Representation frameworks and representations

Chapter 22 on representations is, I feel, the worst written I have come upon up to now. I haven’t yet come to wrap my head around it.

I certainly appreciate the desire to give a general definition of representations in category theory terms. This is how the author puts it:

Let \mathfrak C and \mathfrak C' be two categories. We suppose that we are given the following two things: i) a forgetful functor \mathcal F from the category \mathfrak C to the category of sets, and ii) a rule which assigns, to each object P' in category \mathfrak C', an object Z in category \mathfrak C and an isomorphism (in the category of sets) from set \mathcal F(Z) to the set \Mor(P', P').

I see two issues about this setting. I will “correct” these issues in this post, giving what I hope is a better definition, and come to the issue of direct products and sums of representations and of subrepresentations in another post.

About the functor to the category of sets

About the above complex statement, the author reassures us:

There is an obvious forgetful functor from every category we shall consider to the category of sets, and we shall always use this one for item i). The purpose of i) is to allow us to speak of “elements” of objects in category \mathfrak C.

The existence of this “obvious functor” in the categories we consider is very familiar; indeed, groups, vector spaces and so on are built upon sets. We are clearly used to speaking of an element of a group, though in fact we should speak of an element of the underlying set.

Postulating the existence of this forgetful functor \mathcal F from \mathfrak C to \catSet is, however, not enough to ensure that we are in the “familiar situation”. There is another element of this familiar situation that is not captured by the mere existence of this functor. Namely: in the familiar situation, the morphisms between objects of \mathfrak C are no more and no less than morphisms – i.e., mappings – between the corresponding sets. If O_1 and O_2 are objects of \mathfrak C, then \Mor(O_1, O_2) is literally a subset of \Mor(\mathcal F(O_1), \mathcal F(O_2)). This implies in particular that \mathcal F is “one to one” in its action on morphisms; there cannot be two different morphisms \phi, \psi: O_1 \to O_2 such that \mathcal F(\phi) = \mathcal F(\psi).

That this is not necessarily implied by the mere existence of a functor \mathcal F from \mathfrak C to \catSet can be made clear by considering this trivial fact: for any category \mathfrak C, there is a functor \mathcal F_0 from \mathfrak C to \catSet that takes any object O of \mathfrak C to the set \{O\} and any morphism in \mathfrak C between objects O_1 and O_2 to the one and only mapping that exists from set \{O_1\} to set \{O_2\}. One can check that this is indeed a functor; but clearly, in general it will take many different morphisms in \mathfrak C to just one morphism in \catSet.

To be closer to the familiar situation, we thus need (at least) the following additional condition: that the functor \mathcal F from \mathfrak C to \catSet be “morphism-injective”, that is, such that, for any given objects O_1 and O_2 of \mathfrak C and any \phi, \psi \in \Mor(O_1, O_2), we have \mathcal F(\phi) = \mathcal F(\psi) \implies \phi = \psi.

We will see why this condition is necessary particularly in the case of the direct products and sums of representations. In the book, it is hidden behind the notations of the author, that tend to muddle the distinction between morphisms in \mathfrak C and the corresponding morphisms in \catSet.

Must it really be an isomorphism?

The author postulates that there must be “ii) a rule which assigns, to each object P' in category \mathfrak C', an object Z in category \mathfrak C and an isomorphism (in the category of sets) from set \mathcal F(Z) to the set \Mor(P', P')“.

It makes for a simpler formulation if instead of speaking of an isomorphism, we say that this rule \mathcal R must assign to the object P' of \mathfrak C' an object Z of \mathfrak C such that \mathcal F(Z) = \Mor(P', P').

For instance: if \mathfrak C' is the category of real vector spaces, and \mathfrak C that of real associative algebras, we want our rule \mathcal R to “make the set of endomorphisms of a given real vector space P' into an associative algebra”, that is, \mathcal R(P') = Z is an associative algebra the underlying set of which is \Mor(P', P').

But this doesn’t work well if, instead, we want \mathfrak C to be the category of groups. The author calls this a “special case of the definition of a representaton”, but doesn’t clearly note that in this case, we much change his definition, replacing “isomorphism” by “monomorphism”. In the case of groups, it is not the whole of \Mor(P', P') that is “made into a group”, but only the set of bijective linear mappings from P' to itself.

So we will adopt the modified specification of ii): We need a rule \mathcal R which assigns, to each object P' in category \mathfrak C', an object \mathcal R(P') = Z such that \mathcal F(Z) \subseteq \Mor(P', P).

My definition of a representation framework

A representation framework is made of:

  • Two categories \mathfrak C and \mathfrak C'.
  • A morphism-injective functor from \mathcal C to \catSet.
  • A rule \mathcal R that assigns to each object P' in category \mathfrak C', an object \mathcal R(P') = Z such that \mathcal F(Z) \subseteq \Mor(P', P).

A representation is…

Given such a representation framework, a representation of an object A of \mathfrak C is an ordered pair (P', \psi) where P' is an object of \mathfrak C' and \psi a \mathcal C-morphism A \to \mathcal R(P').

This is very close to the definition given by the author. He then goes on to state:

(…) for each element a of object A, we must have a certain morphism, which we write \psi_a, from object P' to itself.

 

This “\psi_a” is misleading, in that it gives the impression that \psi_a is the image of a by \psi. But \psi is a morphism between A and Z = \mathcal R(P'). It is not (in general) a mapping at all. It is not \psi, but \mathcal F(\psi) that is a mapping and can map a to this \psi_a, element of \mathcal F(Z) \subseteq \Mor(P', P').

Despite the awkwardness, I will often prefer the notation “\mathcal F(\psi)(a)” to the shorter “\psi_a“, because the latter can give rise to the notion that if we happen to have, for each a, a certain \mathfrak C'-morphism P' \to P' that we call \gamma_a, for instance, then necessarily this mapping forms a \mathfrak C-morphism \gamma: A \to Z; while it only means that we have a mapping of sets \mathcal F(A) \to \mathcal F(Z) ‑ if indeed the \gamma_a‘s are all in \mathcal F(Z), which may be only a subset of \Mor(P', P') ‑ and this mapping may or may not be some \mathcal F(\gamma) with \gamma a \mathfrak C-morphism from A to Z. This false reasoning arises precisely in the presentations by the author of direct products of representations, direct sums and subrepresentations. That will be for another post.

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