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show/hide this revision's text 4 throw out that last piece, doesn't make much sense to me anymore, reference doesn't exist anyway

While it's true that there are lots of internal things that a corepresentable homology functor wouldn't support, I think it's also enlightening to see that you wouldn't get the nice sorts of dualities that homology and cohomology theories have. After all, we've already agreed that cohomology theories ought to be somehow representable, so maybe we should start there. Instead of using stable maps $X \to E_n$ to produce $n$-degree $E$-cohomology classes of $X$, you can think of these instead as elements in the stable homotopy groups $\pi_{-*}^S F(X, E)$, where $F(X, E)$ denotes the function spectrum of maps $X \to E$. This presentation makes the right choice for defining $E$-homology somehow much more obvious: the functor $F(X, -)$ has an adjoint, called the smash product (this is the whole point of the smash product -- it plays the role of "tensor product" for spaces!), and so for homology we think about maps $S^n \to E \wedge X$ instead. That homology and cohomology are not (usually) exact duals in a linear algebraic sense is somehow measuring the twist introduced by this adjunction. This does actually turn out to be the right definition for homology; (extraordinary) homology theories in the traditional sense are in fact modeled by functors of the form $\pi_*^S (E \wedge -)$.

This construction has a number of attractive features -- for instance, it means that we can (under some flatness and ringy conditions) think about "homology cooperations" associated to a spectrum, and they look like $E_* E$, a pleasant mirror of cohomology operations living in $E^* E$. We also always get a pairing $E^* X \times E_* X \to E_* E$ of cohomology and homology classes that lands in homology operations, by composing as $S^n \to E \wedge X \to E \wedge E$. (This pairing even gets used occasionally, though I'd be hard-pressed to come up with an obvious citation.)

For the most familiar homology theory, singular homology with $\mathbb{Z}/p$-coefficients, this flatness business does hold, the operations and cooperations even turn out to be $\mathbb{Z}/p$-vector space duals, and the coaction and action line up in the way you'd expect from Milnor's work.

(This belongs as a comment on Lawson's answer, I think, but it looks like I'm too new here to make that happen.)

edit: Additionally, someone on another thread (I believe in this thread, though it's since been deleted) said that cohomology could be thought of as coloring simplices in a certain infinity-category, and I think that that viewpoint is related to what I'm trying to describe. Food for thought.
    Post Undeleted by Eric Peterson
show/hide this revision's text 3 mixed up E and X, i think this is correct now

Rather than pointing out

While it's true that there are lots of internal things that a corepresentable homology functor wouldn't support, I think it's more also enlightening to see that you wouldn't get the nice sorts of dualities that homology and cohomology theories have. After all, we've already agreed that cohomology theories ought to be somehow representable, so maybe we should start there. Instead of using stable maps $X --> E_ n \to E_n$ to produce n-degree E-cohomology $n$-degree $E$-cohomology classes of X, $X$, you can think of these instead as elements in the stable homotopy groups pi_ * ^S $\pi_{-*}^S F(X, E)E)$, where F(X, E) $F(X, E)$ denotes the function spectrum of maps $X --> E\to E$. This presentation makes the right choice for defining E-homology $E$-homology somehow much more obvious: the functor F(-, E) $F(X, -)$ has an adjoint, called the smash product (this is the whole point of the smash product -- it plays the role of "tensor product" for spaces!), and so for homology we think about maps $S^n --> X smash \to E \wedge X$ instead. That homology and cohomology are not (usually) exact duals in a linear algebraic sense is somehow measuring the twist introduced by this adjunction. This turns does actually turn out to really be the right definition as outlined abovefor homology; (extraordinary) homology theories in the traditional sense are in fact modeled by functors of the form pi_*^S $\pi_*^S (-- smash E).E \wedge -)$.

This construction has a number of attractive features -- for instance, it means that we can (under some flatness and ringy conditions) think about "homology cooperations" associated to a spectrum, and they look like E_ * E$E_* E$, a pleasant mirror of cohomology operations living in E^* E$E^* E$. We also always get a pairing E^* $E^* X x E_ \times E_* X --> E_ \to E_* E E$ of cohomology and homology classes that lands in homology operations, by composing as $S^n --homology-> \to E smash \wedge X --id smash cohomology-> E smash \to E \wedge E$. (This pairing even gets used occasionally, though I'd be hard-pressed to come up with an obvious citation.)

For the most familiar homology theory, singular homology with Z/p-coefficients, $\mathbb{Z}/p$-coefficients, this flatness business does hold, the operations and cooperations even turn out to be Z/p-vector $\mathbb{Z}/p$-vector space duals, and the coaction and action line up in the way you'd expect from Milnor's work.

(This belongs as a comment on Lawson's answer, I think, but it looks like I'm too new here to make that happen. Hope happen.)

edit: Additionally, someone on another thread (I believe in this isn't too off the mark!thread, though it's since been deleted) said that cohomology could be thought of as coloring simplices in a certain infinity-category, and I think that that viewpoint is related to what I'm trying to describe. Food for thought.
    Post Deleted by Eric Peterson
show/hide this revision's text 2 missed a ring spectrum condition
show/hide this revision's text 1