\), \(u\le|a|\). Consequently \(v\le|a|\), that is, the map \( \mathrm E_v\to A_j\) with value $a$ is actually a morphism in \(\Setss{\V}\), and we conclude that \(b \colon\mathrm E_v\to B\) factors through \(g_j:A_j\to B\). The essential uniqueness of this factorization follows from the fact that a singleton set is \(\lambda\)-presentable in \(\SET\).

The category \(\Setss{\V}\) is locally presentable.

It suffices to note that the $\SET$-based topological category  \(\Setss{\V}\) is cocomplete, and that every object of its strong generator $\{\mathrm E_v\mid v\in\V\}$ is locally $\lambda$-presentable, with $\lambda$ as in Lemma <ref>.

§ $\V$-NORMED CATEGORIES

A $\V$-normed category $\mathbb X$ is a (\(\Setss{\V}\))-enriched category. This just means that $\mathbb X$ is  an ordinary category with (small) $\V$-normed hom-sets such that, for all objects $x,y,z$, the maps

$$\mathrm E_{\kk}\to \mathbb X(x,x),\;*\mapsto 1_x,\qquad\text{and}\qquad
\mathbb X(x,y)\otimes\mathbb X(y,z)\to \mathbb X(x,z),\; (f,g)\mapsto g\cdot f,$$

are $\V$-normed; equivalently, for all morphisms $f:x\to y$ and $g:y\to z$ one has

$$\kk\leq |1_x|\qquad\text{and}\qquad
|f|\otimes|g|\leq|g\cdot f|.$$

A functor $F:\mathbb X\to \mathbb Y$ is $\V$-normed if it makes its hom maps $\mathbb X(x,y)\to\mathbb Y(Fx,Fy)$ $\V$-normed; that is, if

$$|f|\leq |Ff|$$

holds for all morphisms $f$ in $\mathbb X$. For the emerging categories

and  ()

of all small $\V$-normed categories with their $\V$-normed functors and its (higher-universe) counterpart of all $\V$-normed categories we use respectively the more familiar notation

as justified by the following observation.

(1) Considering the monoid $(\V,\otimes,\kk)$ as a one-object 2-category with its 2-cells given by the order of $\V$, we may describe a $\V$-normed category $\mathbb X$  equivalently as a 2-category with only identical 2-cells, equipped with a lax functor $|\text{-}|:\mathbb X\to\V$. A $\V$-normed functor $F:\mathbb X\to\mathbb Y$ is then a (lax, but necessarily strict) 2-functor producing the lax-commutative diagram

$\xymatrix{\mathbb X\ar[rr]^F\ar[rd]_{|\text{-}|_{\mathbb X}}^{\quad\leq} && \mathbb Y\ar[ld]^{|\text{-}|_{\mathbb Y}}\\
& \V & \\

(2) The (monoidal) functors of the diagram of Remarks <ref>(2) induce the diagram

\V\text{-}\mathsf{Cat}\ar[d]_{\mathrm{forget}}\ar[rr]^{\mathrm i} &&<EMAIL_ADDRESS>s}_{\top}\\
\SET\ar[rr]^{\mathrm{indiscrete}} &&<EMAIL_ADDRESS>
of change-of-base functors. Here an object of $\V\text{-}\mathsf{Cat}$ is (as in [30] and [22]) a set $X$ which, for all $x,y\in X$, comes with a value $X(x,y)\in \V$, satisfying the laws

$$\kk\leq X(x,x)\qquad\text{and}\qquad X(x,y)\otimes X(y,z)\leq X(x,z).$$

The functor $\mathrm i$ describes the $\V$-category $X$ equivalently as an indiscrete category $\mathbb X=\mathrm iX$ with $\mathrm{ob}\mathbb X=X$, putting the $\V$-norm
$$|x\to y|=X(x,y)$$
on the only morphism in $\mathbb X(x,y)$. The functor $\mathrm s$ takes an arbitrary small $\V$-normed category $\mathbb X$ to the $\V$-category $\mathrm s\mathbb X=\mathrm{ob}\mathbb X$ with
$$(\mathrm s\mathbb X)(x,y)=\bigvee\{|f|\;|\;f\in\mathbb X(x,y)\}.$$

(3) The norm-forgetting functor $\Catss{\V}\to\mathsf{Cat}$ must be  carefully distinguished from the functor
$$(-)_{\circ}:\Catss{\V}\to \mathsf{Cat}$$
which sends a small $\V$-normed category $\mathbb X$ to the category $\mathbb X_{\circ},$ defined (as in enriched category theory [26]) to have the same objects as $\mathbb X$, but the morphisms of which are only those morphisms $f:x\to y$ in $\mathbb X$ with $\kk\leq |f|$ (since these are equivalently described by the $(\Setss{\V})$-morphisms $\mathrm E_{\kk}\to\mathbb X(x,y))$. Extending the terminology of [29] from $\mathcal R_+$ to arbitrary $\V$, we call the morphisms of $\mathbb X_{\circ}$ the $\kk$-morphisms of $\mathbb X$, and we say that
the (ordinary and generally non-full) subcategory
$$\mathbb X_{\circ}$$
of $\mathbb X$ is the category of $\kk$-morphisms in $\mathbb X$. An isomorphism $f$ in $\mathbb X_{\circ}$ is called a $\kk$-isomorphism of $\mathbb X$; i.e., $f$ is an isomorphism in $\mathbb X$ such that both, $f$ and $f^{-1}$, are $\kk$-morphisms.

Caution: An isomorphism in the ordinary category $\mathbb X$ may not belong to $\mathbb X_{\circ}$, and even if it does, it may not be an isomorphism in $\mathbb X_{\circ}$: for a (non-symmetric) two-point metric space $X=\{a,b\}$ with $X(a,b)=1$ and all other distances $0$, just consider $\mathbb X=\mathrm iX$, as formed in Examples <ref>(3).

(4) Being of the form $\mathcal W\text{-}\mathsf{Cat}$ for some symmetric monoidal-closed category $\mathcal W$, all four categories of the diagram in (2) are again symmetric monoidal closed (if not cartesian closed). In particular, recall that the tensor product of $X$ and $Y$ in $\V\text{-}\mathsf{Cat}$ is carried by the cartesian product and structured by

$$(X\otimes Y)((x,y),(x',y'))=X(x,x')\otimes Y(y,y'),$$

and that their internal hom, $[X,Y]$, is carried by the hom-set $\V\text{-}\mathsf{Cat}(X,Y)$ and  structured by

$$[X,Y](f,g)=\bigwedge_{x\in X}Y(fx,gx).$$

Of course, $\mathrm E=\{*\}$ with $\mathrm E(*,*)=\kk$ is the monoidal unit in $\V\text{-}\mathsf{Cat}$ (see [22] for details).

We define the tensor product $\mathbb X\otimes\mathbb Y$ of the $\V$-normed categories $\mathbb X$ and $\mathbb Y$ to be carried by the ordinary category $\mathbb X\times \mathbb Y$, structured by

One then routinely shows that their internal hom $[\mathbb X,\mathbb Y]$ is given by the $\V$-normed functors $\mathbb X\to \mathbb Y$ and all natural transformations between them, normed by

$$|\alpha|=\bigwedge_{x\in \mathrm{ob}\mathbb X}|\alpha_x|.$$

The terminal category $\mathbb E$ in $\mathsf{Cat}$ with $\mathrm{ob}\mathbb E=\mathrm E=\{\ast\}$ becomes the monoidal unit when one puts $|1_*|=\kk$ (but note that it is terminal in  $\Catss{\V}$ only if $\kk=\top$). Clearly, the functors $\mathrm i$ and $\mathrm s$ preserve the monoidal structure, and $\mathrm i$ preserves even the closed structure.

We also observe that the forgetful functor $\Catss{\V}\to\mathsf{Cat}$ is, like $\V\text{-}\mathsf{Cat}\to\SET$, topological; that is: for any  (arbitrarily large) family $F_i:\mathbb X\to\mathbb Y_i\;(i\in I)$ of functors of a fixed category $\mathbb X$ to $\V$-normed categories, there is the  “initial” $\V$-normed structure on $\mathbb X$, given by

$$|f|=\bigwedge_{i\in I}|F_if|.$$

Let us summarize the main points of these observations:

The category  $\Catss{\V}$ is symmetric monoidal closed and topological over $\mathsf{Cat}$. In particular, $ \Catss{\V}$ is complete and cocomplete, and the forgetful functor $ \Catss{\V}\to\mathsf{Cat}$ has both, a right and a left adjoint, providing an ordinary small category  respectively with the indiscrete and the discrete $\V$-norm. The restriction to small $\V$-normed categories whose carrier category is indiscrete reproduces the corresponding statements for the category $\V\text{-}\mathsf{Cat}$ and its forgetful  functor to $\SET$.

We also mention that, since $\Setss{\V}$ is locally presentable (Proposition <ref>), by the general result shown in [27] this important property gets inherited by $(\Setss{\V})\text{-}\mathsf{Cat}$:

The category \(\Catss{\V}\) is locally presentable.

(1) A $\mathsf 1$-normed category (for the terminal quantale $\mathsf 1$) is just an ordinary category, and for $\V=\mathsf 1$ the diagram of Facts <ref>(2) flattens to

$$\mathrm s\dashv\mathrm i:\Cats{\mathrm 1}=\SET\longrightarrow\mathsf{Cat}=\Catss{\mathsf 1}.$$

For the Boolean quantale $\V=\two$, the diagram of Facts <ref>(2) takes the form

\ORD\ar[d]_{\mathrm{forget}}\ar[rr]^{\mathrm i} &&<EMAIL_ADDRESS>s}_{\top}\\
\SET\ar[rr]^{\mathrm{indiscrete}} &&<EMAIL_ADDRESS>
Here $\ORD=\Cats{\mathsf 2}$ is the category of preordered sets and monotone maps. Objects in $\Catss{\two}$ may be described as small categories $\mathbb X$ which come with a distinguished class  $\mathcal S$ of morphisms that is closed under composition and contains all identity morphisms; necessarily then, as a category, $\mathcal S=\mathbb X_{\circ}$ as defined in Facts <ref>(3). Morphisms in $\Catss{\two}$ are functors preserving the distinguished morphisms.

(2) In an $\mathcal R_+$-normed category $\mathbb X$, henceforth often called just normed and written with the natural $\leq$ for the real numbers, rather than with the natural $\geq$ as in the Introduction, the norm conditions read as

$$\quad |1_x|=0\qquad\text{and}\qquad
|g\cdot f|\leq |f|+|g|$$

for all $f:x\to y$ and $g:y\to z$. A ($\mathcal R_+$-)normed functor $F:\mathbb X\to\mathbb Y$ must satisfy $|Ff|\leq|f|$ for all morphisms $f$ in $\mathbb X$.

(3) Every Lawvere metric space $X=(X,d)$ is equivalently described as a small indiscrete category $\mathbb X$ whose objects are the points of $X$, such that for every pair $x,y\in X$, the only morphism $x\to y$ in $\mathbb X$ gets normed by $|x\to y|=X(x,y)$, where we have written $X(x,y)$ for $d(x,y)$ in accordance with Facts <ref>(2). Hence, the categorical norm conditions as shown in (2) just generalize the defining conditions of a Lawvere metric space. Non-expanding maps between Lawvere metric spaces are equivalently presented as normed functors of indiscrete small normed categories. This describes the full reflective embedding

$$\mathrm i: \mathsf{Met}_1:=\Cats{\mathcal R_+}\longrightarrow\mathsf{NCat}_1:=\Catss{\mathcal R_+}$$

of Facts <ref>(2) (with the subscript $1$ indicating the Lipschitz constant defining the morphisms). Its left adjoint $\mathrm s$ provides the set $X$ of all objects of a small normed category $\mathbb X$ with the (Lawvere) metric

$$X(x,y)=\inf\{|f|\mid f\in\mathbb X(x,y)\}.$$

(4) (Lawvere [30]) The subsets of a (Lawvere) metric space $X$ are the objects of the small normed category $\mathbb HX$ whose morphisms $\varphi:A\to B$ are arbitrary $\SET$-maps, normed (like in the internal hom $[A,B]$ of $\Setss{\V}$) by

$$|\varphi|=\sup_{x\in A}X(x,\varphi x).$$

The reflector $\mathrm s$ provides the powerset of $X$ with the non-symmetrized Hausdorff metric

$$d(A,B)=\inf_{\varphi:A\to B}\sup_{x\in A}X(x,\varphi x)=\sup_{x\in A}\inf_{y\in B}X(x,y),$$

where the validation of the second equality (presenting the metric in its more usual form) requires the Axiom of Choice.

(5)  (See also [29, 35].) Here is an $\mathcal R_+$-norm that measures the degree to which an arbitrary mapping between metric spaces fails to be 1-Lipschitz (i.e., fails to be non-expanding). Just form the (somewhat strange) category $\mathsf{Met}_{\infty}$ whose objects are Lawvere metric spaces, and whose morphisms $\varphi:X\to Y$ are mere $\SET$-maps, normed by

$$|\varphi|=\sup_{x,x'\in X}\log^{\circ}(\frac{ Y(\varphi x,\varphi x')}{X(x,x')}),$$

where we have used the abbreviation $\log^{\circ}\alpha=\max\{0,\log\alpha\}$ for $\alpha\in[0,\infty]$ and extended the real arithmetic to $[0,\infty]$, the details of which are given in Section 8; see also Corollary <ref>.
If $X$ is a metric space in the classical sense, then this extension may be largely avoided since
$|\varphi|=\log^{\circ}\mathrm L(\varphi),$
$$\mathrm L(\varphi)=\sup\,\{\frac{ Y(\varphi x,\varphi x')}{X(x,x')}\mid x,x'\in X,\; X(x,x')\neq 0\})$$
is the Lipschitz value of $\varphi$ in $[0,\infty]$.
Since the $0$-morphisms in
the normed category $\mathsf{Met}_{\infty}$, i.e., the morphisms $\varphi$ with $|\varphi|=0$, are precisely the $1$-Lipschitz maps, we have

If $X$ and $Y$ are the underlying metric spaces of normed vector spaces and $\varphi$ is linear, then (with $\nv\text{-}\nv$ denoting the given norms of the vector spaces), the above formula reads equivalently as

$$|\varphi|=\sup_{x\neq 0}\,\log^{\circ}(\frac{\nv \varphi x\nv}{\nv x\nv})\;,$$

as recorded in the Introduction.
Hence, for $\varphi$ 1-Lipschitz, $e^{|\varphi|}=\nv\varphi\nv$ is the usual operator norm of $\varphi$.

(6) For every commutative monoid $(M,+,0)$ we have the free quantale $(\mathcal PM,\subseteq, +,\{0\})$ over the monoid $M$, with the powerset of $M$ structured by $A+B=\{a+b\mid a\in A,\, b\in B\}$ for all $A,B\subseteq M$. (Note that this quantale drastically fails to be integral, i.e., here, for the tensor-neutral element $\kk=\{0\}$ one has $\kk<\top=M$, unless $M$ is trivial.) A $\mathcal PM$-normed category $\mathbb X$ may be thought of as a category that comes equipped with a mapping that assigns to every $\alpha\in M$ a class $\mathcal S_{\alpha}$ of morphisms in $\mathbb X$, which we may call “of type $\alpha$”, or “bounded by $\alpha$”, subject to the rules that identity morphisms are bounded by $0$, and that, for composable morphisms $f$ and $g$ that are respectively bounded by $\alpha$ and $ \beta$, one has $g\cdot f$ bounded by $\alpha+\beta$. A $\mathcal PM$-normed functor must preserve bounds.

We note that the trivial monoid $\{0\}$ returns the case $\V=\mathsf 2$ of (1).

(7) Other quantales of interest include the quantale $\Delta$ of distance distribution functions [14, 21, 22], categorically characterized as a coproduct of two copies of $\mathcal{R}_+$ in the category of commutative unital quantales and their homomorphisms [38].
The small categories enriched in $\Delta$ are the probabilistic metric spaces. Its consideration as a value quantale for norms of categories,  however, we leave for future work.

§ THE $\V$-NORMED CATEGORIES $\NSETSS{\V},\;\V\TEXT{-}\MATHSF{LIP},$ AND $\V\TEXT{-}\MATHSF{DIST}$

Every symmetric monoidal-closed category $\mathcal W$ becomes a $\mathcal W$-enriched category with the same objects, qua its internal hom. Exploiting this fact for $\mathcal W=\Setss{\V}$ we obtain a $\V$-normed category whose objects are $\V$-normed sets, but whose hom-sets of morphisms $A\to B$ are given by the internal hom $[A,B]$ of $\Setss{\V}$, i.e., by all $\SET$-maps $A\to B$. The emerging normed category must be carefully distinguished from the category $\Setss{\V}$ and, as it plays an important role in what follows, deserves a separate notation,

not to be confused with its (generally non-full) subcategory $\Setss{\V}$. For clarity, with the proof of Proposition <ref> we summarize these points, as follows.

The category $\nSetss{\V}$ of $\V$-normed sets with arbitrary mappings as morphisms becomes a $\V$-normed category with

In the notation and terminology of Facts <ref>(3), the ordinary category $\Setss{\V}$ is precisely the category of $\kk$-morphisms of the $\V$-normed category $\nSetss{\V}$:

Hence, at the level of sets we have

and  ()_∘(A,B)=()(A,B)

for all $\V$-normed sets $A$ and $B$.

(1) As an ordinary category, $\nSetss{\V}$ is equivalent to the category $\SET$. The introduction of a separate notation is justifiable only when $\nSetss{\V}$ is regarded as a $\V$-normed category.

(2) The monoid $(\V,\otimes,\kk)$ may be regarded as a one-object $\V$-normed category, with an identical norm function. As the monoid acts on itself, we obtain a functor

$$\lambda: \V\longrightarrow \nSetss{\V},\quad u\longmapsto (\lambda_u:\V\to\V, \,v\mapsto u\otimes v),$$

where $\V$, as a domain and codomain of the (left-)translation $\lambda_u$, is regarded just as an identically $\V$-normed set. The functor $\lambda$ is $\V$-normed, actually norm-preserving, since

$$ |\lambda_u|=\bigwedge_{v\in\V}[v,u\otimes v]=u\,.$$

In generalization of Example <ref>(5), next we consider another category in which morphisms are not required to respect the structure of the objects: the objects of  the category $\V\text{-}\mathsf{Lip}$ are small $\V$-categories, with arbitrary maps as morphisms (so that, as an ordinary category, $\V$-$\mathsf{Lip}$ is actually equivalent to $\SET$ again, as in Remarks <ref>(1)). Their $\V$-norm measures to which extent they may fail to be $\V$-functors, as follows.

Defining the Lipschitz $\V$-norm  of a mapping $\varphi: X\to  Y$ between small $\V$-categories by

$$|\varphi|= \bigwedge_{x,x'\in X}[X(x,x'),Y(\varphi x,\varphi x')]$$

makes $\V\text{-}\mathsf{Lip}$ a $\V$-normed category, such that
$\kk\leq|\varphi|$ holds precisely when $\varphi$ is a $\V$-functor:

Furthermore, the forgetful functor

$$\V\text{-}\mathsf{Lip}\longrightarrow \nSetss{\V},\; X\longmapsto X\times X,$$

defined to remember just that every $\V$-category $X$ comes with a function $X\times X\to\V$, is not only $\V$-normed but actually norm-preserving. Restricting it to $\kk$-morphisms gives a faithful functor $\Cats{\V}\longrightarrow\Setss{\V}$.

For arbitrary maps $\varphi :X\to Y$ and $\psi: Y\to Z$ of $\V$-categories $X,Y$ and $Z$, utilizing the fact that $\V$ with its internal hom $[\text{-,-}]$ is a $\V$-category, we obtain

\begin{align*}
|\varphi|\otimes|\psi| &= (\bigwedge_{x,x'\in X} [X(x,x'),Y(\varphi x,\varphi x')])\otimes(\bigwedge_{y,y'\in Y}[Y(y,y'),Z(\psi y,\psi y')])\\
&\leq\bigwedge_{x,x'\in X}[X(x,x'),Y(\varphi x,\varphi x')]\otimes [Y(\varphi x,\varphi x'),Z(\psi\varphi x,\psi\varphi x')] \\
& \leq\bigwedge_{x,x'\in X}[X(x,x'),Z(\psi\varphi x,\psi\varphi x')]=|\psi\cdot\varphi|\;.
\end{align*}

Since trivially $\kk\leq| \mathrm{id}_X|$, this makes $\V\text{-}\mathsf{Lip}$ $\V$-normed. The other statements are even easier to verify.

We will apply the Proposition in Section 9 in order to obtain results for categories of generalized metric spaces.

There is another well-known way of weakening the notion of $\V$-functor. Recall that a $\V$-distributor $\rho:X\lmodto Y$ (also $\V$-(bi)module or -profunctor) of $\V$-categories $X$ and $Y$ is given by a $\V$-functor $\rho:X^{\op}\otimes Y\to\V$, i.e., by a function $\rho$ satisfying

$$X(x',x)\otimes\rho(x,y)\otimes Y(y,y')\leq \rho(x',y')$$

for all $x,x'\in X$ and $y,y'\in Y$. Its composite with $\sigma:Y\lmodto Z$ is defined by

$$(\sigma\cdot\rho)(x,z)=\bigvee_{y\in Y}\rho(x,y)\otimes\sigma(y,z).$$

With the identity $\V$-distributor $1_X^*$ on $X$ given by the structure of $X$, one obtains the category $\V\text{-}\mathsf{Dist}$, together with the identity-on-objects functors

$$\xymatrix{\Cats{\V}\ar[rr]^{-_*} && \V\text{-}\mathsf{Dist} && (\Cats{\V})^{\op}\ar[ll]_{-^*}\;,\\}$$

defined by $f_*(x,y)=Y(fx,y)$ and $f^*(y,x)=Y(y,fx)$ for every $\V$-functor $f:X\to Y$ and all $x\in X, y\in Y$. With the order of $\V$-distributors induced pointwise by the order of $\V$, we can regard $\V\text{-}\mathsf{Dist}$ as a 2-category, with 2-cells given by order. One then has $f_*\dashv f^*$, i.e., $1_X^*\leq f^*\cdot f_*$ and $f_*\cdot f^*\leq 1_Y^*$.

Setting the Hausdorff norm of a $\V$-distributor $\rho:X\lmodto Y$ of $\V$-categories as

$$|\rho|=\bigwedge_{x\in X}\bigvee_{y\in Y}\rho(x,y)$$

one makes $\V\text{-}\mathsf{Dist}$ a $\V$-normed category such that every $\V$-functor $f$, represented as a $\V$-distributor $f_*$ or $f^*$, becomes a $\kk$-morphism. The function $|\text{-}|$ is monotone, thus making

of Facts <ref>(1) a lax 2-functor.

Given $\rho$ and $\sigma:Y\lmodto Z$ one has

\begin{align*}
|\rho|\otimes|\sigma| &=(\bigwedge_{x\in X}\bigvee_{y\in Y} \rho(x,y))\otimes (\bigwedge_{y'\in Y}\bigvee_{z\in Z}\sigma(y',z))     \\
&\leq \bigwedge_x\bigvee_y(\rho(x,y)\otimes \bigvee_z\sigma(y,z))\\
\end{align*}

Since one also has $\kk\leq \bigwedge_xX(x,x)=|1_X^*|$, this proves the principal assertion of the Proposition. The additional claim may also be verified easily.

Alternatively, for a $\V$-distributor $\rho:X\lmodto Y$ one may set (cp. [30])
$$|\!|\rho|\!|=\bigvee_{\varphi:X\to Y}\bigwedge_{x\in X} \rho(x,\varphi x)$$

to make the category $\V\text{-}\mathsf{Dist}$  $\V$-normed; here the join runs over all mappings $\varphi:X\to Y$. The choice-free proof of this claim proceeds similarly to the proof for $|\rho|$. But, under the Axiom of Choice, and if the complete lattice $\V$ is (constructively) completely distributive (see [40, 22]), one has in fact $|\!|\rho|\!|=|\rho|$ for all $\rho$. We note that the underlying lattices of all quantales considered in this paper so far are completely distributive; in particular $\mathsf 2$ and $\mathcal R_+$ are.

For every $\V$-distributor $\rho:X\lmodto Y$ and all subsets $A\subseteq X,\;B\subseteq Y$, denoting their inclusion maps to $X$ and $Y$ by $\mathrm i_A$ and $\mathrm i_B$, respectively, we define

$$(\mathcal H\rho)(A,B):=|\mathrm i_B^*\cdot \rho\cdot(\mathrm i_A)_*|=\bigwedge_{x\in A}\bigvee_{y\in B}\rho(x,y)$$

and use the abbreviation $\mathcal HX=\mathcal H1_X^*$. Applying the norm rules of Proposition <ref> we now show how one easily concludes  (some essential parts of) Theorem 3 in [4] on the Hausdorff monad on $\V\text{-}\mathsf{Cat}$ (identified in [36] as describing its Eilenberg-Moore algebras as the conically cocomplete $\V$-categories), and on the lax extension of that monad to $\V\text{-}\mathsf{Dist}$:

The function $\mathcal HX$ makes the powerset of every $\V$-category $X$ a $\V$-category, denoted again by $\mathcal HX$, such that $\mathcal H\rho:\mathcal HX\lmodto\mathcal HY$ becomes a $\V$-distributor for every $\V$-distributor $\rho:X\lmodto Y$. This defines a $\V$-normed lax 2-functor $\mathcal H$, so that $|\rho|\leq|\mathcal H\rho|$,  and it restricts to a (strict) endofunctor of $\Cats{\V}$ which lifts the powerset functor of $\SET$, so that one has the commutative diagram

$$\xymatrix{\Cats{\V}\ar[d]_{\mathcal H}\ar[rr]^{-_*} && \V\text{-}\mathsf{Dist}\ar[d]^{\mathcal H} && (\Cats{\V})^{\op}\ar[d]^{{\mathcal H}^{\mathrm{op}}}\ar[ll]_{-^*}\\
\Cats{\V}\ar[rr]^{-_*} && \V\text{-}\mathsf{Dist} && (\Cats{\V})^{\op}\ar[ll]_{-^*}\\

For $\V$-distributors $\rho:X\lmodto Y,\;\sigma:Y\lmodto Z$ and all subsets $A\subseteq Y$ and $C\subseteq Z$, we have

\begin{align*}
(\mathcal H\sigma\cdot\mathcal H\rho)(A,C)&=\bigvee_{B\subseteq Y}\mathcal H\rho(A,B)\otimes\mathcal H\sigma(B,C)\\
&=\bigvee_{B\subseteq Y}|\mathrm i_C^*\cdot\sigma\cdot(\mathrm i_B)_*|\otimes|\mathrm i_B^*\cdot\rho\cdot (\mathrm i_A)_*|  \\
&\leq \bigvee_{B\subseteq Y}|\mathrm i_C^*\cdot\sigma\cdot(\mathrm i_B)_*\otimes\mathrm i_B^*\cdot\rho\cdot (\mathrm i_A)_*|  \\
&\leq |\mathrm i_C^*\cdot \sigma\cdot1_Y^*\cdot\rho\cdot(\mathrm i_A)_*|\\&=\mathcal H(\sigma\cdot\rho)(A,C)
\end{align*}

and $\kk\leq |1_A^*|\leq|\mathrm i_A^*\cdot(\mathrm i_A)_*|=\mathcal HX(A,A).$ For $\rho=\sigma=1_X^*$, this shows that $\mathcal HX$ is a $\V$-category. Choosing alternately only one of the $\V$-distributors to be identical will show that $\mathcal H\rho$ is a $\V$-distributor, while the general case confirms that $\mathcal H$ is a lax 2-functor of $\V\text{-}\mathsf{Dist}$. It is $\V$-normed since

$$\mathcal H\rho(A,B)=|\mathrm i_B^*\cdot\rho\cdot(\mathrm i_A)_*|\geq|\mathrm i_B^*|\otimes|\rho|\otimes|(\mathrm i_A)_*|  \geq\kk\otimes|\rho|\otimes\kk=|\rho|\;. $$

For a $\V$-functor $f:X\to Y$ also
$\mathcal Hf:\mathcal HX\to \mathcal H Y$ with $(\mathcal Hf)(A)=f(A)$ is a $\V$-functor since

$$\mathcal H X(A,A')=|(\mathrm i_{A'})^*\cdot(\mathrm i_A)_*|\leq|(\mathrm i_{A'})^*\cdot f^*\cdot f_*\cdot (\mathrm i_A)_*|=|(\mathrm i_{f(A')})^*\cdot(\mathrm i_{f(A)})_*|=\mathcal HY(f(A'),f(A))\;.$$

Finally, the left part of the diagram above commutes since

$$( \mathcal Hf_*)(A,B)=|\mathrm i_B^*\cdot f_*\cdot (\mathrm i_A)_*|=|\mathrm i_B^*\cdot(\mathrm i_{f(A)})_*|=\mathcal HY(f(A),B)=(\mathcal Hf)_*(A,B)\;, $$

and the commutativity of the right part follows by duality.

Generalizing Example <ref>(4) from $\mathcal R_+$ to our general quantale $\V$, and trading the Hausdorff norm $|\text{-}|$ à la Lawvere for $|\!|\text{-}|\!|$ of Remark <ref>, one  obtains the following corollary (without using Choice or assuming complete distributivity of $\V$ !):

For every $\V$-category $X$, the subsets of $X$ are the objects of a small $\V$-normed category $\mathbb HX$ whose morphisms are arbitrary maps $\varphi: A\to B$, normed by $|\varphi| =\bigwedge_{x\in A}X(x,\varphi x)$. The sum functor $\mathrm s$ of Facts <ref>(2) takes $\mathbb HX$ to the $\V$-category $\tilde{\mathcal H}X$  with
$\tilde{\mathcal H}(A,B)=\bigvee_{\varphi:A\to B}|\varphi| $.

As already noted in Examples <ref>(4), with Choice one has $\tilde{\mathcal H}X=\mathcal HX$. However, even then
the passage $X\mapsto\mathbb HX$ (unlike $X\mapsto \tilde{\mathcal H}X$) generally fails to be functorial.

For further investigations on the functor $\mathcal H$ and various restrictions thereof we refer the reader to [20]. The question to which extent completeness properties of the object $X$ get transferred to $\mathcal HX$, without the symmetrization of the structure and/or some restriction on the subsets of $X$ to be considered, such as the (in some sense) compact subsets, remains open. This includes the Cauchy cocompleteness (of the normed category $\mathrm iX$) as introduced in Section 6.

§ NORMED CONVERGENCE AND SYMMETRY

In order to introduce the concept of normed convergence in a $\V$-normed category, we find it useful to remind ourselves how sequential colimits are formed in $\Setss{\V}$. The following easily checked statement is an immediate consequence of $\Setss{\V}$ being topological over $\SET$, amended by an also easily verified second formula for the norm of the colimit object.

The colimit of a sequence \(A_{0}\to A_{1}\to A_{2} \dots\)
in \(\Setss{\V}\) is formed by providing the colimit $A$ of the underlying sequence in $\SET$ with the least norm that makes the colimit cocone $\xymatrix{(A_N\ar[r]^{\kappa_N\;\;} & A)_{N\in\mathbb N}}$ live in $\SET/\!/\mathcal V$:

$$|c|=\bigvee\{|a|\mid a\in \bigcup_{N\in\mathbb N}\kappa_N^{-1}c\}=\bigwedge_{N\in\mathbb N}\;\bigvee_{n\geq N}\;\bigvee_{a\in\kappa_n^{-1}c}|a|$$

for all $c\in A$.

Let $s: \mathbb N\to \mathbb X$ be a sequence[Here the ordered set $\mathbb N$ is treated as a category, discretely $\V$-normed with constant value $\bot$ for all non-identical arrows, so that the sequence $s$ becomes a $\V$-normed functor $\mathbb N\to\mathbb X$.] in a $\mathcal V$-normed category $\mathbb X$, written as

$$ s=\xymatrix{(x_m\ar[r]^-{s_{m,n}}& x_n)_{m\leq n\in\mathbb N}}\;.$$

An object $x$ is a normed colimit of $s$ in $\mathbb X$ if

(C1) $x$ is a colimit of $s$ in the ordinary category $\mathbb X$, with a colimit cocone $\xymatrix{(x_n\ar[r]^-{\gamma_n}&x)_{n\in \mathbb N}}$ s. th.

(C2) for all objects $y$ in $\mathbb X$, the canonical $\SET$-bijections[Note that a colimit $x$ of $s$ in the ordinary category $\mathbb X$ is also a colimit of every restricted sequence $s_{|N}$.]

$$\xymatrix{\mathrm{Nat}(s_{|N},\Delta y)\ar[rr]^-{\kappa_N} && \mathbb X(x,y), & (f\cdot\gamma_n)_{n\geq N} && f\ar@{|->}[ll]_-{\kappa_N^{-1}} & ({N\in\mathbb N})\\}$$

form a colimit cocone in $\SET/\!/\mathcal V$, where $s_{|N}$ is the restriction of $s$ to $\uparrow\! N=\{N, N+1, \dots\}$ and $\mathrm{Nat}(s_{|N},\Delta y)=[\uparrow\! N,\mathbb X](s_{|N},\Delta y)\;(N\in\mathbb N)$ is considered as a sequence in $\Setss{\V}$, with all connecting maps given by restriction.

Keeping the notation of this definition, let us immediately analyze the meaning of (C2):

Condition (C2) says equivalently that, for all morphisms $f$ in $\mathbb X$ with domain $x$, one must have

|f|=⋁_N ∈ℕ⋀_n≥N|f·γ_n| .

The “$\leq$”-part of this equality is satisfied if, and only if,

k≤⋁_N ∈ℕ⋀_n≥N|γ_n| .

Trivially, the natural $\SET$-bijections $\kappa_N\; (N\in \mathbb N)$ form a colimit cocone in $\SET$. In order to make it a colimit cocone in $\Setss{\V}$, by Proposition <ref> the norm on $\mathbb X(x,y)$ must satisfy

$$|f|=\bigvee\{|\beta|\mid \beta=\Delta f\cdot \gamma_{|N}:s_{|N}\to \Delta y,\;N\in\mathbb N\}$$

for all $f:x\to y$ in $\mathbb X$, which, by the norm formula for natural transformations (Proposition <ref>), amounts to the claimed formula for $|f|$.

The second statement of the Proposition follows from the following more general lemma.

For any cocone $\alpha:s\to \Delta x$ over a sequence $s=(x_n)_{n\in\mathbb N}$ with vertex $x$ in a $\V$-normed category $\mathbb X$, the following are equivalent:

(i) $\kk\leq \bigvee_{N \in\mathbb N}\bigwedge_{n\geq N}|\alpha_n|\,;$

(ii) $|1_x|\leq \bigvee_{N \in\mathbb N}\bigwedge_{n\geq N}|\alpha_n|\,;$

(iii) $|f|\leq \bigvee_{N \in\mathbb N}\bigwedge_{n\geq N}|f\cdot\alpha_n|$, for every morphism $f:x\to y$ in $\mathbb X$ .

Trivially, one has (iii)$\Longrightarrow$(ii)$\Longrightarrow$(i). For (i)$\Longrightarrow$(iii), we note

$$|f|=|f|\otimes\kk\leq|f|\otimes\bigvee_{N \in\mathbb N}\bigwedge_{n\geq N}|\alpha_n|\leq\bigvee_{N \in\mathbb N}\bigwedge_{n\geq N}|f|\otimes|\alpha_n|\leq \bigvee_{N \in\mathbb N}\bigwedge_{n\geq N}|f\cdot\alpha_n|.$$

Extending the terminology used for morphisms in Facts <ref>(3), we call a cocone  $\alpha: s\to\Delta x$ over a sequence $s=(x_n)_{n\in\mathbb N}$ in a normed category $\mathbb X$  a $\kk$-cocone if it satisfies condition (i) of Lemma <ref>. We conclude from Proposition <ref>:

An object $x$ is a normed colimit of a sequence $s$ in a $\V$-normed category $\mathbb X$ if, and only if, $x$ is a colimit of $s$ in the ordinary category $\mathbb X$ with a colimit cocone $\gamma$ such that

(C2a)  $\kk\leq\bigvee_{N \in\mathbb N}\bigwedge_{n\geq N}|\gamma_n|$,  i.e., $\gamma$ is a $\kk$-cocone;

$|f|\geq \bigvee_{N\in\mathbb N} \bigwedge_{n\geq N}|f\cdot\gamma_n|$, for every morphism $f:x\to y$ in $\mathbb X$.

A normed colimit of a sequence in a $\V$-normed category $\mathbb X$ is uniquely determined up to a $\kk$-isomorphism,  i.e., up to an isomorphism in $\mathbb X_{\circ}$.

If $\gamma:s\to \Delta x$ and $\delta:s\to\Delta y$ are both colimit cocones representing $x$ and $y$ as normed colimits of $s$, respectively, then the canonical morphism $f:x\to y$ is not only an isomorphism in $\mathbb X$, but also satisfies

$$|f|=\bigvee_{N \in\mathbb N}\bigwedge_{n\geq N}|f\cdot\gamma_n|=\bigvee_{N \in\mathbb N}\bigwedge_{n\geq N}|\delta_n|\geq\kk\;,$$

and likewise $|f^{-1}|\geq\kk$. Hence, $f$ is an isomorphism in $\mathbb X_{\circ}$.

Here is a sufficient, but not necessary, condition on the $\V$-normed category $\mathbb X$ (which will be discussed further in Facts <ref>) to make condition (C2b) of Corollary <ref> redundant, as follows.

Let $\mathbb X$ be a $\V$-normed category satisfying the condition

(S)     $|f\cdot h|\otimes |h|\leq|f|$  for all composable morphisms $h$ and $f$.

Then an object $x$ is a normed colimit of a sequence $s$ in $\mathbb X$ if, and only if, $x$ is a colimit of $s$ in the ordinary category $\mathbb X$, with a colimit cocone that is a $\kk$-cocone.

First we note that, in (S), the morphism $h$ may be replaced equivalently by any cocone $\alpha: D\to \Delta x$, for some diagram $D:\mathbb I\to\mathbb X$ with $\mathbb I\neq\emptyset$, so that (S) then reads as
$|\Delta f\cdot \alpha|\otimes |\alpha|\leq |f|.$
Indeed, for all $i\in\mathbb I$, using (S) and $\mathbb I\neq\emptyset$, one has

$$|\Delta f\cdot \alpha|\otimes |\alpha|=\bigwedge_{i\in\mathbb I}|f\cdot\alpha_i|\otimes\bigwedge_{i\in\mathbb I}|\alpha_i|\leq\bigwedge_{i\in\mathbb I}|f\cdot\alpha_i|\otimes|\alpha_i|\leq\bigwedge_{i\in I}|f|=|f|\;.$$

Having (C2a) and this extended version of (S), we can now show (C2b) of Corollary <ref>, as follows, utilizing also the fact that the occurring joins are directed:

\begin{align*}
\bigvee_{N\in\mathbb N}\bigwedge_{n\geq N}|f\cdot\gamma_n| &\leq(\bigvee_{N\in\mathbb N}\bigwedge_{n\geq N}|f\cdot \gamma_n|)\otimes(\bigvee_{N\in\mathbb N}\bigwedge_{n\geq N}|\gamma_n|) \\
&\leq \bigvee_{N\in\mathbb N}(\bigwedge_{n\geq N}|f\cdot \gamma_n|\otimes\bigwedge_{n\geq N}|\gamma_n|)\\
&\leq \bigvee_{N\in\mathbb N}\bigwedge_{n\geq N}|f\cdot \gamma_n|\otimes|\gamma_n|\\
\end{align*}

(1) Condition (S) is a (strong) symmetry condition on the normed category $\mathbb X$. Indeed, if $\mathbb X=\mathrm iX$ is given by a $\V$-category $X$ as in Facts <ref>(2), then (S) means equivalently that $X$ is symmetric, i.e., that

holds for all $x,y\in X$. We call an arbitrary $\V$-normed category $\mathbb X$ satisfying (S) forward symmetric. The condition generally fails in $\nSetss{\V}$, even for $\V=\mathcal R_+=([0,\infty],\geq,+,0)$. Indeed, considering $\mathbb N$ as an identically $\mathcal R_+$-normed set, then for the endomaps $f$ and $h$ which keep $0$ fixed while $hn=n-1$ and $fn=n+1$ for all $n>0$, one has $|f|=1$ but $|f\cdot h|+|h|=0$.

(2) The dualization of (S) reads as

(S$^{\mathrm{op}}$)     $| g\cdot f|\otimes |g|\leq|f|$  for all composable morphisms $f$ and $g$;

we call $\mathbb X$ backwards symmetric in this case. Indeed
for $\mathbb X=\mathrm iX$ as in (1), condition (S$^{\mathrm{op}}$) again amounts to the $\V$-category $X$ being symmetric, and again, it generally fails in $\nSetss{\V}$. However, for arbitrary $\mathbb X$, conditions (S) and (S$^{\mathrm{op}}$) are far from being equivalent (as already the example in (3) shows). But, as noted for $\V=\mathcal R_+$  in Lemma 2.2 of [29], each of the two conditions implies the inverse of an isomorphism $f$ in the ordinary category $\mathbb X$ to have the same norm as $f$; for example, with (S$^{\mathrm{op}}$) one has

$$|f|\geq|f^{-1}\cdot f|\otimes|f^{-1}|\geq \kk\otimes |f^{-1}|=|f^{-1}|,$$

and likewise for $|f^{-1}|\geq|f|$. In particular, if $\mathrm{(S)}$ or $\mathrm{(S^{\mathrm{op}})}$ holds, a morphism in $\mathbb X_{\circ}$ that is an isomorphism in the ordinary category $\mathbb X$ is also an isomorphism in $\mathbb X_{\circ}$.

(3) For $\V=\mathcal R_+$, in addition to our conditions on a normed category, Kubiś [29] includes condition (S$^{\mathrm{op}}$) as part of his definition of normed category, and then defines the normed convergence of $s$ to $x$ by requiring only conditions (C1) and (C2a), in their $\mathcal R_+$-versions.  This, however, does not make the colimit unique up to a $0$-isomorphism (here $0=\kk$).

Indeed, the following simple witness appears already in [29]. Consider the category given by the preordered set $\mathbb N\cup\{a,b\}$ with new distinct elements $a,b$, and extend the natural order by $n\leq a\leq b$ and $n\leq b\leq a$ for all $n\in \mathbb N$; it gets (Kubiś-)normed by putting $|x\to y|=0$ whenever $x\notin \{a,b\}$, and $|a\to b|=|b\to a|=\infty$. Hence, $a$ and $b$ are ordinary colimits of the sequence $(n)$, both satisfying (C1) and (C2a), and the ambient category satisfies (S$^{\mathrm{op}}$). However, (S) is violated – not even (C2b) holds, which is why $a$ and $b$ fail to be $0$-isomorphic.

If one modifies this example by declaring the norms of morphisms $n\to b$ to be $1$, rather than $0$, one still has a normed category satisfying (S$^{\mathrm{op}}$), but now the ordinary colimit $b$ no longer satisfies (C2a) whilst $a$ still does.

(4) A $\V$-normed monoid is simply a monoid $(A,\cdot, 1)$ which, considered as a one-object category $\mathbb A$ with $A$ as its only hom-set, is $\V$-normed; that is: $A$ comes with a function $|\text{-}|:A\to\V$ satisfying $\kk\leq|1|$ and $|a|\otimes|b|\leq|ab|$ for all $a,b\in A$. In case $\V=\mathcal R_+$, such normed monoids are often called semi-normed [8], but here we will omit the prefix. Every left-invariant Lawvere metric on a monoid $A$ makes $A$ a normed monoid [29]. In fact, more generally, if a (multiplicatively written) monoid $A$ carries a $\V$-category $A$ structure such that, for all $a,b, c\in a$, one has $A(ca,cb)=A(a,b)$, then

makes $A$ a $\V$-normed monoid. Indeed, trivially one has $\kk\leq|1|$ and

$$|a|\otimes |b|=X(1,a)\otimes X(1,b)=X(1,a)\otimes X(a,ab)\leq X(1,ab)=|ab|\;.$$

Conversely, if the $\V$-normed monoid $A$ is, algebraically, a group, then the norm makes $A$ a left-invariant $\V$-category, via

and this actually results into a one-one correspondence between $\V$-norms and left invariant $\V$-category structures on the given group $A$.

We note that the identity norm of the $\V$-normed commutative monoid $(\V,\otimes,\kk)$ of Remark <ref> is induced by its $\V$-category structure, given by its internal hom $[\text{-,-}]$, although this structure fails to be (left-) invariant, unless $\V$ is trivial. And, of course, $\V$ fails to be a group in all but the trivial case.

(5) Further to the case that the $\V$-normed monoid $(A,\cdot,1)$ considered in (4) is actually a group, let us call $A$ a $\V$-normed group if the additional condition
$|a^{-1}| = |a|$ holds for all $a\in A$. (For $\V=\mathcal R_+$, this gives the standard notion of normed group as used in [8].) The induced $\V$-category structure of a $\V$-normed group $A$ is symmetric, so that $A(a,b)=A(b,a)$ holds for all $a,b\in A$. Conversely, if the induced $\V$-category structure of $A$ is symmetric, then the one-object $\V$-normed category $\mathbb A$ induced by $A$ is forward symmetric, i.e.,

(S)        $|ab|\otimes |b|=A(1,ab)\otimes A(1,b)=A(1,ab)\otimes A(ab, a)\leq A(1,a)=|a|$

holds. Moreover, as follows already from (2), condition (S) implies $|a^{-1}|=|a|$ for all $a\in A$ and thus makes $A$ a $\V$-normed group.  Consequently, for a $\V$-normed monoid $A$ that, algebraically, is a group, the following conditions are equivalent:

(i) $A$ is a $\V$-normed group;

(ii) the induced $\V$-category structure of $A$ is symmetric;

(iii) the one-object $\V$-normed category $\mathbb A$ given by $A$ is forward symmetric.

Caution: The one-object $\V$-normed category $\mathbb A$ must not be confused with the (generally) multi-object $\V$-normed category $\mathrm iA$ as considered in (1). In the latter category, conditions (S) and (S$^{\mathrm{op}}$) are equivalent, unlike in the former category, unless the group $A$ is Abelian.

§ CAUCHY COCOMPLETENESS

We now extend (the key) Definition <ref> in the expected way:

For a $\mathcal V$-normed category $\mathbb X$, we say that

* a sequence $ s=\xymatrix{(x_m\ar[r]^{s_{m,n}\quad}& x_n)_{m\leq n\in\mathbb N}}$ in $\mathbb X$ is Cauchy if
$\mathrm k\leq \bigvee_{N\in\mathbb N}\;\bigwedge_{n\geq m\geq N}|s_{m,n}|,$ and

* $\mathbb X$ is Cauchy (norm-)cocomplete if every Cauchy sequence in $\mathbb X$ has a normed colimit in $\mathbb X$.

We will compare this notion for $\V$-normed categories with the well-established notion of idempotent completeness for ordinary categories (and $\V$-enriched categories, see [10]) in Corollary <ref>.

(1) Let us note first that the existence of a normed colimit for a sequence $s$ does not necessitate $s$ to be Cauchy, even when $\V=\mathcal R_+$. For example, considering the ordered set $\mathbb N\cup\{\infty\}$ of natural numbers with an added maximum as a category, normed by $|m\to n|=n-m$ and $|n\to\infty|=0$ for all $m\leq n$ in $\mathbb N$, we obtain a normed category (satisfying (S), but not (S$^{\mathrm{op}}$)) such that $\infty$ is a normed colimit of the sequence $s=(n)_{n\in\mathbb{N}}$, although $s$ badly fails to be Cauchy; in fact, here $\inf_{N\in\mathbb N}\sup_{n\geq m\geq N}|s_{m,n}|=\infty$.

(2) The notions of Definitions <ref> and <ref> are easily dualizable. For a $\V$-normed category $\mathbb X$, the dual $\mathbb X^{\mathrm{op}}$ of the ordinary category $\mathbb X$ becomes $\V$-normed when giving every morphism the same norm as in $\mathbb X$. Now, having an inverse sequence $s:\mathbb N^{\mathrm{op}}\to \mathbb X$, given by morphisms $s_{m,n}:x_n\to x_m$ in $\mathbb X$ for all $m\leq n\in\mathbb N$, the inverse sequence is said to be Cauchy in $\mathbb X$ if the sequence $s^{\mathrm{op}}:\mathbb N\to\mathbb X^{\mathrm{op}}$ is Cauchy in $\mathbb X^{\mathrm{op}}$. Furthermore, an object $x$ is a normed limit of $s$ in $\mathbb X$ if $x$ is a normed colimit of $s^{\mathrm{op}}$ in $\mathbb X^{\mathrm{op}}$; this means that $x$ is a limit of $s$ in the ordinary category $\mathbb X$, with a limit cone $\lambda:\Delta x\to s$ such that

$$|f|=\bigvee_{N\in\mathbb N}\bigwedge_{n\geq N}|\lambda_n\cdot f|,$$

for all morphisms $f:y\to x$ in $\mathbb X$.

(1) For $\V=\mathcal R_+$, we repeat that the condition for a sequence $s$ to be Cauchy in the $\mathcal R_+$-normed category $\mathbb X$ reads as

$$\inf_{N\in\mathbb N}\sup_{n\geq m\geq N}|s_{m,n}|=0,$$

and for the ordinary colimit $x$ with colimit cocone $\gamma$ in $\mathbb X$ to be a normed colimit means that

$$\inf_{N\in\mathbb N}\sup_{n\geq N}|\gamma_n|=0\quad\text{and}\quad |f|\leq\sup_{n\geq N}|f\cdot\gamma_n|,$$

for every morphism $f:x\to y$ in $\mathbb X$ and all $N\in\mathbb N$ (see Corollary <ref>).
In case $\mathbb X=\mathrm iX$ is induced by a (Lawvere) metric space $X$, the sequence $s=(x_n)$ is Cauchy if, and only if,
$$\inf_{N\in\mathbb N}\sup_{n\geq m\geq N}X(x_m,x_n)=0,$$
so that $s$ must be forward Cauchy in the sense of [9]; furthermore, $x$ is a normed colimit of $s$ if, and only if,

$$X(x,y)=\inf_{N\in\mathbb N}\sup_{n\geq N}X(x_n,y)$$

for all $y\in X$, which means that $x$ is a forward limit of $s$ in the language of [9]. (Note that here the ordinary colimit condition for $x$ comes for free since $\mathrm iX$ is a groupoid.) The notions of backward Cauchy sequence and backward limit in $X$ come about by dualization according to Facts <ref>(2), i.e., by interchanging the arguments of $X$(-,-).

Of course, if X is symmetric, there is no need to distinguish between forward and backward, and one obtains the standard notions of Cauchy sequence and sequential convergence.

(2) Expanding on Examples <ref>(4), for $\V=\mathsf 1$ we have $ \mathsf{Cat}/\!/\mathcal V\cong\mathsf{Cat}$, and every sequence in a category $\mathbb X$ is Cauchy, and  $\mathbb X$ is Cauchy cocomplete if, and only if, $\mathbb X$ has colimits of sequences.

For $\mathcal V=\mathsf 2$, describing an object in \(\Catss{\V}\) or  $\mathsf{CAT}/\!/\V$ as a category $\mathbb X$ with a distinguished class $\mathcal S$ of morphisms satisfying $\mathsf{Id}(\mathbb X)\subseteq\mathcal S$ and $\mathcal S\cdot\mathcal S\subseteq \mathcal S$, a sequence $s$ in $\mathbb X$ is Cauchy if, and only if, eventually all of its connecting maps lie in $\mathcal S$; and $\mathbb X$ is Cauchy cocomplete if, and only if, every Cauchy sequence $s$ has a colimit $x$ with a colimit cocone $(\gamma_n)_n$ lying eventually in $\mathcal S$, such that any morphism $f:x\to y$ belongs to $\mathcal S$ as soon as eventually all morphisms $f\cdot \gamma_n$ do so.

(3) More generally, for $\mathcal V=\mathcal PM$ with a commutative monoid $(M,+,0)$ and  $\mathsf{Cat}/\!/\mathcal V$ described as in Examples <ref>(5), i.e., as containing all small categories $\mathbb X$ equipped with a class of morphisms “bounded by $\alpha$” for every $\alpha\in M$, a sequence $s$ in $\mathbb X$ is Cauchy if, and only if, eventually all connecting maps are bounded by $0$; and $\mathbb X$ is Cauchy cocomplete if, and only if, every Cauchy sequence $s$ has a colimit $x$ with a colimit cocone $(\gamma_n)_n$ eventually bounded by $0$ such that, for every $\alpha \in M$, any morphism $f:x\to y$ is bounded by $\alpha$ as soon as eventually all morphisms $f\cdot \gamma_n$ are bounded by $\alpha$.
The symmetry conditions (S) and (S$^{\mathrm{op}}$) of Corollary <ref> and Facts <ref>(2) amount to the weak cancellations conditions

(S)$\quad\, f\cdot h\in\mathcal S\quad\&\quad h\in\mathcal S\Longrightarrow f\in\mathcal S\;,$

(S$^{\mathrm{op}}$) $\quad\, g\cdot f\in\mathcal S\quad\&\quad g\in\mathcal S\Longrightarrow f\in\mathcal S\;,$

which typically hold for classes of epimorphisms and classes of monomorphisms, respectively.

(4) For a $\V$-normed monoid $(A,\cdot,1)$, considered as a one-object $\V$-normed category $\mathbb A$ (as in Facts <ref>(4)), a sequence $s$ in $\mathbb A$ is simply a sequence $(a_n)_n$ of elements in $A$, and a cocone $\alpha$ over $s$ is given by elements $\alpha_n\in A$ satisfying $\alpha_{n+1}a_n=\alpha_n$ for all $n\in\mathbb N$.
If $A$ is a group, the cocone is already determined by  $\alpha_0$,
since necessarily  $\alpha_n=\alpha_0a_0^{-1}a_1^{-1} ... a_{n-1}^{-1}$ (where the product of an empty string of elements is $1$). In particular, we may consider the cocone $\gamma$ with $\gamma_0=1$ which, since every morphism in $\mathbb A$ is an isomorphism, presents the only object of  $\mathbb A$ trivially as an ordinary colimit of $s$: the factorizing morphism induced by an arbitrary cocone $\alpha$ is simply $\alpha_0$:

<EMAIL_ADDRESS>& \ast\ar[r]^-{a_1} & \ast\ar[r]^-{a_2} &\ast ......\ast\ar[r]^-{a_{n-1}}&\ast\ar[rr]^{\gamma_n}\ar[d]_{\alpha_n} && \ast\ar[lld]^{\alpha_0} \\
&&&& \ast \\

If the group $A$ is actually a $\V$-normed group, so that $\mathbb A$ enjoys the symmetry condition (S) (as shown in Facts <ref>(5)), for $\gamma$ to present $\ast$ as a normed colimit, by  Corollary <ref> it is necessary and sufficient that $\gamma$ is $\kk$-cocone, which means (since $|a|=|a^{-1}|$ for all $a\in A$) that

$$\kk\leq\bigvee_{N\in\mathbb N}\bigwedge_{n\geq N}|a_{n-1}....a_0|$$

holds. This, however, is a steep hurdle. For example, for $\V=\mathcal R_+$ consider the multiplicative group $\mathbb Q_{>0}$ of positive rationals, which becomes a normed group when one sets

$$ |r|:=\sum_p\max\{n_p,-n_p\} \;\text{whenever}\;r=\prod_p p^{n_p}\; \text{with}\; n_p\in\mathbb Z, $$

where $p$ runs through the set of prime numbers (and the products and sums are only nominally infinite). Note that $|r|=0$ only if $r=1$. The convergence condition for the sequence reads as $\inf_{N\in\mathbb N}\sup_{n\geq N}|a_{n-1}....a_0|$ $=0$, and since the norms are always integer valued, it is easy to see that this equivalently means that the sequence $s$ must have some initial segment $a_0,...,a_{N-1}$ of rational numbers whose product equals $1$, followed by an infinite tail that is constantly $1$. By contrast, for $s$ to be Cauchy, an arbitrary initial segment is allowed, followed by a tail that is constantly $1$. As a consequence, $\mathbb Q_{>0}$ fails to be Cauchy cocomplete.

§ A NOTE ON IDEMPOTENT COMPLETENESS

Instigated by Example <ref>(4), for a general $\V$-normed category $\mathbb X$ we briefly examine the question of whether constant sequences in $\mathbb X$ are Cauchy and have a normed colimit in $\mathbb X$.
Here a sequence $s:\mathbb N\to \mathbb X$ is understood to be constant if $s_{m,n}=e:x\to x$ for all $m<n$ in $\mathbb N$. Such morphism $e$ must necessarily be idempotent in the ordinary category $\mathbb X$, and every idempotent morphism defines a constant sequence. Recall that the idempotent $e$ splits in $\mathbb X$ if $e= t\cdot r$ for some morphisms $r, t$ with $r\cdot t=1$ (which already exist when $\mathbb X$ has epi-mono factorizations, or equalizers, or coequalizers). Such factorization of $e$ is unique, up to a uniquely determined isomorphism.

The constant sequence given by an idempotent $e:x\to x$ has a colimit in the ordinary category $\ncatX$ if, and only if, $e$ splits.

Given a colimit cocone $\rho_n:x\to y\,(n\in\mathbb N)$ of the constant sequence defined by $e$, one has

$$\rho_0=\rho_1\cdot e=\rho_2\cdot e\cdot e=\rho_2\cdot e =\rho_1$$

and, inductively, $\rho_1=\rho_2=\dots=:r$. The colimit cocone makes this morphism epic. Furthermore, the cocone $\eta$ with $\eta_n=e$ for all $n$ corresponds to a morphism $t:y\to x$ with $t\cdot r=e$, which also satisfies $r\cdot t=1_y$ since $r\cdot t\cdot r=r\cdot e=r$.

Conversely, given the splitting $r,t$ of $e$, the cocone $\rho$ with $\rho_n:=r:x\to y$ for all $n$ which we call related to the splitting, exhibits $y$ as a colimit of the constant sequence $e$:
since any cocone $\alpha:s\to\Delta z$ is easily seen to satisfy $\alpha_0\cdot t=\alpha_n\cdot t$, we obtain $(\alpha_0\cdot t)\cdot \rho_n=\alpha_n$ for all $n\in \mathbb N$, and furthermore, any morphism $f:y\to z$ with $\Delta f\cdot \rho =\alpha$ necessarily satisfies $f\cdot r=\alpha_0$, so that $f=\alpha_0\cdot t$.

Recall that
$\mathbb X$ (as an ordinary category) is said to be idempotent complete[We adopt here the terminology used in the recent paper [19]. Other terms used in the literature are Karoubi complete, Lawvere complete or, most frequently, Cauchy complete. We avoid the latter term, not to risk confusion with the dualization of our term of Cauchy cocompleteness for normed categories which is far more directly modelled after Cauchy's original ideas than idempotent completeness (of any flavour) is. Besides, as  a concept that gained its recognition through various important contributions in different contexts, it may indeed be difficult to attach just one person's name to idempotent completeness.] if all idempotents split in $\mathbb X$. Idempotent completeness of the category $\mathbb X_{\circ}$ suffices to provide an affirmative answer to the question raised at the beginning of this section. More precisely:

The constant sequence $s$ in a $\V$-normed category $\mathbb X$ given by an idempotent morphism $e$ is Cauchy precisely when $e$ is a $\kk$-morphism. In this case, the constant cocone related to a given splitting $e=r\cdot t$ of $e$ in $\mathbb X$ gives a normed colimit of $s$ in $\mathbb X$ if, and only if, the morphisms $r$ and $t$ are  both $\kk$-morphisms.

The first claim is obvious. Also, trivially the constant cocone $\rho$ related to the splitting $r, t$ of $e$ is a $\kk$-cocone if, and only if, $r$ is a $\kk$-morphism. Since $\rho$ is an ordinary colimit cocone, for the proof of the second claim, assuming $\kk\leq |r|$, we just need to show that $\kk\leq |t|$ holds if, and only if,
$$\bigvee_{N\in\mathbb N}\bigwedge_{n\geq N}|f\cdot \rho_n|=|f\cdot r|\leq|f|$$
for all $f:y\to z$ in $\mathbb X$. Indeed, from $\kk\leq|t|$ one obtains
$|f\cdot r|\leq|f\cdot r|\otimes|t|\leq|f\cdot r\cdot t|=|f|\;,$
and conversely, exploiting this inequality for $f=t$, from $\kk\leq|e|$ we obtain $\kk\leq|t\cdot r|\leq |t|$.

For a $\V$-normed category $\mathbb X$, if the category $\mathbb X_{\circ} $ is idempotent complete, every constant Cauchy sequence in $\mathbb X$ has a normed colimit.

§ THE PRINCIPAL EXAMPLE: SEMI-NORMED AND NORMED VECTOR SPACES

When a norm function $\nv\text{-}\nv$ on a (real, say) vector space $X$ satisfies the standard axioms for a norm, except that non-zero vectors are allowed to have norm $0$, one usually calls $X$ semi-normed. Here, in order to ensure that the category to be formed becomes Cauchy cocomplete, just like for the metric of a Lawvere metric space we should abandon not only the separation condition, but also the finiteness condition for norms. This then necessitates the extension of the real multiplication to $\infty$, so that we can maintain the norm axiom for scalar multiples of vectors with norm $\infty$.
To that end we first introduce the quantale
$$\mathcal R_{\times} = ([0,\infty],\geq,\cdot,1),$$
in such a way that the exponential function $e:\mathcal R_+\to\mathcal R_{\times}$, extended by $e^{\infty}=\infty$, becomes a homomorphism of quantales, i.e., a homomorphism of monoids which preserves infima (with respect to the natural order $\leq$ of the extended non-negative real line).

The monotonicity of the multiplication on $[0,\infty]$ in each variable certainly necessitates $\alpha\cdot\infty=\infty$ for $\alpha>0$, and the preservation of infima then forces the equality

$$ 0\cdot \infty=\inf\{\alpha\cdot \infty\mid\alpha>0\}=\infty\;.$$

Since it extends the usual fractions in case $\alpha, \beta\notin\{0,\infty\}$,
we denote the internal hom $[\beta,\alpha]$ in $\mathcal R_{\times}$ by $\frac{\alpha}{\beta}$ for all $\alpha, \beta\in[0,\infty]$. Hence, its value is given by the adjunction rule

$$\frac{\alpha}{\beta}\leq\gamma\iff\alpha\leq\beta\cdot\gamma \;,$$

for all $\gamma\in[0,\infty]$, that is:  $\frac{\alpha}{\beta}=\inf\{\gamma\in[0,\infty]\mid \alpha\leq\beta\cdot\gamma\}$. This gives in particular

$$ \frac{0}{0}=0,\quad\frac{\alpha}{0}=\infty\;(\alpha>0),\quad  \frac{\alpha}{\infty}=\frac{\infty}{\infty}=0\;.$$

The inf-preserving map $e:[0,\infty]\to [0,\infty]$ has a left adjoint, $\log^{\circ}$, whose values are ruled by adjunction, i.e.,

$$\log^{\circ}\alpha\leq\beta\iff\alpha\leq e^{\beta}\;;$$

As a mapping $\log^{\circ}:\mathcal R_{\times}\to \mathcal R_+$ we obtain only a lax homomorphism of quantales, since the easily established inequality

$$\log^{\circ}(\alpha\cdot\beta)\leq \log^{\circ}\alpha+\log^{\circ}\beta$$

is generally strict.

We are ready to define our category of semi-normed vector spaces:

A semi-norm on a (real) vector space $X$ is a function $\nv\text{-}\nv:X\to [0,\infty]$ satisfying

(N0) $\nv0\nv=0$,

(N1) $\nv a x\nv=|a|\nv x\nv$,

(N2) $\nv x+y\nv\leq\nv x\nv +\nv y\nv$,

for all $x\in X$ and $a\in\mathbb R, a\neq 0$. The thus defined semi-normed vector spaces are the objects of the category

whose morphisms are arbitrary linear maps. It contains as a full subcategory the category $\mathsf{NVec}_{\infty}$ mentioned in the Introduction; see Definition <ref> for details.

We define the logarithmic norm of a morphism $f:X\to Y$ in $\mathsf{SNVec}_{\infty}$ by

$$|f|=\sup_{x\in X}\,(\log^{\circ}\frac{\nv fx\nv}{\nv x\nv})\;.$$

Here are some immediate consequences of this definition.

Let $f:X\to Y$ be a linear map of semi-normed vector spaces. Then:

(1) If $X$ contains a vector $x_0$ with $\nv x_0\nv=0$ and $\nv fx_0\nv\neq 0$, then $|f|=\infty$.

(2) If $\nv x\nv=0$ always implies $\nv fx\nv=0$, then $|f|=\sup_{\nv x\nv=1}\,(\log^{\circ}\nv fx\nv)\;.$

(3) For all $x\in X$ one has $\nv fx\nv\leq e^{|f|}\, \nv x\nv$.

(4) One has $|f|=0$ if, and only if, $\nv fx\nv\leq\nv x\nv$ holds for all $x\in X$.

(1) From $\log^{\circ}\frac{\nv fx_0\nv}{\nv x_0\nv}\leq|f|$ one obtains $\infty=\frac{\nv fx_0\nv}{\nv x_0\nv}\leq e^{|f|}$ and, hence, $|f|=\infty$.

(2) Trivially $t:=\sup_{\nv x\nv=1}\,(\log^{\circ}\nv fx\nv)\leq|f|$. For “$\geq$” consider any $x\in X$. If $\nv x\nv=0$, also $\nv fx\nv=0$ by hypothesis, and $\log^{\circ}\frac{\nv fx\nv}{\nv x\nv}=0\leq t$ follows; likewise if $\nv x\nv=\infty$. In all other cases one considers $x_1:=\frac{1}{\nv x\nv}x$ in a standard manner.

(3) The claim follows from $\log^{\circ}\frac{\nv fx\nv}{\nv x\nv}\leq|f|$
by the adjunction rules for fractions and for $\log^{\circ}\dashv e$.

(4) Again, by adjunction, $\nv fx\nv \leq\nv x\nv $ implies $\frac{\nv fx\nv}{\nv x\nv}\leq 1$ for all $x\in X$, so that “if” follows, while  “only if” is obtained from (3).

With its logarithmic norm, $\mathsf{SNVec}_{\infty}$ is a Cauchy-cocomplete normed category whose $0$-morphisms are precisely the $1$-Lipschitz linear maps: $(\mathsf{SNVec}_{\infty})_{\circ}=\mathsf{SNVec}_1$.

Let us first confirm that $\mathsf{SNVec}_{\infty}$ is indeed a normed category. By Lemma <ref>(4), one has $|\mathrm{id}_X|=0$ for every semi-normed space $X$.
For morphisms $f:X\to Y$ and $g:Y\to Z$ in  $\mathsf{SNVec}_{\infty}$, we may assume that every $x\in X$ with $\nv x\nv=0$ satisfies $\nv g(fx)\nv=0$ since the existence of $x_0\in X$ with $\nv x_0\nv=0$ but $\nv g(fx_0)\nv\neq 0$ would, by Lemma <ref>(1), imply $|g|=\infty$ or $|f|=\infty$ (depending on whether $\nv fx_0\nv=0$ or not), so that  $\infty=|gf|\leq |f |+|g|$ would hold trivially. Consequently, by Lemma <ref> (2),(3) we may restrict ourselves to considering only vectors $x\in X$ with $\nv x\nv=1$ and obtain the required inequality from

$$\log^{\circ}\nv g(fx)\nv\leq\log^{\circ}(e^{|g|}\cdot\nv fx\nv)\leq \log^{\circ}e^{|g|}+ \log^{\circ}\nv fx\nv \leq |g|+\sup_{y\in X}(\log^{\circ}\frac{\nv fy\nv}{\nv y\nv})=|g|+|f|\;.  $$

Also, by Lemma <ref>(4), the $0$-morphisms are precisely the $1$-Lipschitz morphisms of $\mathsf{SNVec}_{\infty}$.

Let us now consider a Cauchy sequence $s=(\xymatrix{X_m\ar[r]^{s_{m,n}}&X_n\\})_{m\leq n}$ in $\mathsf{SNVec}_{\infty}$, so that

$$\inf_{n\in \mathbb N}\sup_{n\geq m\geq N}|s_{m,n}|=0\;,$$

and form its (ordinary) colimit $X$ in the category $\mathsf{Vec}$ of (real) vector spaces and linear maps, with colimit cocone  $\gamma=(\xymatrix{X_n\ar[r]^{\gamma_n}&X\\})_{ n\in\mathbb N}$. First we define for all $x\in X$

$$\nv x\nv:= \sup_{N\in \mathbb N}\,\inf_{n\geq N}\inf_{z\in\gamma_n^{-1}x}\nv z\nv_n\;,$$

and will now show that this makes $X$ a semi-normed vector space. (In what follows, we will no longer use subscripts to indicate where norms are being taken, as the context should give sufficient clarity.) We trivially have $\nv 0\nv=0$, as well as $\nv x+y\nv\leq\nv x\nv+\nv y\nv$ whenever $x,y\in X$ are such that $\nv x\nv=\infty$  or $\nv y\nv=\infty$. Hence we may assume that the norms $\nv x\nv$ and $\nv y\nv$ are finite and will establish the triangle inequality for them by showing that, for all $N\in \mathbb N$ and any real $\varepsilon$ with $0<\varepsilon\leq 1$, one has
$\inf_{n\geq N}\inf_{z\in\gamma_n^{-1}(x+y)}\nv z\nv\leq \nv x\nv+\nv y\nv+\varepsilon r$ with $r:=\nv x\nv+\nv y\nv+3$.

Putting $\eta:=\log(1+\varepsilon)>0$, the Cauchyness of $s$ gives us some $M\in \mathbb N$ with $|s_{m,n}|\leq\eta$ for all $n\geq m\geq M$. The definition of the norms of $x$ and $y$ guarantees the existence of $m,n\geq\max\{M,N\}$ and $u\in X_m, v\in X_n$ with
$\gamma_m(u)=x,\;\gamma_n(v)=y$ and $\nv u\nv\leq \nv x\nv+\varepsilon,\;\nv v\nv\leq \nv y\nv+\varepsilon$.
Without loss of generality we may assume $m\leq n$ and, setting $w:=s_{m,n}(u)$, have  $\nv w\nv\leq e^{\eta}\nv u\nv=\nv u\nv+\varepsilon\nv u\nv$. Consequently, $\gamma_n(w+v)=x+y$ and, with  the triangle inequality holding in $X_n$, we obtain

\begin{align*}
\inf_{z\in\gamma_n^{-1}(x+y)}\nv z\nv\leq \nv w+v\nv &\leq \nv w\nv +\nv v\nv\\
&\leq \nv u\nv+\varepsilon\nv u\nv+\nv v\nv\\
& \leq \nv x\nv+\varepsilon +\varepsilon(\nv x\nv+\varepsilon)+\nv y\nv+\varepsilon\\
&\leq\nv x\nv+\nv y\nv+\varepsilon r\;,
\end{align*}

This concludes the proof of (N3).
The validity of (N2), i.e., of $\nv ax\nv=|a|\nv x\nv$ for all real $a\neq 0$ and $x\in X$, is an immediate consequence of the equivalence $(z\in \gamma_n^{-1}(ax)\iff w\in \gamma_n^{-1}x)$ whenever $z=aw$, and of the fact that multiplication in $[0,\infty]$ by the positive real  number $|a|$ preserves both, infima and suprema.

We are left with having to confirm conditions (C2a) and (C2b) of Corollary <ref>.

(C2a) To show $\inf_N\sup_{m\geq N}|\gamma_m|=0$, we consider any $\varepsilon>0$ and, since $s$ is Cauchy, obtain some $N\in\mathbb N$ with
$\sup_{n\geq m\geq N}|s_{m,n}|\leq\varepsilon.   $

Claim: For all $m\ge N$ and $z\in X_m$ one has $\nv \gamma_mz\nv\leq \sup_{n\geq m}\nv s_{m,n}z\nv$. Indeed, by definition one has
$\nv\gamma_mz\nv=\sup_{K\in \mathbb N}\phi(K)$ where $\phi(K):=\inf\{\nv w\nv\mid k\geq K, w\in X_k, \gamma_kw=\gamma_mz\}$. Since $\phi(K)$ is monotonely increasing in $K$, we can write

$$\nv\gamma_mz\nv=\sup_{n\geq m}\inf\{\nv w\nv\mid k\geq n, w\in X_k, \gamma_kw=\gamma_mz\}\leq\sup_{n\geq m}\nv s_{m,n}z\nv,$$

with the inequality holding since $s_{m,n}z$ is one of the admissible vectors $w$: $\gamma_n(s_{m,n}z)=\gamma_mz$.

A consequence of the Claim is that one has $\nv \gamma_mz\nv=0$ whenever $\nv z\nv=0$, since Lemma <ref>(3) implies

$$\nv s_{m,n}z\nv\leq e^{|s_{m,n}|}\nv z\nv\leq e^{\varepsilon}\nv z\nv $$

for all $n\geq m$. Therefore, with Lemma <ref>(2), we obtain

$$|\gamma_m|=\sup_{z\in X_m,\nv z\nv=1}\log^{\circ}\nv\gamma_mz\nv\leq \log^{\circ}(e^{\varepsilon})=\varepsilon,$$

as desired.

(C2b) For every linear map $f:X\to Y$ to a semi-normed vector space $Y$, we must show $|f|\leq \inf_N \sup_{n\geq N}|f_n|$, where $f_n=f\cdot\gamma_n$.

Case 1: For every $x\in X$ with $\nv x\nv=0$ we have $\nv fx\nv=0$. Then Lemma <ref>(2) applies for the computation of $|f|$, so that it suffices for us to show $\log^{\circ}\nv fx\nv\leq\sup_{n\geq N}|f_n|$ whenever  $\nv x\nv=1$ and $N\in\mathbb N$; we may actually also assume $\nv fx\nv>1$ since otherwise $\log^{\circ}\nv fx\nv=0$ . Given any $\varepsilon>0$, the definition of $\nv x\nv$ guarantees the existence of some $m\geq N$ and $z\in \gamma_m^{-1}x$ with $\nv z\nv\leq 1+\varepsilon$. Since $fx= f_mz$, the case $\nv fx\nv=\infty$ would imply $|f_m|=\infty$. Hence, it suffices to consider the case $\nv fx\nv<\infty$ and, without loss of generality, we may assume $0<\varepsilon\leq\nv fx\nv-1$. Then

$$ |f_m|\geq\log^{\circ}\frac{\nv f_mz\nv}{\nv z\nv}\geq\log\frac{\nv fx\nv}{1+\varepsilon}=\log\nv fx\nv+\log\frac{1}{1+\varepsilon} $$

for all such $\varepsilon$, which implies the desired inequality.

Case 2: For some $x_0\in X$ with $\nv x_0\nv=0$ we have $\nv fx_0\nv\neq 0$. Then Lemma <ref>(1) gives $|f|=\infty$, and we must show $ \inf_N \sup_{n\geq N}|f_n|\geq\infty$. Similarly to Case 1, given any $N\in \mathbb N$ and $\varepsilon>0$, we have some $m\geq N$ and $z\in\gamma_m^{-1}x_0$ with $\nv z\nv\leq\varepsilon$, which implies

$$ |f_m|\geq\log^{\circ}\frac{\nv f_mz\nv}{\nv z\nv}\geq\log^{\circ}\frac{\nv fx_0\nv}{\varepsilon} $$

for all such $\varepsilon$. Again, the desired inequality follows.

The full normed subcategory $\mathsf{NVec}_{\infty}$ of  $\mathsf{SNVec}_{\infty}$ as considered in the Introduction fails to be closed under the formation of normed colimits of Cauchy sequences. Even for a Cauchy sequence $s$ of (strictly contractive) linear maps $s_{m,n}: X_m\to X_n$ of normed vector spaces (with all norms finite), the normed colimit in  $\mathsf{SNVec}_{\infty}$ may fail to be a normed vector space. Indeed, consider the sequence already mentioned in the Introduction; that is: $X_n:=\mathbb R$ normed by $\nv 1\nv_n=\frac{1}{n}$ and $s_{n,m}=\mathrm{id}_{\mathbb R}$ for all $m\leq n$. The normed colimit of $s$ in $\mathsf{SNVec}_{\infty}$
may again be formed by identity maps, $\gamma_n:X_n\to X=\mathbb R$, with the norm in $X$ given by

$$\nv 1\nv=\sup_N\inf_{n\geq N}\nv 1\nv_n=0,$$

i.e., all norms in $X$ are $0$, so that separation fails to the largest extent possible.

In general, the colimit maps $\gamma_n:X_n\to X$ presenting $X$ as a normed colimit of the sequence $s=(X_m\to X_n)_{m\leq n}$ in $\mathsf{SNVec}_{\infty}$ have an important extra property (which follows from the Claim formulated in the Proof of Theorem <ref>) that deserves to be named:

A linear map $f:X\to Y$ of semi-normed vector spaces is called a zero-to-zero morphism if $\nv fx\nv=0$ holds for all $x\in X$ with $\nv x\nv=0$. (We note that every bounded operator is a zero-to-zero morphism.) We denote by

the (non-full) subcategory of $\mathsf{SNVec}_{\infty}$ containing all semi-normed vector spaces and their zero-to-zero morphisms. The objects of the category

mentioned in the Introduction are precisely the separated semi-normed vector spaces; i.e., its objects $X$ satisfy the separation condition

$$\nv x\nv=0\Longrightarrow x=0$$

for all $x\in X$, and they fall short of being normed vector spaces in the classical sense only insofar as vectors are permitted to have infinite norms. The separation condition of its domain makes a morphism in $\mathsf{NVec}_{\infty}$, i.e., an arbitrary linear map, automatically a zero-to-zero morphism. Therefore, $\mathsf{NVec}_{\infty}$ is a full subcategory of $\mathsf{SNVec}_{00}$.

The normed category $\mathsf{NVec}_{\infty}$ is reflective in the normed category $\mathsf{SNVec}_{00}$, as $(\Setss{\mathcal R_+})$-enriched categories.

For $X\in\mathsf{SNVec}_{00}$ consider its subspace $X_0:=\{x\in X\mid \nv x\nv=0\}$ and let $p:X\to X/X_0$ be the projection.
$\nv x\nv=\nv y+(x-y)\nv\leq\nv y \nv+\nv x-y\nv$
one has $(\nv x-y\nv=0\iff \nv x \nv=\nv y\nv)$ for all $x,y\in X$, so that $\nv px\nv:=\nv x\nv$ makes $X/X_0$ a well-defined object of $\mathsf{NVec}_{\infty}$ and $p$ a zero-to-zero morphism – in fact, an isometry. Furthermore, for all $Y\in \mathsf{NVec}_{\infty}$ we have the natural bijection

$$-\cdot p:\mathsf{NVec}_{\infty}(X/X_0,Y)\to\mathsf{SNVec}_{00}(X,Y),$$

whose surjectivity is guaranteed by our restriction to zero-to-zero morphisms (as opposed to all linear maps of semi-normed vector spaces). In fact, this bijection is a $(\Setss{\V})$-isomorphism since, for every linear map $f:X/X_0\to Y$, one has

$$|f|=\sup_{z\in X/X_0}(\log^{\circ}\frac{\nv fz\nv}{\nv z\nv})=\sup_{x\in X}\;(\log^{\circ}\frac{\nv f(px)\nv}{\nv px\nv})=\sup_{x\in X}\;(\log^{\circ}\frac{\nv f(px)\nv}{\nv x\nv})=|f\cdot p|\;.  $$

The normed category $\mathsf{NVec}_{\infty}$ has normed colimits of all those Cauchy sequences whose normed colimit in $\mathsf{SNVec}_{\infty}$ is a colimit in the ordinary category  $\mathsf{SNVec}_{00}$.

Keeping the notation of the proof of Theorem <ref>, we consider a Cauchy sequence $s$ in $\mathsf{NVec}_{\infty}$  and obtain its normed colimit $X$ in $\mathsf{SNVec}_{\infty}$, with a colimit cocone $\gamma$ formed by zero-to-zero morphisms. A fortiori, by hypothesis, $\gamma$ is a colimit cocone in $\mathsf{SNVec}_{00}$, so that any linear map $f:X\to Y$ must be a zero-to-zero morphism. To arrive at a colimit of $s$ in $\mathsf{NVec}_{\infty}$, according to Proposition <ref>, we must apply the reflector to $X$ and obtain the colimit cocone $(p\cdot\gamma_n: X_n\to X/X_{0})_n$. As one easily checks (or formally derives with Proposition <ref> proved below), since the adjunction of Proposition <ref> is ($\Setss{\mathcal R_+}$)-enriched, this cocone presents $X/X_0$ in fact as a normed colimit of $s$ in $\mathsf{NVec}_{00}$.

By contrast:

The normed category $\mathsf{NVec}_{\infty}$ is not Cauchy cocomplete.

The sequence presented in Remark <ref> is a Cauchy sequence in $\mathsf{NVec}_{\infty}$. If it had a normed colimit in $\mathsf{NVec}_{\infty}$, this would have to be also a colimit in the ordinary category of vector spaces and linear maps, so that we may assume that it is given by $\mathbb R$, with a cocone of identity maps and normed by $|1|=c$ for some $c>0$. But then $|\gamma_n|=\log^{\circ}(nc)$ for all $n\in \mathbb N$, which would give $\sup_N\inf_{n\geq N}|\gamma_n|=\infty$, in contradiction to the normed colimit condition (C2a).

Let us also point out that existing normed colimits in $\mathsf{NVec}_{\infty}$ of Cauchy sequences of Banach spaces need not be Banach:

Consider the sequence

$$\xymatrix{\{0\}\ar[r] &\mathbb R\ar[r] & \mathbb R^{2}\ar[r] & \mathbb R^{3}\ar[r] & ...\ar[r] & \mathrm{colim}_n\;\mathbb R^{n}=\bigoplus_n\mathbb R^n\\

of isometric embeddings of Euclidean spaces whose normed colimit is given by the direct sum (in $\mathsf{Vec}$) of its objects, normed accordingly. In the direct sum, we have the Cauchy sequence $(x_n)_n$, where the $i$-th component of $x_n$ is $\frac{1}{i+1}$ for $i\leq n$, and $0$ otherwise, but the sequence does not converge in
$\bigoplus_n\mathbb R^n$.

§ CHANGE OF BASE FOR NORMED CATEGORIES,  METRIC SPACES

Our next goal is to show that the category $\MET_{\infty}$ of all Lawvere metric spaces, with arbitrary maps between them as morphisms and normed as in  Example <ref>(5), is Cauchy cocomplete. We will do so in three steps, by first giving  conditions on our quantale $(\V,\leq,\otimes,\kk)$ guaranteeing that the normed category $V\text{-}\mathsf{Lip}$ of all small $\V$-categories with arbitrary maps as defined in Proposition <ref> is Cauchy cocomplete. With the benefit of the methods used in [13, 14], the proof extends the “epsilon techniques” used in the proof of Theorem <ref> to a fairly general quantalic context. Then we will briefly discuss how Cauchy cocompleteness for $\V$-normed categories fares under changing the “base”  $\V$, before applying our findings to the adjunction $e\dashv \log^{\circ}:\mathcal R_{\times}\to \mathcal R_+$ to obtain the Cauchy cocompleteness of $\MET_{\infty}$.

Recall that for $u,v\in\V$ one says that $u$ is totally below $v$, written as $u\ll v$, if $v\leq\bigvee W$ with $W\subseteq \V$ can hold only if $u\leq w$ for some $w\in W$. We say that $v$ is approximated from totally below if $v=\bigvee\Downarrow\!\!v$, where $ \Downarrow\!\!v=\{u\in\V\mid u\ll v\}$. Recall that the complete lattice $\V$ is constructively completely distributive [40, 22] if every element in $\V$ is approximated from totally below. (In the presence of Choice this property implies complete distributivity in the standard sense.) We say that the tensor product of $\V$ preserves the totally below relation if $u\ll v$ implies $u\otimes w\ll v\otimes w$ for all $w>\bot$ in $\V$. In the presence of this preservation property, and if $\kk=\top$ is approximated from totally below, then $\V$ is constructively completely distributive.

Let the totally below relation in $\V$ be preserved by $\otimes$, and let the tensor-neutral element $\kk$ be approximated from totally below, with the set $\Downarrow\!\!\kk$ being up-directed in $\V$. Then the normed category $\V\text{-}\mathsf{Lip}$ is Cauchy cocomplete.

For a given Cauchy sequence $s=(\xymatrix{X_m\ar[r]^{s_{m,n}}&X_n\\})_{m\leq n}$ in $\V\text{-}\mathsf{Lip}$, we form its (ordinary) colimit $X$ in the category $\SET$ with colimit cocone  $\gamma=(\xymatrix{X_n\ar[r]^{\gamma_n}&X\\})_{ n\in\mathbb N}$ and now want to define a $\V$-category structure on $X$ by

$$X(x,y):= \bigwedge_{N\in \mathbb N}\;\bigvee\{X_n(x',y')\mid n\geq N, x'\in\gamma_n^{-1}x, y'\in\gamma_n^{-1}y\},$$

for all $x,y\in X$. Trivially, $\kk\leq X(x,x)$. In order to establish the inequality
$X(x,y)\otimes X(y,z)\leq X(x,z)$ in $X$, we may disregard the trivial case $X(x,y)=\bot$ or $X(y,z)=\bot$. We consider any $\varepsilon\in \V$ with $\bot<\varepsilon\ll\kk$. Then the Cauchyness of $s$ gives us some $M\in \mathbb N$ with $\varepsilon \leq |s_{m,\ell}|$ for all $\ell\geq m\geq M$. By the assumed preservation property we have
$X(x,y)\otimes \varepsilon \ll X(x,y)$ and $X(y,z)\otimes \varepsilon \ll X(y,z)$. Therefore, for every $N\in\mathbb N$, the definitions of $X(x,y)$ and $X(y,z)$ let us pick
$m,n\geq\max\{M,N\}$ and $x'\in\gamma_m^{-1}x, \;y'\in\gamma_m^{-1}y,\;y''\in\gamma_n^{-1}y,\;z'\in\gamma_n^{-1}z$, such that

$$X(x,y)\otimes\varepsilon\leq X_m(x',y')\quad\text{and}\quad X(y,z)\otimes\varepsilon\leq X_n(y'',z')\;.$$

Now, since $\gamma_my'=\gamma_ny''$, the construction of $X$ as a directed colimit in $\SET$ allows us to find $\ell\geq m,n$ with $s_{m,\ell}y'=s_{n,\ell}y''$.
Since, by the definition of the $\V$-norm in $\V\text{-}\mathsf{Lip}$, the inequalities $\varepsilon\leq |s_{m,\ell}|$ and $\varepsilon\leq |s_{n,\ell}|$ imply $X_m(x',y')\otimes\varepsilon\leq X_{\ell}(s_{m,\ell}x',s_{m,\ell}y')$ and $X_n(y'',z')\otimes\varepsilon\leq X_{\ell}(s_{n,\ell}y'',s_{n,\ell}z')$, we obtain

\begin{align*}
X(x,y)\otimes\varepsilon\otimes\varepsilon\otimes X(y,z)\otimes\varepsilon\otimes\varepsilon &\leq X_{\ell}(s_{m,\ell}x',s_{m,\ell}y')\otimes X_{\ell}(s_{n,\ell}y'',s_{n,\ell}yz')\\
&\leq X_{\ell}(s_{m,\ell}x',s_{n,\ell}z')  \\
&\leq X(x,z),
\end{align*}

with the last inequality holding since $N$ was given arbitrarily. The desired inequality $X(x,y)\otimes X(y,z)\leq X(x,z)$ follows since, with the up-directed set $\Downarrow\!\!\kk$, we have

and then, likewise, $\kk=\bigvee_{\varepsilon\ll\kk}\varepsilon\otimes\varepsilon\otimes\varepsilon\otimes\varepsilon$.

(C2a) In order to show $\kk\leq\bigvee_{N\in\mathbb N}\bigwedge_{n\geq N}|\gamma_n|$, let $\varepsilon\ll\kk$ and, as above, pick $M\in \mathbb N$ with $\varepsilon\leq\bigwedge_{n\geq m\geq M}|s_{m,n}|$. For every $m\geq M$ and all $z,w\in X_m$, we have (similarly as in the proof of Theorem <ref>)
$ X(\gamma_mz,\gamma_mw)=\bigwedge_{K\in\mathbb N} \Phi(K)$, where
$$\Phi(K)=\bigvee\{X_{\ell}(z',w')\mid\ell\geq K, z',w'\in X_{\ell}, \gamma_{\ell}z'=\gamma_mz, \gamma_{\ell}w'=\gamma_mw\}$$
is monotonely
decreasing in $K$. Therefore,

\begin{align*}
X(\gamma_mz,\gamma_mw)=\bigwedge_{n\geq m}\Phi(n)&\geq\bigwedge_{n\geq m}X_n(s_{m,n}z,s_{m,n}w)\\
&\geq\bigwedge_{n\geq m}(|s_{m,n}|\otimes X_m(z,w))\\
&\geq(\bigwedge_{n\geq m}|s_{m,n}|)\otimes X_m(z,w).
\end{align*}

This gives

\varepsilon\leq\bigwedge_{n\geq m\geq M}|s_{m,n}|\leq\bigwedge_{m\geq M}[X_m(z,w),X(\gamma_mz,\gamma_mw)]\leq\bigwedge_{m\geq M}|\gamma_m|\leq\bigvee_N\bigwedge_{n\geq N}|\gamma_n|\

and, with $\kk=\bigvee_{\varepsilon\ll\kk}\varepsilon$, the desired inequality.

(C2b) For any mapping $f: X\to Y$ of $\V$-categories, we must show
$$\bigvee_N\bigwedge_{n\geq N}|f\cdot\gamma_n|\;\leq\;|f|=\bigwedge_{x,y\in X}[X(x,y),Y(fx,fy)]\;;$$

equivalently, $\bigwedge_{n\geq N}|f\cdot\gamma_n|\otimes X(x,y)\leq Y(fx,fy)$, for all $N\in\mathbb N$ and $x,y\in X$. To this end, discarding the trivial case $X(x,y)=\bot$, we consider $\varepsilon\ll\kk$ and have $X(x,y)\otimes \varepsilon\ll X(x,y)$ and may pick $m\geq N$ and $x'\in\gamma_m^{-1}x,\; y'\in\gamma_m^{-1}y$ with $X(x,y)\otimes \varepsilon \leq X_m(x',y')$. Therefore,

$$\bigwedge_{n\geq N}|f\cdot\gamma_n|\otimes X(x,y)\otimes\varepsilon\leq |f\cdot\gamma_m|\otimes X(x,y)\otimes\varepsilon\leq|f\gamma_M|\otimes X_m(x',y')\leq Y(fx,fy),$$

which implies the desired inequality.

The quantales $\mathsf 2,\; \mathcal R_+$ and $\mathcal R_{\times}=([0,\infty],\geq,\cdot, 1)$ satisfy the hypotheses of Theorem <ref>. Therefore, the  $\mathsf 2$-normed category $\mathsf 2\text{-}\mathsf{Lip}$ of preordered sets and arbitrary maps is Cauchy-cocomplete, and likewise for $\mathcal R_+\text{-}\mathsf{Lip}$ and $\mathcal R_{\times}\text{-}\mathsf{Lip}$.

Recall that a monotone map $\varphi: \V\to \mathcal W$ to a quantale $(\mathcal W,\leq,\boxtimes,\mathrm n)$ is a lax homomorphism of quantales if $\mathrm n\leq\varphi \kk$ and $\varphi v\boxtimes\varphi v'\leq\varphi(v\otimes v')$ for all $v,v'\in \V$. Such lax homomorphism induces the change-of-base functor

$$\mathcal B_{\varphi}:\mathsf{CAT}/\!/\V\longrightarrow\mathsf{CAT}/\!/\mathcal W,\quad (\ncatX,|\text{-}|)\longmapsto(\ncatX,|\text{-}|_{\varphi}),$$

which regards a $\V$-normed category $\ncatX$ as a $\mathcal W$-normed category via $|f|_{\varphi}:=\varphi(|f|)$ for all morphisms $f$ in $\ncatX$, and which makes $\V$-normed functors become $\mathcal W$-normed.

Furthermore, if we also have a lax homomorphism $\psi:\mathcal W\to V$ which, as a monotone map, is right adjoint to $\varphi$, then we have the induced adjunction
$\mathcal B_{\varphi}\dashv\mathcal B_{\psi}$. Indeed, a $\V$-normed functor $F:\ncatX\to\mathcal B_{\psi}\ncatY$ may be considered equivalently as a $\mathcal W$-normed functor $F:\mathcal B_{\varphi}\ncatX\to\ncatY$, since the adjunction $\varphi\dashv\psi$ facilitates, for all morphism $f$ in $\ncatX$, the equivalence

Let $\varphi\dashv\psi:\mathcal W\to\V$ be adjoint lax homomorphisms of quantales, with $\psi$ preserving joins of monotone sequences in $\mathcal W$. Then, if the $\mathcal W$-normed category $\ncatY$ is Cauchy cocomplete, so is the $\V$-normed category $\mathcal B_{\psi}\ncatY$.

Let $s=(s_{m,n})_{m\leq n}$ be a Cauchy sequence in $\mathcal B_{\psi}.$ So, with $|\text{-}|$ denoting the norm in the given $\mathcal W$-normed category $\ncatY$, we have $\kk\leq\bigvee_N\bigwedge_{n\geq m\geq N}|s_{n,m}|_{\psi}$ and, since the left adjoint $\varphi$ preserves joins, obtain

$$\mathrm n\leq \varphi\kk=\bigvee_N\varphi(\bigwedge_{n\geq m\geq N}\psi(|s_{m,n}|))\leq \bigvee_N\bigwedge_{n\geq m\geq N}\varphi\psi(|s_{m,n}|)\leq\bigvee_N\bigwedge_{n\geq m\geq N}|s_{m,n}|\;. $$

So, $s$ is Cauchy in $\ncatY$ and, hence, has a normed colimit $x$ in $\ncatY$, with colimit cocone $\gamma$. We claim that $x$ is also a normed colimit of $s$ in $\mathcal B_{\psi}\ncatY$. Indeed, this is an immediate consequence of the assumed preservation of joins of monotone sequences in $\mathcal W$ and the preservation of all meets  by the right adjoint  $\psi$ since, for all morphisms $f:x\to y$ in $\ncatY$, from $|f|=\bigvee_N\bigwedge_{n\geq N}|f\cdot\gamma_n|$ one obtains $|f|_{\psi}=\bigvee_N\bigwedge_{n\geq N}|f\cdot\gamma_n|_{\psi}$.

Let us now specialize the adjunction $\varphi\dashv\psi$ of lax homomorphisms of quantales as in the Proposition to the adjunction

$$e\dashv \log^{\circ}:\mathcal R_{\times}=([0,\infty],\geq,\cdot,1)\longrightarrow \mathcal R_+=([0,\infty],\geq,+,0)\;.$$
The exponential function is a (strict) homomorphism of quantales, and $\log^{\circ}$ is lax and preserves infima (w.r.t. the natural order). Since $\mathcal R_+\text{-}\mathsf{Lip}$ is Cauchy cocomplete by Theorem <ref>, the same is true for $\mathcal B_{\log^{\circ}}(\mathcal R_{\times}\text{-}\mathsf{Lip})=\MET_{\infty}$, by the Proposition. Therefore:

The normed category $\MET_{\infty}$ is Cauchy cocomplete.

§ REMARKS ON CAUCHY COCOMPLETENESS FOR $\MATHSF 2$-NORMED CATEGORIES

We consider the Boolean quantale $\V=\mathsf{2}$ and recall that
$\mathsf 2$-normed categories are ordinary categories $\mathbb X$ which come with a wide subcategory $\mathcal S$. In simplified form, the Cauchy cocompleteness of $(\mathbb X,\mathcal S)$ as described in Examples <ref>(2) equivalently means that the colimit of any sequence $s$ with all connecting morphisms in $\mathcal S$ exists in $\mathbb X$ and has a tail $s_{|N}=(s_{m,n})_{n\geq m\geq N}$ (with some $N\in \mathbb N$) which is actually a colimit in the subcategory $\mathcal S$. This, and some variations and generalizations thereof, is a concept that has been widely studied, primarily in the context of directed colimits and, dually, of inverse limits, rather than that of (co)limits of sequences. We limit ourselves to mentioning only a few instances and pointers to the literature.

For the following categories $\mathbb X$ and their wide subcategories $\mathcal S$, the $\mathsf 2$-normed category $(\mathbb X,\mathcal S)$ is Cauchy cocomplete:

(1) $\mathbb X$ is any locally finitely presentable category (in particular, any variety of finitary universal algebras), and $\mathcal S$ is the class of monomorphisms in $\mathbb X$: see Proposition 1.62 in [3]. The same source confirms that one may take for $\mathcal S$ also the class of regular monomorphisms whenever this class is closed under composition. But this latter provision frequently fails since, in any category with cokernel pairs and their equalizers, it means equivalently that all strong (or extremal) monomorphisms in $\mathbb X$ must be regular, or that $\mathbb X$ must have (epi, regular mono)-factorizations. For example, already since [25] the provision is known to fail in the category of semigroups; see Exercise 14I in [1] for a simple finite counter-example. However, various generalizations of the so-called amalgamation property guarantee in a general categorical context that extremal monomorphisms be regular: see Theorem 2 of [37].

(2) $\mathbb X$ is the category of topological spaces, and $\mathcal S$ is either the class of monomorphisms (injective continuous maps) or the class open embeddings: see Proposition 3.2 of [2]. However, we may not consider the class $\mathcal S^*$ of regular monomorphisms (arbitrary embeddings), even though $\mathcal S^*$ is trivially closed under composition and enjoys the property that the colimit cocone of any directed system with connecting maps in $\mathcal S^*$ consists of maps in $\mathcal S^*$ again (by Proposition 3.4 of [2]). Indeed, Example 3.5 of [2] gives a sequence of embeddings whose colimit in $\mathbb X$ is not a colimit in $\mathcal S^*$. For an extension of these observations from $\mathbb X$ to some of its full subcategories we also refer to [2].

(3)$\mathbb X$ is the opposite category of the categories of compact Hausdorff spaces, or of all Hausdorff spaces, and $\mathcal S$ is the class of epimorphisms in these two categories, i.e. the class of continuous surjections or of dense continuous maps, respectively: for compact Hausdorff spaces, see Corollary 2 of Section 9.6 in [11], and for Hausdorff spaces see Example 5 in [6]. The latter reference offers a wide array of categories from commutative algebra whose opposites provide further examples of Cauchy-cocomplete categories.

§ PRESHEAF CATEGORIES ARE CAUCHY COCOMPLETE

We continue to work with a quantale $(\V,\leq,\otimes,\kk)$ and first consider an arbitrary sequence $s=\xymatrix{(A_m\ar[r]^-{s_{m,n}} & A_n)_{m\leq n}}$ in  $\nSetss{\V}$. So, while the sets $A_n$ are $\V$-normed, the maps $s_{m,n}$ may not be. Still, with the forgetful functor $U:\nSetss{\V}\to\SET$, we can form the colimit $A$ of $Us$ in $\SET$, with cocone $\xymatrix{(A_n\ar[r]^{\gamma_n} & A)_n}$. Trivially, any norm on $A$ makes the resulting $\V$-normed set a colimit of $s$ in $\nSetss{\V}$, since there is no constraint on the morphisms in that category. But there is one norm on $A$ that distinguishes itself by a special property, as follows.
$$ |c|=\bigwedge_{N\in\mathbb N}\bigvee_{n\geq N}\bigvee_{a\in\gamma_n^{-1}c}|a|;$$

that is, we  employ the same formula as the one established for colimits of sequences in $\SET/\!/\V$ (see Proposition <ref>), but now without any a-priori expectation that it would make the maps $\gamma_n$ $\V$-normed. We call the above norm on $A$ the $\gamma$-induced Cauchy norm since it has the important property (C2b) (see Corollary <ref>):

Let the set $A$ be provided with the $\gamma$-induced Cauchy norm as above. Then, for any mapping $f:A\to B$ to a $\V$-normed set $B$, one has

$$|f|\geq\bigvee_{N\in\mathbb N} \bigwedge_{n\geq N}|f\cdot\gamma_n|\,.$$

Since, in its first (contravariant) variable, the internal hom $[\text{-,-}]$ of $\V$ transforms arbitrary joins into meets, we have:

\begin{align*}
|f|=\bigwedge_{c\in A}[|c|,|fc|] & =\bigwedge_{c\in A}[\bigwedge_{N}\bigvee_{n\geq N}\bigvee_{a\in\gamma_n^{-1}c}|a|,\,|fc|]   \\
& \geq \bigwedge_{c\in A}\bigvee_{N}[\bigvee_{n\geq N}\bigvee_{a\in\gamma_n^{-1}c}|a|,\,|fc|]\\
& \geq \bigvee_N\bigwedge_{c\in A}[\bigvee_{n\geq N}\bigvee_{a\in\gamma_n^{-1}c}|a|,\,|fc|]\\
& = \bigvee_N\bigwedge_{n\geq N}\bigwedge_{c\in A}\bigwedge_{a\in\gamma_n^{-1}c}[|a|,\,|fc|]\\
& = \bigvee_N\bigwedge_{n\geq N}\bigwedge_{a\in A_n}[|a|,\,|f(\gamma_na)|]\\
& = \bigvee_N\bigwedge_{n\geq N}|f\cdot\gamma_n|\;.
\end{align*}

In order to strengthen the assertion of Lemma <ref> and show that, in fact, $\nSetss{\V}$, and even all $\nSetss{\V}$-valued presheaf categories, are Cauchy cocomplete, we need a small additional hypothesis on the $\otimes$-neutral element $\kk$ of the quantale $\V$. Actually, we offer two alternative possibilities, (A) or (B), for suitably augmenting our general quantalic setting, as follows:

(A)   $\kk$ is approximated from totally below (see Section 9),  that is:
$$\kk=\bigvee\{\varepsilon\in\V \mid \varepsilon\ll\kk\}.$$

(B)   $\kk$  $\wedge$-distributes over arbitrary joins, that is:
$$\kk\wedge\bigvee_{i\in I}v_i=\bigvee_{i\in I}\kk\wedge v_i\;.$$

(1) Condition (A) certainly holds when the lattice $\V$ is (constructively) completely distributive in the sense of [40] since, in that case, every element in $\V$ is approximated from totally below by definition.

(2) Condition (B) trivially holds when the quantale $\V$ is integral, i.e., when $\kk=\top$, and also when  the underlying lattice of the quantale $\V$ is a frame since, in the latter case every element in $\V$ $\wedge$-distributes over arbitrary  joins by definition, whilst in the former case the map $\kk\wedge(-)$ is just the identity map on $\V$.

(3) All examples of quantales mentioned in this paper thus far satisfy both conditions, (A) and (B). We discuss the two conditions further in Section <ref> and in particular confirm their logical independence.

We are now ready to prove the main general theorem of the paper.

When the quantale $\V$ satisfies condition (A) or (B), then the $\V$-normed category $[\mathbb X,\nSetss{\V}]$ is Cauchy cocomplete, for every small $\V$-normed category $\mathbb X$.

Considering a Cauchy sequence $ \sigma=\xymatrix{(P_m\ar[r]^-{\sigma_{m,n}}& P_n)_{m\leq n\in\mathbb N}}$ in the category
$[\mathbb X,\nSetss{\V}]$ (given by all $\V$-normed functors $\mathbb X\to\nSetss{\V}$ and their natural transformations),
with the forgetful functor $U:\nSetss{\V}\to\SET$ we form the colimit $P$ of $U\sigma$ in the ordinary functor category $\SET^{\mathbb X}$, with cocone $\gamma=(\xymatrix{P_n\ar[r]^{\gamma_n} & P})_n$. Then, for every object $x$ in $\mathbb X$, the colimit $Px$ of the sequence $(U\sigma_{m,n}^x)_{m\leq n}$ in $\SET$ may be provided with the Cauchy norm induced by the cocone $(\gamma_n^x)_n$ (see Lemma <ref>), and in this way $P$ is then considered as a $\nSetss{\V}$-valued functor.

In order to establish $P$ as a normed colimit of $\sigma$, by Corollary <ref>, we must show:

(C1) The functor $P:\mathbb X\to \nSetss{\V}$ is $\V$-normed (so that it serves as a colimit of $\sigma$ in the ordinary full subcategory $[\mathbb X,\nSetss{\V}]$ of $(\nSetss{\V})^{\mathbb X}$, formed by all $\V$-normed functors $\mathbb X\to\nSetss{\V}$);

(C2a) $\gamma$ is a $\kk$-cocone, i.e., $\kk\leq \bigvee_N\bigwedge_{n\geq N}|\gamma_n|$;

(C2b) $|\alpha|\geq \bigvee_N\bigwedge_{n\geq N}|\alpha\cdot\gamma_n|$, for every natural transformation $\alpha:P\to Q$.

For showing (C1) we use (C2a) (the proof of which is presented further below, independently of (C1)) and, since every $P_n$ is $\V$-normed and every $\gamma_n=(\gamma_n^x)_{x\in\mathbb X}:P_n\to P$ is natural, obtain for all morphisms $f:x\to y$ in $\mathbb X$

\begin{align*}
|f|=\kk\otimes|f|&\leq(\bigvee_N\bigwedge_{n\geq N}|\gamma_n|)\otimes|f|\\
&\leq\bigvee_N(\bigwedge_{n\geq N}|\gamma_n^y|\otimes|f|)\\
&\leq \bigvee_N\bigwedge_{n\geq N}|\gamma_n^y|\otimes |P_nf|\\
&\leq \bigvee_N\bigwedge_{n\geq N}|\gamma_n^y\cdot P_nf|\\
&= \bigvee_N\bigwedge_{n\geq N}|Pf\cdot\gamma_n^x|\leq|Pf|\,,
\end{align*}

with the last inequality following from the fact that $Px$ carries the $(\gamma_n^x)_n$-induced Cauchy norm, so that Lemma <ref> applies.

Using this last argument again for every $x\in\mathbb X$, and before turning to the more cumbersome proof of (C2a), we can immediately show that condition (C2b) holds, as follows:

$$|\alpha|=\bigwedge_{x\in X}|\alpha_x|\geq\bigwedge_{x\in\mathbb X}\bigvee_N\bigwedge_{n\geq N}|\alpha_x\cdot\gamma_n^x|\geq\bigvee_N\bigwedge_{n\geq N}\bigwedge_{x\in\mathbb X}|\alpha_x\cdot\gamma_n^x|=\bigvee_N\bigwedge_{n\geq N}|\alpha\cdot\gamma_n|\,.    $$

For the proof of (C2a), we first calculate

\begin{align*}
\bigvee_N\bigwedge_{n\geq N}|\gamma_n| &= \bigvee_N\bigwedge_{n\geq N}\bigwedge_{x\in\mathbb X}|\gamma_n^x| \\
& =  \bigvee_N\bigwedge_{n\geq N}\bigwedge_{x\in\mathbb X}\bigwedge_{a\in P_nx}[|a|,|\gamma_n^xa|] \\
&=  \bigvee_N\bigwedge_{x\in\mathbb X}\bigwedge_{c\in Px}\bigwedge_{n\geq N}\bigwedge_{a\in(\gamma_n^x)^{-1}c}[|a|,|c|] \\
&=  \bigvee_N\bigwedge_{x\in\mathbb X}\bigwedge_{c\in Px}[\;\bigvee_{n\geq N}\bigvee_{a\in(\gamma_n^x)^{-1}c}|a|,\; \bigwedge_M \bigvee_{m\geq M}\bigvee_{b\in(\gamma_m^x)^{-1}c}|b|\; ] \\
&=  \bigvee_N\bigwedge_M\bigwedge_{x\in\mathbb X}\bigwedge_{c\in Px}[\,|\!|c|\!|_N, |\!|c|\!|_M\,]\,,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(*)
\end{align*}

where, for the last equality, we have used the abbreviation $|\!|c|\!|_N:=\bigvee_{n\geq N}\bigvee_{a\in(\gamma_n^x)^{-1}c}|a|$ for all $N\in \mathbb N, x\in\mathbb X$, and $c\in Px$.

We now consider the alternative hypotheses (A) and (B) and finish the proof under each of them separately, as follows.

(A) Since the Cauchy sequence $\sigma$ satisfies $\kk\leq\bigvee_N\bigwedge_{n\geq m\geq N}|\sigma_{m,n}|$, for every $\varepsilon\ll \kk$ in $\V$ we find an $N\in\mathbb N$ with $\varepsilon\leq\bigwedge_{n\geq m\geq N}|\sigma_{m,n}|$, i.e.,

$$\varepsilon\leq |\sigma_{m,n}^x|$$

for all $n\geq m\geq N$ and $x\in \mathbb X$. Now, given any $c\in Px$ and  $M\in \mathbb N$, in the case $M\leq N$ we trivially have $|\!|c|\!|_N\leq|\!|c|\!|_M$ and obtain $\varepsilon\ll\kk\leq [ |\!|c|\!|_N,|\!|c|\!|_M]$, so certainly $\varepsilon\leq [ |\!|c|\!|_N,|\!|c|\!|_M]$   . If $M\geq N$, with $\ell:=M-N$ we have

\begin{align*}
|\!|c|\!|_M =\bigvee_{m\geq M}\bigvee_{b\in(\gamma_m^x)^{-1}c}|b| & \geq \bigvee_{n\geq N}\bigvee_{a\in(\gamma_n^x)^{-1}c}|\sigma_{n,n+\ell}^x\,a|  \\
& \geq \bigvee_{n\geq N}\bigvee_{a\in(\gamma_n^x)^{-1}c}|a|\otimes|\sigma_{n,n+\ell}^x| \\
& \geq (\bigvee_{n\geq N}\bigvee_{a\in(\gamma_n^x)^{-1}c}|a|\,)\otimes u=|\!|c|\!|_N\otimes u\,,
\end{align*}

which again implies $\varepsilon\leq [ |\!|c|\!|_N,|\!|c|\!|_M]$. Consequently, since $\kk=\bigvee\{\varepsilon\mid \varepsilon\ll\kk\}$, with $(*)$ we obtain $\kk\leq \bigvee_N\bigwedge_{n\geq N}|\gamma_n|$, as desired.

(B) Analyzing further the equality $(*)$,

$$\bigvee_N\bigwedge_{n\geq N}|\gamma_n|= \bigvee_N\bigwedge_{x\in\mathbb X}\bigwedge_{c\in Px}[\,|\!|c|\!|_N, \bigwedge_M|\!|c|\!|_M\,]\,,  $$

we note that we have

\begin{align*}
[\,|\!|c|\!|_N, \bigwedge_M|\!|c|\!|_M\,] &=[\,|\!|c|\!|_N,\bigwedge_{M\leq N} |\!|c|\!|_M\wedge \bigwedge_{M\geq N}|\!|c|\!|_M\,]\\
&=[\,|\!|c|\!|_N,\bigwedge_{M\leq N} |\!|c|\!|_M\,]\;\wedge\; [\,|\!|c|\!|_N,\bigwedge_{M\geq N}|\!|c|\!|_M\,]  \\
& \geq \kk \wedge [\,|\!|c|\!|_N,\bigwedge_{M\geq N}  |\!|c|\!|_M\,] \\
& =\bigwedge_{M\geq N}(\kk\wedge[\,|\!|c|\!|_N,  |\!|c|\!|_M\,])\,.
\end{align*}

Here, for $M\geq N$, as in part (A), setting $\ell =M-N$ one has

$$|\!|c|\!|_M\geq  \bigvee_{n\geq N}\bigvee_{a\in(\gamma_n^x)^{-1}c}|a|\otimes|\sigma_{n,n+\ell}^x| = |\!|c|\!|_N\otimes \bigvee_{n\geq N}|\sigma_{n,n+\ell}^x|\geq |\!|c|\!|_N\otimes |\sigma_{N,M}^x|$$

and, hence, $ [\,|\!|c|\!|_N, |\!|c|\!|_M\,]\geq |\sigma_{N,M}^x| $.  Consequently, with hypothesis (B) and the Cauchyness of $\sigma$ we obtain

\begin{align*}
\bigvee_N\bigwedge_{n\geq N}|\gamma_n| & \geq \bigvee_N\bigwedge_{x\in\mathbb X}\bigwedge_{M\geq N}(\kk\wedge|\sigma_{N,M}^x|) \\
&=\bigvee_N(\kk\wedge\bigwedge_{M\geq N}\bigwedge_{x\in\mathbb X}|\sigma_{N,M}^x|)   \\
&=\kk\wedge\bigvee_N\bigwedge_{M\geq N}|\sigma_{N,M}| \\
& \geq\kk\wedge\kk=\kk\,,
\end{align*}

which concludes the proof.

In conjunction with Remarks <ref> we conclude:

For every small $\V$-normed category $\mathbb X$, the $\V$-normed category $[\mathbb X,\nSetss{\V}]$ is Cauchy cocomplete under any of the following hypotheses:

* the quantale $\V$ is integral;

* the lattice $\V$ is a frame;

* the lattice $\V$ is (constructively) completely distributive.

Is there a quantale $\V$ for which the $\V$-normed category $\nSetss{\V}$ fails to be Cauchy cocomplete?

§ NORMED COLIMITS AS WEIGHTED COLIMITS

In this section we assume that the quantale $\V$ satisfies condition (A) or (B) so that we can apply Theorem <ref>. Under this condition, we show that normed colimits of Cauchy sequences can be equivalently described as weighted (formerly indexed) colimits in the sense of [26], for an appropriate class of weights. By definition, a normed colimit of a sequence \(s\) in a \(\V\)-normed category \(\ncatX\) is given by an object \(x\) of \(\ncatX\) together with bijections

naturally in \(z\), so that the induced cocone

is a colimit in \(\Setss{\V}\).

We start with the following observation.

Every left adjoint \(\V\)-normed functor \(F \colon\ncatX \to\ncatY\) preserves normed colimits of sequences.

Let \(G \colon \ncatY\to\ncatX\) be right adjoint of \(F\) in \(\Catss{\V}\), so that we have isomorphisms

in \(\Setss{\V}\), naturally in \(x\) and \(y\). Therefore, for every sequence $s=(x_m\xrightarrow{s_{m,n}} x_n)_{m\leq n\in\NN}$ in \(\ncatX\), every \(N\in\NN\) and every object \(y\) in \(\ncatY\), we also have an isomorphism

in \(\Setss{\V}\), naturally in \(s\) and \(y\). Let \(x\) be a normed colimit of \(s\) in \(\ncatX\). For every object \(y\) in \(\ncatY\), the diagram

commutes. Since the cocone \((\Nat(s|_{N},\Delta Gy)\to\ncatX(x,Gy))_{N\in\NN}\) is a colimit in \(\Setss{\V}\), so is the cocone \((\Nat(Fs|_{N},\Delta y)\to\ncatY(Fx,y))_{N\in\NN}\). This proves that \(Fx\) is a normed colimit of \(Fs\) in \(\ncatY\).

Recall from [26] that, for \(\V\)-normed functors \(F \colon\ncatA\to\ncatX\) and \(\phi \colon\ncatA^{\op}\to\nSetss{\V}\), a \(\phi\)-weighted colimit of \(F\) is given by an object \(x\) in \(\ncatX\) together with \(\V\)-normed isomorphisms

naturally in \(y\). In this context it is convenient to use the language of \((\nSetss{\V})\)-valued distributors \(\ncatX\lmodto\ncatY\) which, just like the $\V$-valued distributors in Section 4,  are defined as \(\V\)-normed functors \(\ncatX^{\mathrm{op}}\otimes\ncatY\to\nSetss{\V}\). Every \(\V\)-normed functor \(F \colon\ncatX\to\ncatY\) induces a pair of distributors

\begin{align*}
F_{*}& \colon\ncatX\lmodto\ncatY,\quad F_{*}(x,y)=\ncatY(Fx,y),\\
F^{*}& \colon\ncatY\lmodto\ncatX,\quad F^{*}(y,x)=\ncatY(y,Fx).
\end{align*}

In particular, interpreting an object \(x\) in \(\ncatX\) as a \(\V\)-normed functor \(x \colon\ncatOne\to\ncatX\), one obtains

x_*,  x_*=(x,-)
and  x^*,  x^*=(-,x).

For distributors \(\phi \colon\ncatX\lmodto\ncatA\) and \(\psi \colon\ncatA\lmodto\ncatX\) and objects \(x\) in \(\ncatX\) and \(y\) in \(\ncatY\), one considers

whenever this coend exists (see [33, 32]). This is certainly the case when \(\ncatA\) is small (since $\nSetss{\V}$, being equivalent to $\SET$, is small-cocomplete), and then the formula above defines the composite distributor \(\psi\cdot\phi \colon\ncatX\lmodto\ncatY\). Another important case is \(\phi=F_{*}\) for a \(\V\)-normed functor \(F \colon\ncatX\to\ncatA\), since then one simply has

for all objects \(x\) in \(\ncatX\) and \(y\) in \(\ncatY\). Hence, the presheaf \(\ncatX(F-,y)\) can be written as the composite \(\ncatX(F-,y)=y^{*}\cdot F_{*}\colon\ncatA\lmodto\ncatOne\). Moreover, for \(\ncatA\) small, any presheaf \(\phi \colon\ncatA\lmodto\ncatOne\) may be composed with \(F^{*}\colon\ncatX\lmodto\ncatA\) to yield \(\phi\cdot F^{*}\colon\ncatX\lmodto\ncatOne\), and \(-\cdot F^{*}\dashv -\cdot F_{*}\) is an adjunction between \([\ncatA^{\op},\nSetss{\V}]\) and the higher-universe \(\V\)-normed category \([\ncatX^{\op},\nSetss{\V}]\). Therefore we have natural isomorphisms

It follows that  \(x\) is a \(\phi\)-weighted colimit of \(F\) if, and only if, \(x\) is a \((\phi\cdot F^{*})\)-weighted colimit of the identity functor \(\ncatX\to\ncatX\), and in that case we simply speak of a \((\phi\cdot F^{*})\)-weighted colimit in \(\ncatX\).

For a $\V$-normed category $\ncatX$ we consider the Yoneda embedding

_→[^,],  x↦x^*=(-,x),

whose codomain (irrespective of potentially having to be formed in a higher universe) is $\V$-normed again. Actually, $\yoneda_{\ncatX}$ preserves norms since, for every $f:x\to y$ in $\ncatX$, one has
$$|\yoneda_{\ncatX}f|=|\ncatX(\text{-},f)|=\bigwedge_{z\in\ncatX}|\ncatX(z,f)|=\bigwedge_{h:z\to x}[|h|,|f\cdot h|]=|f|.$$

We now let
\(\Psh\ncatX\) denote the full $\V$-normed subcategory of \([\ncatX^{\op},\nSetss{\V}]\) defined by all accessible presheaves (see [28]). By definition, these are the small-weighted colimits of representables. Viewing a presheaf \(\phi \colon\ncatX^{\op}\to\nSetss{\V}\) as a distributor \(\phi \colon\ncatX\lmodto\ncatOne\), this means that \(\phi\) belongs to \(\Psh\ncatX\) if, and only if, there is a fully faithful $\V$-normed functor \(F \colon\ncatA\to\ncatX\) with \(\ncatA\) small and a distributor \(\psi \colon\ncatA\lmodto\ncatOne\) with \(\phi=\psi\cdot F^{*}\). Of course, for \(\ncatX\) small, one has
\(\Psh\ncatX= [\ncatX^{\op},\nSetss{\V}]\).

For every \(\V\)-normed category \(\ncatX\), the $\V$-normed category  \(\Psh\ncatX\) is Cauchy cocomplete. Moreover, for every \(\V\)-normed functor \(F \colon\ncatX\to\ncatY\), the \(\V\)-normed functor \(\Psh F \colon \Psh\ncatX\to\Psh\ncatY\) preserves normed colimits of sequences.

Let \(\sigma\colon\NN\to \Psh\ncatX\) be a Cauchy sequence in \(\Psh\ncatX\). Since \(\NN\) is a (countable) set, there is a small full $\V$-normed subcategory \(\ncatA\) of \(\ncatX\) (with inclusion functor \(I \colon\ncatA\to\ncatX\)) such that \(\sigma\) factors as

[column sep=large]

and \(\sigma_{0}\colon\NN\to\Psh\ncatA\) is Cauchy in \(\Psh\ncatA\). By Theorem <ref>, \(\Psh\ncatA\) is Cauchy cocomplete; we let \(Q\) be a normed colimit of \(\sigma_{0}\) in \(\Psh\ncatA\). By Proposition <ref>, \(\Psh I(Q)\) is a normed colimit of \(\sigma\) in \(\Psh\ncatX\). Finally, since \(\Psh F\cdot\Psh I\) is left adjoint, \(\Psh F(\Psh I(Q))\) is a normed colimit of \(\Psh F\cdot\sigma\).

For a Cauchy sequence \(s\) in \(\ncatX\), we let \(\phi_{s}\) denote the normed colimit of \(\yoneda_{\ncatX}\cdot s\) in \(\Psh\ncatX\). Then, for every object \(y\) in \(\ncatX\), the cocone

⟶(ϕ_s,_y), N∈,

is a colimit in \(\Setss{\V}\).

Let \(s\) be a Cauchy sequence in \(\ncatX\). Then \(\ncatX\) has a normed colimit of \(s\) if, and only if, \(\ncatX\) has a \(\phi_{s}\)-weighted colimit.

Assume first that \(x\) is a normed colimit of \(s\). Then, for every object $y\in\ncatX$,  the cocone

(s|_N,Δy)⟶(x,y), N∈,

is a colimit in \(\Setss{\V}\). Therefore we obtain an isomorphism

\begin{equation}\label{d:eq:1}
\ncatX(x,y)\to \Nat(\phi_{s},\yoneda_{\ncatX}y),
\end{equation}

naturally in \(y\). This tells us that \(x\) is also a \(\phi_{s}\)-weighted colimit.

Conversely, if \(x\) is a \(\phi_{s}\)-weighted colimit so that there is a natural isomorphism (<ref>), then also

(s|_N,Δy)→(x,y), N∈
is a colimit cocone in \(\Setss{\V}\), and therefore \(x\) is a normed colimit of \(s\) in \(\ncatX\).

We conclude that normed colimits of Cauchy sequences are equivalently described as certain weighted colimits. Below we explain that it suffices to consider “countable diagram shapes”. As usual, we call a ($\V$-normed) category \(\ncatX\) countable whenever the class of arrows of \(\ncatX\) is actually a countable set.

Let \(\sigma \colon\NN\to\ncatX\) be a Cauchy sequence in a $\V$-normed category \(\ncatX\), and let \(\phi \colon\ncatX\lmodto\ncatOne\) be a colimit of \(\yoneda_{\ncatX}\cdot\sigma\) in \(\Psh\ncatX\).

* Consider the $\V$-normed subcategory \(\ncatA\) of \(\ncatX\) generated by the image of \(\sigma\), with inclusion functor \(I \colon\ncatA\to\ncatX\). By construction, \(\ncatA\) is countable. Moreover, with \(\sigma_{0}\colon\NN\to\ncatA\) denoting the sequence in \(\ncatA\) with \(I\cdot\sigma_o=\sigma\), also \(\sigma_0\) is a Cauchy sequence in \(\ncatA\). Letting \(\phi_{0}\colon\ncatA\lmodto\ncatOne\) be the normed colimit of \(\yoneda_{\ncatA}\cdot \sigma_0\) in \(\ncatA\), we have  \(\phi=\phi_0\cdot I^{*}\). Therefore, \(\ncatX\) has a \(\phi\)-weighted colimit if, and only if, \(\ncatX\) has a \(\phi_{0}\)-weighted colimit of \(I \colon\ncatA\to\ncatX\).

* Consider \(\NN\) just as an ordinary category (given by its order). By Proposition <ref>, $\NN$ may be equipped with the initial normed structure

with respect to the ordinary functor \(\sigma \colon\NN\to\ncatX\) and the given norm of $\ncatX$. Then, since $\sigma$ is Cauchy in $\ncatX$, the sequence

becomes Cauchy in \(\NN\) as well, and we can form the normed colimit \(\phi_0\) of \(\yoneda_{\NN}\) in \(\Psh\NN\). With $\phi$ defined as above, \(\ncatX\) has a \(\phi\)-weighted colimit if, and only if, \(\ncatX\) has a \(\phi_{0}\)-weighted colimit of \(I \colon\NN\to\ncatX\).

All told, we have the following characterisation of Cauchy cocompleteness.

Let \(\ncatX\) be a \(\V\)-normed category. Then the following assertions are equivalent.

$\mathrm{(i)}$ \(\ncatX\) is Cauchy cocomplete.

$\mathrm{(ii)}$   \(\ncatX\) has all weighted colimits of normed colimits of Cauchy sequences of representables.

$\mathrm{(iii)}$   \(\ncatX\) has all weighted colimits of diagrams \(F\colon\ncatA\to\ncatX\), \(\Phi \colon\ncatA\lmodto\ncatOne\), where \(\ncatA\) is countable and \(\Phi\) is a normed colimit of a Cauchy sequence of representables in \(\Psh\ncatA\).

\(\ncatX\) has all weighted colimits of diagrams \(F\colon\ncatA\to\ncatX\), \(\Phi \colon\ncatA\lmodto\ncatOne\), where the underlying category of \(\ncatA\) is \(\NN\) and \(\Phi\) is a normed colimit of a Cauchy sequence of representables in \(\Psh\ncatA\).

§ CAUCHY COCOMPLETION OF $\V$-NORMED CATEGORIES

In the previous section we have shown that normed colimits of Cauchy sequences can be equivalently described as weighted colimits, for a certain choice of weights – under the assumption that the quantale $\V$ satisfies condition (A) or (B), which we also make in this section. Following the nomenclature of [28, 5], for every small \(\V\)-normed category \(\ncatA\) we consider the class \(\Phi[\ncatA]\) of presheaves \(\phi\in \Psh\ncatA\) that are normed colimits of Cauchy sequences of representables, and put

Φ=∑_ smallΦ[].

A \(\V\)-normed category \(\ncatX\) is called \(\Phi\)-cocomplete whenever, for all \(\V\)-normed functors \(F \colon\ncatA\to\ncatX\) and \(\phi \colon\ncatA^{\op}\to\nSetss{\V}\) with \(\ncatA\) small and \(\phi\in\Phi[\ncatA]\), the \(\phi\)-weighted colimit of \(F\) in \(\ncatX\) exists. Moreover, a \(\V\)-normed functor is called \(\Phi\)-cocontinuous whenever it preserves all weighted colimits with weight in \(\Phi\).

By Corollary <ref>, a normed category \(\ncatX\) is Cauchy cocomplete if, and only if, \(\ncatX\) is \(\Phi\)-cocomplete. Furthermore, since the diagram

commutes (up to isomorphism) for every \(\V\)-normed functor \(F \colon\ncatX\to\ncatY\), using the notation of Proposition <ref> and writing ncolim instead of just colim to stress the normedness of a colimit, for every  Cauchy sequence \(s \colon\NN\to\ncatX \) we have

Therefore, \(F\) preserves normed colimits of Cauchy sequences if, and only if, \(F\) is \(\Phi\)-cocontinuous. We define

to be the 2-category of \(\Phi\)-cocomplete small \(\V\)-normed categories, \(\Phi\)-cocontinuous \(\V\)-normed functors, and their natural transformations, and write

for its higher universe counterpart. For every \(\V\)-normed category \(\ncatX\), we let \(\Phi(\ncatX)\) denote the smallest replete full \(\V\)-normed subcategory of \(\Psh\ncatX\) containing \(\ncatX\) and closed under \(\Phi\)-colimits. Then the Yoneda functor of \(\ncatX\) restricts to

and we have that  \(\Phi(\ncatX)\) is \(\Phi\)-cocomplete and the inclusion functor  \(\Phi(\ncatX)\to\Psh\ncatX\) is \(\Phi\)-cocontinuous.
We now show that \(\Phi(\ncatX)\) serves as a correct-size Cauchy cocompletion of the $\V$-normed category $\ncatX$, both for small and large $\ncatX$.

For each small normed category \(\ncatX\), the presheaf category \(\Psh\ncatX\) is small.

Φ_0=∑_ countableΦ[],

hence \(\Phi_0\) is small. By Corollary <ref>, \(\Phi(\ncatX)=\Phi_{0}(\ncatX)\) for every normed category \(\ncatX\). By <cit.>, \(\Phi_0(\ncatX)\) is small.

For every \(\V\)-normed category \(\ncatX\) and every Cauchy cocomplete \(\V\)-normed category \(\ncatY\), the composition with \(\yoneda_{\ncatX}\colon\ncatX\to\Phi(\ncatX)\) defines an equivalence

that is, \(\Phi(-)\) provides a left biadjoint to the inclusion 2-functor \(\Cocts{\Phi}\to\CATss{\V}\). This equivalence restricts to

when \(\ncatX\) and \(\ncatY\) are small.

§ THE BANACH FIXED POINT THEOREM FOR NORMED CATEGORIES

At first, letting the quantale $\V$ remain general, but then specializing it to $\V=\mathcal R_+$, we consider a $\V$-normed category $\mathbb X$ and a $\V$-normed endofunctor $F$ of $\mathbb X$ and give sufficient conditions guaranteeing the existence of an object $x$ with $x\cong Fx$, in such a way that they reproduce Banach's Fixed Point Theorem when $\mathbb X=\mathrm iX$ for a metric space $X$. The following terminology makes precise what $x\cong Fx$ may mean in the $\V$-normed context.

For a $\V$-normed functor $F:\mathbb X\to\mathbb X$, we say that an object $x$ in $\mathbb X$ is

* a forward fixed point of $F$ if there is an isomorphism $f:x\to Fx$ of (the ordinary category) $\mathbb X$ with $\kk\leq |f|$;

* a backward fixed point of $F$ if there is an isomorphism $f:Fx\to x$ of (the ordinary category) $\mathbb X$ with $\kk\leq |f|$;