This is part 3 in a series. See the previous part about internal languages for indexed monoidal categories upon which this part heavily depends.

In category theory, the hom-sets between two objects can often be equipped with some extra structure which is respected by identities and composition. For example, the set of group homomorphisms between two abelian groups is itself an abelian group by defining the operations pointwise. Similarly, the set of monotonic functions between two partially ordered sets (posets) is a poset again by defining the ordering pointwise. Linear functions between vector spaces form a vector space. The set of functors between small categories is a small category. Of course, the structure on the hom-sets can be different than the objects. Trivially, with the earlier examples a vector space is an abelian group, so we could say that linear functions form an abelian group instead of a vector space. Likewise groups are monoids. Less trivially, the set of relations between two sets is a partially ordered set via inclusion. There are many cases where instead of hom-sets we have hom-objects that aren’t naturally thought of as sets. For example, we can have hom-objects be non-negative (extended) real numbers from which the category laws become the laws of a generalized metric space. We can identify posets with categories who hom-objects are elements of a two element set or, even better, a two element poset with one element less than or equal to the other.

This general process is called enriching a category in some other
category which is almost always called |\V| in the generic
case. We then talk about having |\V|-categories and |\V|-functors, etc. In a specific case, it will be something
like |\mathbf{Ab}|-categories for an |\mathbf{Ab}|-enriched category, where |\mathbf{Ab}| is the category of abelian groups.
Unsurprisingly, not just *any* category will do for |\V|. However, it turns out very little structure is needed to define
a notion of |\V|-category, |\V|-functor, |\V|-natural transformation, and |\V|-profunctor. The
usual “baseline” is that |\V| is a monoidal category. As mentioned in the previous
post, paraphrasing Bénabou, notions of “families of objects/arrows” are ubiquitous and fundamental in category theory. It is useful for
our purposes to make this structure explicit. For very little cost, this will also provide a vastly more general notion that will readily
capture enriched categories, indexed categories, and categories that are simultaneously
indexed and enriched, of which internal categories are an example. The tool for this is
a (Grothendieck) fibration aka a fibered category or the mostly equivalent concept
of an indexed category.^{1}

To that end, instead of just a monoidal category, we’ll be using indexed monoidal categories. Typically, to get an experience as much like ordinary category theory as possible, additional structure is assumed on |\V|. In particular, it is assumed to be an (indexed) cosmos which means that it is an indexed symmetric monoidally closed category with indexed coproducts preserved by |\otimes| and indexed products and fiberwise finite limits and colimits (preserved by the indexed structure). This is quite a lot more structure which I’ll introduce in later parts. In this part, I’ll make no assumptions beyond having an indexed monoidal category.

The purpose of the machinery of the previous posts is to make this section seem boring and pedestrian. Other than being a little more explicit and formal, for most of the following concepts it will look like we’re restating the usual definitions of categories, functors, and natural transformations. The main exception is profunctors which will be presented in a quite different manner, though still in a manner that is easy to connect to the usual presentation. (We will see how to recover the usual presentation in later parts.)

While I’ll start by being rather explicit about indexes and such, I will start to suppress that detail over time as most of it is inferrable. One big exception is that right from the start I’ll omit the explicit dependence of primitive terms on indexes. For example, while I’ll write |\mathsf F(\mathsf{id}) = \mathsf{id}| for the first functor law, what the syntax of the previous posts says I should be writing is |\mathsf F(s, s; \mathsf{id}(s)) = \mathsf{id}(s)|.

To start, I want to introduce two different notions of |\V|-category, small |\V|-categories and large |\V|-categories, and talk about what this distinction actually means. I will proceed afterwards with the “large” notions, e.g. |\V|-functors between large |\V|-categories, as the small case will be an easy special case.

The **theory of a small |\V|-category** consists of:

- an index type |\O|,
- a linear type |s, t : \O \vdash \A(t, s)|,
- a linear term |s : \O; \vdash \mathsf{id} : \A(s, s)|, and
- a linear term |s, u, t : \O; g : \A(t, u), f : \A(u, s) \vdash g \circ f : \A(t, s)|

satisfying $$\begin{gather} s, t : \O; f : \A(t, s) \vdash \mathsf{id} \circ f = f = f \circ \mathsf{id} : \A(t, s)\qquad \text{and} \\ \\ s, u, v, t : \O; h : \A(t, v), g : \A(v, u), f : \A(u, s) \vdash h \circ (g \circ f) = (h \circ g) \circ f : \A(t, s) \end{gather}$$

In the notation of the previous posts, I’m saying |\O : \mathsf{IxType}|, |\A : (\O, \O) \to \mathsf{Type}|, |\mathsf{id} : (s : \O;) \to \A(s, s)|, and |\circ : (s, u, t : \O; \A(t, u), \A(u, s)) \to \A(t, s)| are primitives added to the signature of the theory. I’ll continue to use the earlier, more pointwise presentation above to describe the signature.

A **small |\V|-category** for an |\mathbf S|-indexed monoidal category |\V| is then an interpretation of
this theory. That is, an object |O| of |\mathbf S| as the interpretation of |\O|, and an
object |A| of |\V^{O\times O}| as the interpretation of |\A|. The interpretation of |\mathsf{id}|
is an arrow |I_O \to \Delta_O^* A| of |\V^O|, where |\Delta_O : O \to O\times O| is the diagonal
arrow |\langle id, id\rangle| in |\mathbf S|. The interpretation of |\circ| is an
arrow |\pi_{23}^* A \otimes \pi_{12}^* A \to \pi_{13}^* A| of |\V^{O\times O \times O}|
where |\pi_{ij} : X_1 \times X_2 \times X_3 \to X_i \times X_j| are the appropriate projections.

Since we can prove in the internal language that the choice of |()| for |\O|, |s, t : () \vdash I|
for |\A|, |s, t : (); x : I \vdash x : I| for |\mathsf{id}|,
and |s, u, t : (); f : I, g: I \vdash \mathsf{match}\ f\ \mathsf{as}\ *\ \mathsf{in}\ g : I| for |\circ|
satisfies the laws of the theory of a small |\V|-category, we know we have a |\V|-category which I’ll call |\mathbb I|
for any |\V|^{2}.

For |\V = \mathcal Fam(\mathbf V)|, |O| is a set of objects. |A| is an |(O\times O)|-indexed family of objects of |\mathbf V| which we can write |\{A(t,s)\}_{s,t\in O}|. The interpretation of |\mathsf{id}| is an |O|-indexed family of arrows of |\mathbf V|, |\{ id_s : I_s \to A(s, s) \}_{s\in O}|. Finally, the interpretation of |\circ| is a family of arrows of |\mathbf V|, |\{ \circ_{s,u,t} : A(t, u)\otimes A(u, s) \to A(t, s) \}_{s,u,t \in O}|. This is exactly the data of a (small) |\mathbf V|-enriched category. One example is when |\mathbf V = \mathbf{Cat}| which produces (strict) |2|-categories.

For |\V = \mathcal Self(\mathbf S)|, |O| is an object of |\mathbf S|. |A| is an arrow of |\mathbf S| into |O\times O|, i.e. an object of |\mathbf S/O\times O|. I’ll write the object part of this as |A| as well, i.e. |A : A \to O\times O|. The idea is that the two projections are the target and source of the arrow. The interpretation of |\mathsf{id}| is an arrow |ids : O \to A| such that |A \circ ids = \Delta_O|, i.e. the arrow produced by |ids| should have the same target and source. Finally, the interpretation of |\circ| is an arrow |c| from the pullback of |\pi_2 \circ A| and |\pi_1 \circ A| to |A|. The source of this arrow is the object of composable pairs of arrows. We further require |c| to produce an arrow with the appropriate target and source of the composable pair. This is exactly the data for a category internal to |\mathbf S|. An interesting case for contrast with the previous paragraph is that a category internal to |\mathbf{Cat}| is a double category.

For |\V = \mathcal Const(\mathbf V)|, the above data is exactly the data of a monoid object in |\mathbf V|. This is a formal illustration that a (|\mathbf V|-enriched) category is just an “indexed monoid”. Indeed, |\mathcal Const(\mathbf V)|-functors will be monoid homomorphisms and |\mathcal Const(\mathbf V)|-profunctors will be double-sided monoid actions. In particular, when |\mathbf V = \mathbf{Ab}|, we get rings, ring homomorphisms, and bimodules of rings. The intuitions here are the guiding ones for the construction we’re realizing.

One aspect of working in a not-necessarily-symmetric (indexed) monoidal category is the choice of the standard order of composition or diagrammatic order is not so trivial since it is not possible to even state what it means for them to be the equivalent. To be clear, this definition isn’t really taking a stance on the issue. We can interpret |\A(t, s)| as the type of arrows |s \to t| and then |\circ| will be the standard order of composition, or as the type of arrows |t \to s| and then |\circ| will be the diagrammatic order. In fact, there’s nothing in this definition that stops us from having |\A(t, s)| being the type of arrows |t \to s| while still having |\circ| be standard order composition as usual. The issue comes up only once we consider |\V|-profunctors as we will see.

A **large |\V|-category** is a model of a theory of the following form. There is

- a collection of index types |\O_x|,
- for each pair of index types |\O_x| and |\O_y|, a linear type |s : \O_x, t : \O_y \vdash \A_{yx}(t, s)|,
- for each index type |\O_x|, a linear term |s : \O_x; \vdash \mathsf{id}_x : \A_{xx}(s, s)|, and
- for each triple of index types, |\O_x|, |\O_y|, and |\O_z|, a linear term |s: \O_x, u : \O_y, t : \O_z; g : \A_{zy}(t, u), f : \A_{yx}(u, s) \vdash g \circ_{xyz} f : \A_{zx}(t, s)|

satisfying the same laws as small |\V|-categories, just with some extra subscripts. Clearly, a small |\V|-category is just a large |\V|-category where the collection of index types consists of just a single index type.

The typical way of describing the difference between small and large (|\V|-)categories would be to say something like: “By
having a collection of index types in a large |\V|-category, we can have a proper class of them. In a
small |\V|-category, the index type of objects is interpreted as an object in a category, and a proper class can’t
be an object of a category^{3}.” However, for us, there’s
a more directly relevant distinction. Namely, while we had a single theory of small |\V|-categories, there is no
single theory of large |\V|-categories. Different large |\V|-categories correspond to models of (potentially) different theories.
In other words, the notion of a small |\V|-category is able to be captured by our notion of theory but not the concept
of a large |\V|-category. This extends to |\V|-functors, |\V|-natural transformations,
and |\V|-profunctors. In the small case, we can define a single theory which captures each of these concepts,
but that isn’t possible in the large case. In general, notions of “large” and “small” are about what we can internalize
within the relevant object language, usually a set theory. Arguably, the only reason we speak of “size” and of proper
classes being “large” is that the Axiom of Specification
outright states that any subclass of a set is a set, so proper classes in **ZFC** can’t be subsets of any set.
As I’ve mentioned elsewhere, you can definitely have set theories with proper classes that are
contained in even finite sets, so the issue isn’t one of “bigness”.

The above discussion also explains the hand-wavy word “collection”. The collection is a collection in the meta-language in
which we’re discussing/formalizing the notion of theory. When working *within* the theory of a particular
large |\V|-category, all the various types and terms are just available ab initio and are independent. There is
no notion of “collection of types” within the theory and nothing indicating that some types are part of a “collection” with others.

Another perspective on this distinction between large and small |\V|-categories is that small |\V|-categories have
a *family* of arrows, identities, and compositions with respect to the notion of “family” represented by our internal
language. If we hadn’t wanted to bother with formulating the internal language of an *indexed* monoidal category, we
could have still defined the notion of |\V|-category with respect to the internal language of a (non-indexed) monoidal
category. It’s just that all such |\V|-categories (except for monoid objects) would have to be large |\V|-categories. That is,
the indexing and notion of “family” would be at a meta-level. Since most of the |\V|-categories of interest will be
large (though, generally a special case called a |\V|-fibration which reins in the size a bit), it may seem that there
was no real benefit to the indexing stuff. Where it comes in, or rather where small |\V|-categories come in, is that
our notion of (co)complete means “has all (co)limits of *small* diagrams” and small diagrams are |\V|-functors from
small |\V|-categories.^{4} There are several other places, e.g. the notion
of presheaf, where we implicitly depend on what we mean by “small |\V|-category”. So while we won’t usually be
focused on small |\V|-categories, which |\V|-categories are small impacts the structure of the whole theory.

The formulation of |\V|-functors is straightforward. As mentioned before, I’ll only present the “large” version.

Formally, we can’t formulate a theory of just a |\V|-functor, but rather we need to formulate a theory of “a pair of |\V|-categories and a |\V|-functor between them”.

A **|\V|-functor between (large) |\V|-categories |\mathcal C| and |\mathcal D|** is a model of
a theory consisting of a theory of a large |\V|-category, of which |\mathcal C| is a model, and a theory
of a large |\V|-category which I’ll write with primes, of which |\mathcal D| is a model, and model of
the following additional data:

- for each index type |\O_x|, an index type |\O’_{F_x}| and an index term, |s : \O_x \vdash \mathsf F_x(s) : \O’_{F_x}|, and
- for each pair of index types, |\O_x| and |\O_y|, a linear term |s : \O_x, t : \O_y; f : \A_{yx}(t, s) \vdash \mathsf F_{yx}(f) : \A’_{F_yF_x}(\mathsf F_y(t), \mathsf F_x(s))|

satisfying $$\begin{gather} s : \O_x; \vdash \mathsf F_{xx}(\mathsf{id}_x) = \mathsf{id}'_{F_x}: \A'_{F_xF_x}(F_x(s), F_x(s))\qquad\text{and} \\ s : \O_x, u : \O_y, t : \O_z; g : \A_{zy}(t, u), f : \A_{yx}(u, s) \vdash \mathsf F_{zx}(g \circ_{xyz} f) = F_{zy}(g) \circ'_{F_xF_yF_z} F_{yx}(f) : \A'_{F_zF_x}(F_z(t), F_x(s)) \end{gather}$$

The assignment of |\O’_{F_x}| for |\O_x| is, again, purely metatheoretical. From within the theory, all we know is that we happen to have some index types named |\O_x| and |\O’_{F_x}| and some data relating them. The fact that there is some kind of mapping of one to the other is not part of the data.

Next, I’ll define **|\V|-natural transformations** . As before, what we’re really doing is defining |\V|-natural transformations
as a model of a theory of “a pair of (large) |\V|-categories with a pair of |\V|-functors between them and a |\V|-natural transformation
between those”. As before, I’ll use primes to indicate the types and terms of the second of each pair of subtheories. Unlike
before, I’ll only mention what is added which is:

- for each index type |\O_x|, a linear term |s : \O_x; \vdash \tau_x : \A’_{F’_xF_x}(\mathsf F’_x(s), \mathsf F_x(s))|

satisfying $$\begin{gather} s : \O_x, t : \O_y; f : \A_{yx}(t, s) \vdash \mathsf F'_{yx}(f) \circ'_{F_xF'_xF'_y} \tau_x = \tau_y \circ'_{F_xF_yF'_y} \mathsf F_{yx}(f) : \A'_{F'_yF_x}(\mathsf F'_y(t), \mathsf F_x(s)) \end{gather}$$

In practice, I’ll suppress the subscripts on all but index types as the rest are inferrable. This makes the above equation the much more readable $$\begin{gather} s : \O_x, t : \O_y; f : \A(t, s) \vdash \mathsf F'(f) \circ' \tau = \tau \circ' \mathsf F(f) : \A'(\mathsf F'(t), \mathsf F(s)) \end{gather}$$

Here’s where we need to depart from the usual story. In the usual story, a |\mathbf V|-enriched profunctor |P : \mathcal C \proarrow \mathcal D| is a |\mathbf V|-enriched functor |P : \mathcal C\otimes\mathcal D^{op}\to\mathbf V| (or, often, the opposite convention is used |P : \mathcal C^{op}\otimes\mathcal D \to \mathbf V|). There are many problems with this definition in our context.

- Without symmetry, we have no definition of opposite category.
- Without symmetry, the tensor product of |\mathbf V|-enriched categories doesn’t make sense.
- |\mathbf V| is not itself a |\mathbf V|-enriched category, so it doesn’t make sense to talk about |\mathbf V|-enriched functors into it.
- Even if it was, we’d need some way of converting between arrows of |\mathbf V| as a category and arrows of |\mathbf V| as a |\mathbf V|-enriched category.
- The equation |P(g \circ f, h \circ k) = P(g, k) \circ P(f, h)| requires symmetry. (This is arguably 2 again.)

All of these problems are solved when |\mathbf V| is a symmetric monoidally closed category.

Alternatively, we can reformulate the notion of a |\V|-profunctor so that it works in our context
and is equivalent to the usual one when it makes sense.^{5} To this end, at a low level a |\mathbf V|-enriched profunctor is a family of
arrows $$\begin{gather}P : \mathcal C(t, s)\otimes\mathcal D(s', t') \to [P(s, s'), P(t, t')]\end{gather}$$ which
satisfies $$\begin{gather}P(g \circ f, h \circ k)(p) = P(g, k)(P(f, h)(p))\end{gather}$$ in the internal language of a symmetric
monoidally closed category among other laws. We can uncurry |P| to eliminate the need for closure,
getting $$\begin{gather}P : \mathcal C(t, s)\otimes \mathcal D(s', t')\otimes P(s, s') \to P(t, t')\end{gather}$$
satisfying $$\begin{gather}P(g \circ f, h \circ k, p) = P(g, k, P(f, h, p))\end{gather}$$ We see that we’re always going to need
to permute |f| and |h| past |k| unless we move the third argument to the second producing the
nice $$\begin{gather}P : \mathcal C(t, s)\otimes P(s, s') \otimes \mathcal D(s', t') \to P(t, t')\end{gather}$$ and the
law $$\begin{gather}P(g \circ f, p, h \circ k) = P(g, P(f, p, h), k)\end{gather}$$ which no longer requires symmetry. This is
also where the order of the arguments of |\circ| drives the order of the arguments of |\V|-profunctors.

A **|\V|-profunctor**, |P : \mathcal C \proarrow \mathcal D|, is a model of the theory (containing
subtheories for |\mathcal C| and |\mathcal D| etc. as in the |\V|-functor case) having:

- for each pair of index types |\O_x| and |\O’_{x’}|, a linear type |s : \O_x, t : \O’_{x’} \vdash \mathsf P_{x’x}(t, s)|, and
- for each quadruple of index types |\O_x|, |\O_y|, |\O’_{x’}|, and |\O’_{y’}|, a linear term |s : \O_x, s’ : \O’_{x’}, t : \O_y, t’ : \O’_{y’}; f : \A_{yx}(t, s), p : \mathsf P_{xx’}(s, s’), h : \A’_{x’y’}(s’, t’) \vdash \mathsf P_{yxx’y’}(f, p, h) : \mathsf P_{yy’}(t, t’)|

satisfying $$\begin{align} & s : \O_x, s' : \O'_{x'}; p : \mathsf P(s, s') \vdash \mathsf P(\mathsf{id}, p, \mathsf{id}') = p : \mathsf P(s, s') \\ \\ & s : \O_x, s' : \O'_{x'}, u : \O_y, u' : \O'_{y'}, t : \O_z, t' : \O'_{z'}; \\ & g : \A(t, u), f : \A(u, s), p : \mathsf P(s, s'), h : \A'(s', u'), k : \A'(u', t') \\ \vdash\ & \mathsf P(g \circ f, p, h \circ' k) = \mathsf P(g, \mathsf P(f, p, h), k) : \mathsf P(t, t') \end{align}$$

This can also be equivalently presented as a pair of a left and a right action satisfying bimodule laws. We’ll make the following definitions |\mathsf P_l(f, p) = \mathsf P(f, p, \mathsf {id})| and |\mathsf P_r(p, h) = \mathsf P(\mathsf{id}, p ,h)|.

A **|\V|-presheaf** on |\mathcal C| is a |\V|-profunctor |P : \mathbb I \proarrow \mathcal C|. Similarly,
a **|\V|-copresheaf** on |\mathcal C| is a |\V|-profunctor |P : \mathcal C \proarrow \mathbb I|.

Of course, we have the fact that the term $$\begin{gather}
s : \O_x, t : \O_y, s' : \O_z, t' : \O_w; h : \A(t, s), g : \A(s, s'), f : \A(s', t')
\vdash h \circ g \circ f : \A(t, t')
\end{gather}$$ witnesses the interpretation of |\A| as a |\V|-profunctor |\mathcal C \proarrow \mathcal C|
for any |\V|-category, |\mathcal C|, which we’ll call the **hom |\V|-profunctor**. More generally, given
a |\V|-profunctor |P : \mathcal C \proarrow \mathcal D|, and |\V|-functors |F : \mathcal C’ \to \mathcal C|
and |F’ : \mathcal D’ \to \mathcal D|, we have the |\V|-profunctor |P(F, F’) : \mathcal C’ \proarrow \mathcal D’|
defined as $$\begin{gather}
s : \O_x, s' : \O'_{x'}, t : \O_y, t' : \O'_{y'}; f : \A(t, s), p : \mathsf P(\mathsf F(s), \mathsf F'(s')), f' : \A'(s', t')
\vdash \mathsf P(\mathsf F(f), p, \mathsf F'(f')) : \mathsf P(\mathsf F(t), \mathsf F'(t'))
\end{gather}$$ In particular, we have the **representable |\V|-profunctors** when |P| is the hom |\V|-profunctor and
either |F| or |F’| is the identity |\V|-functor, e.g. |\mathcal C(Id, F)| or |\mathcal C(F, Id)|.

There’s a natural notion of morphism of |\V|-profunctors which we could derive either via passing the notion of natural transformation of the bifunctorial view through the same reformulations as above, or by generalizing the notion of a bimodule homomorphism. This would produce a notion like: a |\V|-natural transformation from |\alpha : P \to Q| is a |\alpha : P(t, s) \to Q(t, s)| satisfying |\alpha(P(f, p, h)) = Q(f, \alpha(p), h)|. While there’s nothing wrong with this definition, it doesn’t quite meet our needs. One way to see this is that it would be nice to have a bicategory whose |0|-cells were |\V|-categories, |1|-cells |\V|-profunctors, and |2|-cells |\V|-natural transformations as above. The problem there isn’t the |\V|-natural transformations but the |1|-cells. In particular, we don’t have composition of |\V|-profunctors. In the analogy with bimodules, we don’t have tensor products so we can’t reduce multilinear maps to linear maps; therefore, linear maps don’t suffice, and we really want a notion of multilinear maps.

So, instead of a bicategory what we’ll have is a virtual bicategory (or, more generally, a virtual double category).
A virtual bicategory is to a bicategory what a multicategory is to a monoidal category, i.e. multicategories are “virtual monoidal
categories”. The only difference between a virtual bicategory and a multicategory is that instead of our multimorphisms having
arbitrary lists of objects as their sources, our “objects” (|1|-cells) themselves have sources and targets (|0|-cells) and our
multimorphisms (|2|-cells) have *composable sequences* of |1|-cells as their sources.

A **|\V|-multimorphism** from a composable sequence of |\V|-profunctors |P_1, \dots, P_n| to
the |\V|-profunctor |Q| is a model of the theory consisting of the various necessary subtheories and:

- a linear term, |s_0 : \O_{x_0}^0, \dots, s_n : \O_{x_n}^n; p_1 : \mathsf P_{x_0x_1}^1(s_0, s_1), \dots, p_n : \mathsf P_{x_{n-1}x_n}^n(s_{n-1}, s_n) \vdash \tau_{x_0\cdots x_n}(p_1, \dots, p_n) : \mathsf Q_{x_0x_n}(s_0, s_n)|

satisfying $$\begin{align} & t, s_0 : \O^0, \dots, s_n : \O^n; f : \A^0(t, s_0), p_1 : \mathsf P^1(s_0, s_1), \dots, p_n : \mathsf P^n(s_{n-1}, s_n) \\ \vdash\ & \tau(\mathsf P_l^0(f, p_1), \dots, p_n) = \mathsf Q_l(f, \tau(p_1, \dots, p_n)) : \mathsf Q(t, s_n) \\ \\ & s_0 : \O^0, \dots, s_n, s : \O^n; p_1 : \mathsf P^1(s_0, s_1), \dots, p_n : \mathsf P^n(s_{n-1}, s_n), f : \A^n(s_n, s) \\ \vdash\ & \tau(p_1, \dots, \mathsf P_r^n(p_n, f)) = \mathsf Q_r(\tau(p_1, \dots, p_n), f) : \mathsf Q(s_0, s) \\ \\ & s_0 : \O^0, \dots, s_n : \O^n; \\ & p_1 : \mathsf P^1(s_0, s_1), \dots, p_i : \mathsf P^i(s_{i-1}, s_i), f : \A^i(s_i, s_{i+1}), p_{i+1} : \mathsf P^{i+1}(s_i, s_{i+1}), \dots, p_n : \mathsf P^n(s_{n-1}, s_n) \\ \vdash\ & \tau(p_1, \dots, \mathsf P_r^i(p_i, f), p_{i+1}, \dots, p_n) = \tau(p_1, \dots, p_i, \mathsf P_l^{i+1}(f, p_{i+1}) \dots, p_n) : \mathsf Q(s_0, s_n) \end{align}$$ except for the |n=0| case in which case the only law is $$\begin{gather} t, s : \O^0; f : \A^0(t, s) \vdash \mathsf Q_l(f, \tau()) = \mathsf Q_r(\tau(), f) : \mathsf Q(t, s) \end{gather}$$

The laws involving the action of |\mathsf Q| are called **external equivariance**, while the remaining law is called **internal
equivariance**. We’ll write |\V\mathbf{Prof}(P_1, \dots, P_n; Q)| for the set of |\V|-multimorphisms from the
composable sequence of |\V|-profunctors |P_1, \dots, P_n| to the |\V|-profunctor |Q|.

As with multilinear maps, we can characterize composition via a universal property. Write |Q_1\diamond\cdots\diamond Q_n|
for the **composite |\V|-profunctor** (when it exists) of the composable sequence |Q_1, \dots, Q_n|. We then have
for any pair of composable sequences |R_1, \dots, R_m| and |S_1, \dots, S_k| which compose with |Q_1, \dots, Q_n|,
$$\begin{gather}
\V\mathbf{Prof}(R_1,\dots, R_m, Q_1 \diamond \cdots \diamond Q_n, S_1, \dots, S_k; -)
\cong \V\mathbf{Prof}(R_1,\dots, R_m, Q_1, \dots, Q_n, S_1, \dots, S_k; -)
\end{gather}$$ where the forward direction is induced by precomposition with a |\V|-multimorphism
|Q_1, \dots, Q_n \to Q_1 \diamond \cdots \diamond Q_n|. A |\V|-multimorphism with this property
is called **opcartesian**. The |n=0| case is particularly important and, for a |\V|-category |\mathcal C|,
produces the **unit |\V|-profunctor**, |U_\mathcal C : \mathcal C \proarrow \mathcal C| as the composite
of the empty sequence. When we have all composites, |\V\mathbf{Prof}| becomes an actual bicategory rather
than a virtual bicategory. |\V\mathbf{Prof}| always has all units, namely the hom |\V|-profunctors. Much like we
can define the tensor product of modules by quotienting the tensor product of their underlying abelian groups by internal
equivariance, we will find that we can make composites when we have enough (well-behaved) colimits^{6}.

Related to composites, we can talk about left/right closure of |\V\mathbf{Prof}|. In this case we have the natural isomorphisms: $$\begin{gather} \V\mathbf{Prof}(Q_1,\dots, Q_n, R; S) \cong \V\mathbf{Prof}(Q_1, \dots, Q_n; R \triangleright S) \\ \V\mathbf{Prof}(R, Q_1, \dots, Q_n; S) \cong \V\mathbf{Prof}(Q_1, \dots, Q_n;S \triangleleft R) \end{gather}$$ Like composites, this merely characterizes these constructs; they need not exist in general. These will be important when we talk about Yoneda and (co)limits in |\V|-categories.

A |\V|-natural transformation |\alpha : F \to G : \mathcal C \to \mathcal D| is the same as |\alpha\in\V\mathbf{Prof}(;\mathcal D(G, F))|.

Just as an example, let’s prove a basic fact about categories for arbitrary |\V|-categories. This will use an informal style.

The fact will be that full and faithful functors reflect isomorphisms. Let’s go through the typical proof for the ordinary category case.

Suppose we have an natural transformation |\varphi : \mathcal D(FA, FB) \to \mathcal C(A, B)| natural in |A| and |B| such that |\varphi| is an inverse to |F|, i.e. the action of the functor |F| on arrows. If |Ff \circ Fg = id| and |Fg \circ Ff = id|, then by the naturality of |\varphi|, |\varphi(id) = \varphi(Ff \circ id \circ Fg) = f \circ \varphi(id) \circ g| and similarly with |f| and |g| switched. We now just need to show that |\varphi(id) = id| but |id = F(id)|, so |\varphi(id) = \varphi(F(id)) = id|. |\square|

Now in the internal language. We’ll start with the theory of a |\V|-functor, so we have |\O|, |\O’|, |\A|, |\A’|, and |\mathsf F|. While the previous paragraph talks about a natural transformation, we can readily see that it’s really a multimorphism. In our case, it is a |\V|-multimorphism |\varphi| from |\A’(\mathsf F, \mathsf F)| to |\A|. Before we do that though, we need to show that |\mathsf F| itself is a |\V|-multimorphism. This corresponds to the naturality of the action on arrows of |F| which we took for granted in the previous paragraph. This is quickly verified: the external equivariance equations are just the functor law for composites. The additional data we have is two linear terms |\mathsf f| and |\mathsf g| such that |\mathsf F(\mathsf f) \circ \mathsf F(\mathsf g) = \mathsf{id}| and |\mathsf F(\mathsf g) \circ \mathsf F(\mathsf f) = \mathsf{id}|. Also, |\varphi(\mathsf F(h)) = h|. The result follows through almost identically to the previous paragraph. |\varphi(\mathsf{id}) = \varphi(\mathsf F(\mathsf f) \circ \mathsf F(\mathsf g)) = \varphi(\mathsf F(\mathsf f) \circ \mathsf{id} \circ \mathsf F(\mathsf g))|, we apply external equivariance twice to get |\mathsf f \circ \varphi(\mathsf{id}) \circ \mathsf g|. The functor law for |\mathsf{id}| gives |\varphi(\mathsf{id}) = \varphi(\mathsf F(\mathsf{id})) = \mathsf{id}|. A quick glance verifies that all these equations use their free variables linearly as required. |\square|

As a warning, in the above |\mathsf f| and |\mathsf g| are not free variables but constants, i.e. primitive linear terms. Thus there is no issue with an equation like |\mathsf F(\mathsf f) \circ \mathsf F(\mathsf g) = \mathsf{id}| as both sides have no free variables.

This is a very basic result but, again, the payoff here is how boring and similar to the usual case this is. For contrast, the definition of an internal profunctor is given here. This definition is easier to connect to our notion of |\V|-presheaf, specifically a |\mathcal Self(\mathbf S)|-presheaf, than it is to the usual |\mathbf{Set}|-valued functor definition. While not hard, it would take me a bit of time to even formulate the above proposition, and a proof in terms of the explicit definitions would be hard to recognize as just the ordinary proof.

For fun, let’s figure out what the |\mathcal Const(\mathbf{Ab})| case of this result says explicitly. A |\mathcal Const(\mathbf{Ab})|-category is a ring, a |\mathcal Const(\mathbf{Ab})|-functor is a ring homomorphism, and a |\mathcal Const(\mathbf{Ab})|-profunctor is a bimodule. Let |R| and |S| be rings and |f : R \to S| be a ring homomorphism. An isomorphism in |R| viewed as a |\mathcal Const(\mathbf{Ab})|-category is just an invertible element. Every ring, |R|, is an |R|-|R|-bimodule. Given any |S|-|S|-bimodule |P|, we have an |R|-|R|-bimodule |f^*(P)| via restriction of scalars, i.e. |f^*(P)| has the same elements as |P| and for |p \in f^*(P)|, |rpr’ = f(r)pf(r’)|. In particular, |f| gives rise to a bimodule homomorphism, i.e. a linear function, |f : R \to f^*(S)| which corresponds to its action on arrows from the perspective of |f| as a |\mathcal Const(\mathbf{Ab})|-functor. If this linear transformation has an inverse, then the above result states that when |f(r)| is invertible so is |r|. So to restate this all in purely ring theoretic terms, given a ring homomorphism |f : R \to S| and an abelian group homomorphism |\varphi : S \to R| satisfying |\varphi(f(rst)) = r\varphi(f(s))t| and |\varphi(f(r)) = r|, then if |f(r)| is invertible so is |r|.

Indexed categories are equivalent to

*cloven*fibrations and, if you have the Axiom of Choice, all fibrations can be cloven. Indexed categories can be viewed as*presentations*of fibrations.↩︎This suggests that we could define a small |\V|-category |\mathcal C \otimes \mathcal D| where |\mathcal C| and |\mathcal D| are small |\V|-categories. Start formulating a definition of such a |\V|-category. You will get stuck. Where? Why? This implies that the (ordinary, or better, 2-)category of small |\V|-categories does not have a monoidal product with |\mathbb I| as unit in general.↩︎

With a good understanding of what a class is, it’s clear that it doesn’t even make sense to have a proper class be an object. In frameworks with an explicit notion of ”class”, this is often manifested by saying that a class that is an element of another class is a set (and thus not a proper class).↩︎

This suggests that it might be interesting to consider categories that are (co)complete with respect to this monoid notion of “small”. I don’t think I’ve ever seen a study of such categories. (Co)limits of monoids are not trivial.↩︎

This is one of the main things I like about working in weak foundations. It forces you to come up with better definitions that make it clear what is and is not important and eliminates coincidences. Of course, it also produces definitions and theorems that are inherently more general too.↩︎

This connection isn’t much of a surprise as the tensor product of modules is exactly the (small) |\mathcal Const(\mathbf{Ab})| case of this.↩︎