# What minimal structure is required to define Clifford modules in a way as abstract as possible?

Start with a quadratic form $$q$$ on a vector space $$V$$. A module $$M$$ over the corresponding Clifford algebra is determined by a map $$\cdot:V\otimes M\to M$$ satisfying $$v\cdot(v\cdot m)=-q(v)m$$.

Now try to abstract this as follows. The bilinear form $$B(x,y)=q(x)+q(y)-q(x+y)$$ determines a natural transformation $$\varepsilon:TT\to\text{identity}$$, where $$T$$ is the endofunctor $$T=V\otimes-$$ on vector spaces. A Clifford module structure on $$M$$ in these terms is a morphism $$\mu:TM\to M$$, and - here starts my question - certain relationship between the composite $$\mu\circ T\mu:TTM\to TM\to M$$, and $$\varepsilon_M:TTM\to M$$.

The question is what minimal structure does one need to capture this relationship. Seemingly either some kind of nonadditive transformation $$\delta:T\to TT$$ is needed to express $$v\mapsto v\otimes v$$, or some kind of self-distributive law $$\text{switch}:TT\to TT$$. In the latter case however one seemingly needs the additive structure to express $$x\cdot(y\cdot m)+y\cdot(x\cdot m)=B(x,y)m$$.

Has any of this been carried out somewhere? Is it possible to avoid the additive structure, at least using some restrictions? For example, if $$B$$ is nondegenerate, the endofunctor $$T$$ will become self-adjoint, maybe one can use this somehow, I don't know how.

As Liviu Nicolaescu points out, this probably needs some motivation. My motivation is purely abstract-nonsensical in this case. It is known that the category of modules over any algebra can be uniquely (up to equivalence) determined by an abstract category-theoretic universal property. This is because for an algebra $$A$$ the functor $$A\otimes-$$ gets a monad structure, and the category of algebras over a monad is a lax limit in the well known way.

Now for a Clifford algebra, the monad is very special, so that the category of algebras over this monad is equivalent to another category with objects determined by more concise data. I have only described part of these data, but morally this looks like (lax (left)) categorification of the fixed point set of an involution. And the question can be reformulated as follows - given a natural transformation $$\varepsilon:TT\to\text{identity}$$, is there a category-theoretic universal construction (some sort of lax limit again, presumably) that would yield the category of Clifford modules in the particular case when $$T=V\otimes-$$ and $$\varepsilon$$ is induced by a $$q$$ as above?

• The minimal definition you are suggesting seems to obscure the concept you are trying to define. There might be a good reason why you want to do this but I cannot imagine it. Maybe you ought to sketch a motivation for your question. – Liviu Nicolaescu Dec 28 '18 at 10:42
• @LiviuNicolaescu Well actually I cannot suggest any definition at all, the question stems precisely from that. As for the motivation - you probably won't believe but there is indeed a good reason I want to do this but I cannot imagine it either :D – მამუკა ჯიბლაძე Dec 28 '18 at 11:56
• Actually this is an exaggeration. You are right, I will add some motivation – მამუკა ჯიბლაძე Dec 28 '18 at 12:03
• Feedback would be nice for the downvote - could I improve it or, if you think the question itself is bad, why? – მამუკა ჯიბლაძე Dec 29 '18 at 6:50