On Adjoint and Brain Functors

Abstract

There is some consensus among orthodox category theorists that the concept of adjoint functors is the most important concept contributed to mathematics by category theory. We give a heterodox treatment of adjoints using heteromorphisms (object-to-object morphisms between objects of different categories) that parses an adjunction into two separate parts (left and right representations of heteromorphisms). Then these separate parts can be recombined in a new way to define a cognate concept, the brain functor, to abstractly model the functions of perception and action of a brain. The treatment uses relatively simple category theory and is focused on the interpretation and application of the mathematical concepts .

Mathematical Appendix: Are Hets Really Necessary in Category Theory?

Since the concept of a brain functor requires hets for its formulation, it is important to consider the role of hets in category theory. The homomorphisms or homs between the objects of a category ${\mathbb {X}}$ are given by a hom bifunctor ${\text {Hom}}_{{\mathbb {X}}}:{\mathbb {X}}^{op}\times {\mathbb {X}}\rightarrow Sets$. In the same manner, the heteromorphisms or hets from the objects of a category ${\mathbb {X}}$ to the objects of a category ${\mathbb {A}}$ are given by a het bifunctor $\hbox {Het}:{\mathbb {X}} ^{op}\times {\mathbb {A}}\rightarrow Sets$.⁷

The $\hbox {Het}$-bifunctor gives the rigorous way to handle the composition of a het $f:x\rightarrow a$ in $\hbox {Het}\left( x,a\right)$ [thin arrows $\rightarrow$ for hets] with a homomorphism or hom $g:x^{\prime }\Longrightarrow x$ in X [thick Arrows $\Longrightarrow$ for homs] and a hom $h:a\Longrightarrow a^{\prime }$ in A. For instance, the composition $x^{\prime }\overset{g}{\Longrightarrow }x\overset{f}{\rightarrow }a$ is the het that is the image of f under the map: $\hbox {Het}\left( g,a\right) :\hbox {Het}\left( x,a\right) \rightarrow \hbox {Het}\left( x^{\prime },a\right)$. Similarly, the composition $x\overset{f}{\rightarrow }a\overset{h}{\Longrightarrow }a^{\prime }$ is the het that is the image of f under the map: $\hbox {Het}\left( x,h\right) :\hbox {Het}\left( x,a\right) \rightarrow \hbox {Het}\left( x,a^{\prime }\right)$ (Fig. 15).⁸

Fig. 15

Composition of a het with a hom on either end

This is all perfectly analogous to the use of $\hbox {Hom}$-functors to define the composition of homs. Since both homs and hets (e.g., injection of generators into a group) are common morphisms used in mathematical practice, both types of bifunctors formalize standard mathematical machinery.

1.1 Chimeras in the Wilds of Mathematical Practice

The homs-only orientation may go back to the original conception of category theory “as a continuation of the Klein Erlanger Programm, in the sense that a geometrical space with its group of transformations is generalized to a category with its algebra of mappings.” (Eilenberg and MacLane 1945, p. 237) While chimeras do not appear in the orthodox “ontological zoo” of category theory, they abound in the wilds of mathematical practice. In spite of the reference to “Working Mathematician” in the title of MacLane’s text (MacLane 1971), one might seriously doubt that any working mathematician would give, say, the universal mapping property of free groups using the “device” of the underlying set functor U instead of the traditional description given in the left representation diagram (which does not even mention U) as can be seen in most any non-category-theoretic text that treats free groups. For instance, consider the following description in Nathan Jacobson’s text (Jacobson 1985, p. 69).

To summarize: given the set $X=\{x_{1},\ldots ,x_{r}\}$ we have obtained a map $x_{i}\rightarrow \bar{x}_{i}$ of X into a group $FG^{\left( r\right) }$ such that if G is any group and $x_{i}\rightarrow a_{i},\, 1\le i\le r$ is any map of X into G then we have a unique homomorphism of $FG^{\left( r\right) }$ into G, making the following diagram commutative:

$$\begin{aligned} \begin{array}{ccc} X &{} &{} \\ \downarrow &{} \searrow &{} \\ FG^{\left( r\right) } &{} \Longrightarrow &{} G \end{array}. \end{aligned}$$

In Jacobson’s diagram, only the $FG^{\left( r\right) }\Longrightarrow G$ morphism is a group homomorphism; the vertical and diagonal arrows are called “maps” and are set-to-group hets so it is the diagram for a left representation.⁹

1.2 Hets as “Homs” in a Collage Category

The notion of a homomorphism is so general that hets can always be recast as “homs” in a larger category variously called a directly connected category (Pareigis 1970, p. 58) (since Pareigis calls the het bifunctor a “connection”), a cograph category (Shulman 2011), or, more colloquially, a collage category (since it combines quite different types of objects and morphisms into one category in total disregard of any connection to the Erlangen Program). The collage category of a het bifunctor $\hbox {Het}:{\mathbb {X}}^{op}\times {\mathbb {A}}\rightarrow Sets$, denoted ${\mathbb {X}}\bigstar ^{{\text {Het}}}{\mathbb {A}}$ (Lurie 2009, p. 96), has as objects the disjoint union of the objects of ${\mathbb {X}}$ and ${\mathbb {A}}$. The homs of the collage category are defined differently according to the two types of objects. For x and $x^{\prime }$ objects in ${\mathbb {X}}$, the homs $x\Rightarrow x^{\prime }$ are the elements of $\hbox {Hom}_{{\mathbb {X}}}\left( x,x^{\prime }\right)$, the hom bifunctor for ${\mathbb {X}}$, and similarly for objects a and $a^{\prime }$ in ${\mathbb {A}}$, the homs $a\Rightarrow a^{\prime }$ are the elements of $\hbox {Hom}_{{\mathbb {A}}}\left( a,a^{\prime }\right)$. For the different types of objects such as x from ${\mathbb {X}}$ and a from ${\mathbb {A}}$, the “homs” $x\Rightarrow a$ are the elements of $\hbox {Het}\left( x,a\right)$ and there are no homs $a\Rightarrow x$ in the other direction in the collage category.

Does the collage category construction show that “hets” are unnecessary in category theory and that homs suffice? Since all the information given in the het bifunctor has been repackaged in the collage category, any use of hets can always be repackaged as a use of the “${\mathbb {X}}$-to-${\mathbb {A}}$ homs” in the collage category ${\mathbb {X}}\bigstar ^{{\text {Het}}}{\mathbb {A}}$. In any application, like the previous example of the universal mapping property (UMP) of the free-group functor as a left representation, one must distinguish between the two types of objects and the three types of “homs” in the collage category.

Suppose in Jacobson’s example, one wanted to “avoid” having the different “maps” and group homomorphisms by formulating the left representation in the collage category formed from the category of Sets, the category of groups Grps, and the het bifunctor, ${\text {Het}}:Sets^{op}\times Grps\rightarrow Sets$, for set-to-group maps. Since the UMP does not hold for arbitrary objects and homs in the collage category, $Sets\bigstar ^{{\text {Het}}}Grps$, one would have to differentiate between the “set-type objects” X and the “group-type objects” G as well as between the “mixed-type homs” in $\hbox {Hom}\left( X,G\right)$ and the “pure-type homs” in $\hbox {Hom}\left( FG^{(r)},G\right)$. Then the left representation UMP of the free-group functor could be formulated in the het-free collage category $Sets\bigstar ^{{\text {Het}}}Grps$ as follows.

For every set-type object X, there is a group-type object $F\left( X\right)$ and a mixed-type hom $\eta _{X}:X\Rightarrow F\left( X\right)$ such that for any mixed-type hom $f:X\Rightarrow G$ from the set-type object X to any group-type object G, there is a unique pure-type hom $f_{*}:F\left( X\right) \Rightarrow G$ such that $f=f_{*}\eta _{X}$.

Thus the answer to the question “Are hets really necessary?” is “No!”–since one can always use sufficient circumlocutions with the different types of “homs” in a collage category. Jokes aside, the collage category formulation is essentially only a reformulation of the left representation UMP using clumsy circumlocutions. Working mathematicians use phrases like “mappings” or “morphisms” to refer to hets in contrast to homomorphisms–and “mixed-type homs” does not seem to be improved phraseology for hets.

There is, however, a more substantive point, i.e., the general UMPs of left or right representations show that the hets between objects of different categories can be represented by homs within the codomain category or within the domain category, respectively. If one conflates the hets and homs in a collage category, then the point of the representation is rather obscured (since it is then one set of “homs” in a collage category being represented by another set of homs in the same category).

1.3 What About the Homs-Only UMPs in Adjunctions?

There is another het-avoidance device afoot in the homs-only treatment of adjunctions. For instance, the left-representation UMP of the free-group functor can, for each $X\in Sets$, be formulated as the natural isomorphism: $\hbox {Hom}_{Grps}\left( F\left( X\right) ,G\right) \cong \hbox {Het}\left( X,G\right)$. But if we fix G and use the underlying set functor $U:Grps\rightarrow Sets$, then there is trivially the right representation: $\hbox {Het}\left( X,G\right) \cong \hbox {Hom}_{Sets}\left( X,U\left( G\right) \right)$. Putting the two representations together, we have the heteromorphic treatment of an adjunction first formulated by Pareigis (1970):

$${\text{Hom}}_{Grps}\left( F\left( X\right) ,G\right)\cong{\text {Het}}\left( X,G\right)\cong{\text {Hom}}_{Sets}\left( X,U\left( G\right) \right).$$

If we delete the het middle term, then we have the usual homs-only formulation of the free-group adjunction,

$${\text {Hom}}_{Grps}\left( F\left( X\right) ,G\right) \cong {\text {Hom}}_{Sets}\left( X,U\left( G\right) \right),$$

without any mention of hets. Moreover, the het-avoidance device of the underlying set functor U allows the UMP of the free group functor to be reformulated with sufficient circumlocutions to avoid mentioning hets.

For each set X, there is a group $F\left( X\right)$ and a set hom $\eta _{X}:X\Rightarrow U\left( F\left( X\right) \right)$ such that for any set hom $f:X\Rightarrow U\left( G\right)$ from the set X to the underlying set $U\left( G\right)$ of any group G, there is a unique group hom $f_{*}:F\left( X\right) \Rightarrow G$ over in the other category such that the set hom image $U\left( f_{*}\right)$ of the group hom $f_{*}$ back in the original category satisfies $f=U\left( f_{*}\right) \eta _{X}$(Fig. 16).¹⁰

Fig. 16

Over-and-back diagram for free group adjunction

Such het-avoidance circumlocutions have no structural significance since there is a general adjunction representation theorem (Ellerman 2006, p. 147) that all adjoints can be represented, up to isomorphism, as arising from the left and right representations of a het bifunctor.

1.4 Are all UMPs Part of Adjunctions?

Even though the homs-only formulation of an adjunction only ignores the underlying hets (due to the adjunction representation theorem), is that formulation sufficient to give all UMPs? Or are there important universal constructions that are not either left or right adjoints?

Probably the most important example is the tensor product. The universal mapping property of the tensor product is particularly interesting since it is a case where the heteromorphic treatment of the UMP is forced (under one disguise or another). The tensor product functor $\otimes :\left\langle A,B\right\rangle \longmapsto A\otimes B$ is not a left adjoint so the usual device of using the other functor (e.g., a forgetful or diagonal functor) to avoid mentioning hets is not available.

For A, B, C modules (over some commutative ring R), one category is the product category $Mod_{R}\times Mod_{R}$ where the objects are ordered pairs $\left\langle A,B\right\rangle$ of R-modules and the other category is just the category $Mod_{R}$ of R-modules. The values of the ${\text {Het}}$-bifunctor ${\text {Het}}\left( \left\langle A,B\right\rangle ,C\right)$ are the bilinear functions $A\times B\rightarrow C$. Then the tensor product functor $\otimes :Mod_{R}\times Mod_{R}\rightarrow Mod_{R}$ given by $\left\langle A,B\right\rangle \longmapsto A\otimes B$ gives a left representation:

$${\text {Hom}}_{Mod_{R}}\left( A\otimes B,C\right) \cong {\text {Het}}\left( \left\langle A,B\right\rangle ,C\right)$$

that characterizes the tensor product. The canonical het $\eta _{\left\langle A,B\right\rangle }:A\times B\rightarrow A\otimes B$ is the image under the left-representation isomorphism of the identity hom $1_{A\otimes B}$ obtained by taking $C=A\otimes B$, so we have:

$$\begin{aligned}&\qquad \qquad \qquad \qquad \qquad \begin{array}{ccc} \left\langle A,B\right\rangle &{} &{} \\ ^{\eta _{\left\langle A,B\right\rangle }}\downarrow ^{{}} &{} \searrow ^{f} &{} \\ A\otimes B &{} \underset{\exists !\text { }f_{*}}{\Longrightarrow } &{} C \end{array} \\&\hbox {Left representation diagram to characterize tensor products} \end{aligned}$$

where the single arrows are the bilinear hets and the thick Arrow is a module homomorphism within the category $Mod_{R}$.

For instance, in MacLane and Birkhoff’s Algebra textbook (MacLane and Birkhoff 1988), they explicitly use hets (bilinear functions) starting with the special case of an R-module A (for a commutative ring R) and then stating the universal mapping property of the tensor product $A\otimes R\cong A$ using the left representation diagram (MacLane and Birkhoff 1988, p. 318)–like any other working mathematicians. For any R-module A, there is an R-module $A\otimes R$ and a canonical bilinear het $h_{0}:A\times R\rightarrow A\otimes R$ such that given any bilinear het $h:A\times R\rightarrow C$ to an R-module C, there is a unique R-module hom $t:A\otimes R\Longrightarrow C$ such that the following diagram commutes.

$$\begin{aligned}&\qquad \qquad \qquad \qquad \qquad \qquad \begin{array}{ccc} A\times R &{} &{} \\ h_{0}\downarrow ^{{}} &{} \searrow ^{h} &{} \\ A\otimes R &{} \overset{\exists !t}{\Longrightarrow } &{} C \end{array}\\&\hbox {Left representation diagram of special case of tensor product}. \end{aligned}$$