A cospectrum (in the context of homotopy theory) is defined to be a sequence of spaces $X_0, X_1, \ldots, X_n, \ldots, $ equipped with maps $X_{n+1}\to \Sigma X_n$, for each $n$. So cospectra are similar to spectra, except that the structure maps point in the opposite direction.

Cospectra were first introduced by Elon Lima in 1959. I learned about them form Browder's classic paper on the Kervaire invariant problem. Browder seems to make rather essential use of this construction. I had never seen the concept before, so tried to search the literature on cospectra. I found virtually nothing. There are a couple of papers developing some properties of cospectra, but they did not seem to lead to any further activity. As far as I can see, practically no one else investigated cospectra or used them for anything.

So we have a definition that is a more or less natural variant of a very influential one. A definition that was used once in an important paper, and nowhere else. I find this curious, therefore I want to ask

Question 1 Is there a reasonable way to rewrite Browder's proof without cospectra?

According to my limited understanding, the reason for introducing cospectra is that they provide an alternative approach to Spanier-Whitehead duality. Lima's motivation for the definition was to define Spanier-Whitehead dual of spaces more general than finite CW complexes. As far as I could make out, Browder's motivation was similar. He needed to have a notion of Spanier-Whitehead dual that was well-behaved for non-finite spectra, and cospectra seem to do the job for him. Therefore I wonder if the same could be accomplished using the duality between spectra and pro-spectra, or maybe by just using finite approximations to spectra.

If question 1 does not have an obvious positive answer, then there is a natural follow up:

Question 2 How come no one else found use for cospectra? Is there some good mathematics lying that way, waiting to be discovered?

A cospectrum (in the context of homotopy theory) is defined to be a sequence of spaces $X_0, X_1, \ldots, X_n, \ldots, $ equipped with maps $X_{n+1}\to \Sigma X_n$, for each $n$. So cospectra are similar to spectra, except that the structure maps point in the opposite direction.

Cospectra were first introduced by Elon Lima in 1959. I learned about them form Browder's classic paper on the Kervaire invariant problem. Browder seems to make rather essential use of this construction. I had never seen the concept before, so tried to search the literature on cospectra. I found virtually nothing. There are a couple of papers developing some properties of cospectra, but they did not seem to lead to any further activity. As far as I can see, practically no one else investigated cospectra or used them for anything.

So we have a definition that is a more or less natural variant of a very influential one. A definition that was used once in an important paper, and nowhere else. I find this curious, therefore I want to ask

Question 1 Is there a reasonable way to rewrite Browder's proof without cospectra?

According to my limited understanding, the reason for introducing cospectra is that they provide an alternative approach to Spanier-Whitehead duality. Lima's motivation was to define Spanier-Whitehead dual of spaces more general than finite CW complexes. As far as I could make out, Browder's motivation was similar. He needed to have a notion of Spanier-Whitehead dual that was well-behaved for non-finite spectra, and cospectra seem to do the job for him. Therefore I wonder if the same could be accomplished using the duality between spectra and pro-spectra, or maybe by just using finite approximations to spectra.

If question 1 does not have an obvious positive answer, then there is a natural follow up:

Question 2 How come no one else found use for cospectra? Is there some good mathematics lying that way, waiting to be discovered?

A cospectrum (in the context of homotopy theory) is defined to be a sequence of spaces $X_0, X_1, \ldots, X_n, \ldots, $ equipped with maps $X_{n+1}\to \Sigma X_n$, for each $n$. So cospectra are similar to spectra, except that the structure maps point in the opposite direction.

Cospectra were first introduced by Elon Lima, in 1959. I learned about them form Browder's classic paper on the Kervaire invariant problem. Browder seems to make rather essential use of this construction. I had never heard about them before, so tried to search the literature on cospectra. I found virtually nothing. There are a couple of papers developing some properties of cospectra, but they did not seem to lead to any further activity. As far as I can see, practically no one else investigated cospectra or used them for anything.

So we have a definition that is a more or less natural variant of a very influential one. A definition that was used once in an important paper, and nowhere else. I find this curious, therefore I want to ask

Question 1 Is there a reasonable way to rewrite Browder's proof without cospectra?

According to my limited understanding, the reason for introducing cospectra is that they provide an alternative approach to Spanier-Whitehead duality. Lima's motivation for the definition was to define Spanier-Whitehead dual of spaces more general than finite CW complexes. As far as I could make out, Browder's motivation was similar. He needed to have a notion of Spanier-Whitehead dual that was well-behaved for non-finite spectra, and cospectra seem to do the job for him. Therefore I wonder if the same could be accomplished using the duality between spectra and pro-spectra, or maybe by just using finite approximations to spectra.

If question 1 does not have an obvious positive answer, then there is a natural follow up question

Question 2 How come no one else found use for cospectra? Is there some good mathematics lying that way, waiting to be discovered?

A cospectrum (in the context of homotopy theory) is defined to be a sequence of spaces $X_0, X_1, \ldots, X_n, \ldots, $ equipped with maps $X_{n+1}\to \Sigma X_n$, for each $n$. So cospectra are similar to spectra, except that the structure maps point in the opposite direction.

Cospectra were first introduced by Elon Lima in 1959. I learned about them form Browder's classic paper on the Kervaire invariant problem. Browder seems to make rather essential use of this construction. I had never seen the concept before, so tried to search the literature on cospectra. I found virtually nothing. There are a couple of papers developing some properties of cospectra, but they did not seem to lead to any further activity. As far as I can see, practically no one else investigated cospectra or used them for anything.

So we have a definition that is a more or less natural variant of a very influential one. A definition that was used once in an important paper, and nowhere else. I find this curious, therefore I want to ask

Question 1 Is there a reasonable way to rewrite Browder's proof without cospectra?

According to my limited understanding, the reason for introducing cospectra is that they provide an alternative approach to Spanier-Whitehead duality. Lima's motivation for the definition was to define Spanier-Whitehead dual of spaces more general than finite CW complexes. As far as I could make out, Browder's motivation was similar. He needed to have a notion of Spanier-Whitehead dual that was well-behaved for non-finite spectra, and cospectra seem to do the job for him. Therefore I wonder if the same could be accomplished using the duality between spectra and pro-spectra, or maybe by just using finite approximations to spectra.

If question 1 does not have an obvious positive answer, then there is a natural follow up:

Question 2 How come no one else found use for cospectra? Is there some good mathematics lying that way, waiting to be discovered?