🍕(–, B) : C -> Set denotes the contravariant hom functor, normally written Hom(–, B).
In this case, C is a category, and B is a fixed object in that category. The – can be replaced by either an object or morphism of C, and that defines a map from C to Set.
For any given object X in C, the hom-set Hom(X, C) is the set of morphisms X -> B in C. For a morphism f : X -> Y in C, the Set morphism Hom(f, B) : Hom(Y, B) -> Hom(X, B) is defined by sending each g : Y -> B to gf : X -> B. This is the mapping C -> Set defined by Hom(–, C), and it’s a (contravariant) functor because it respects composition: if h : X -> Y and f : Y -> Z then fh : X -> Z and Hom(fh, C) = Hom(h, C)Hom(f, C) sends g : Z -> B to gfh : X -> B.
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P^(n)® AKA RP^n is the n-dimensional real projective space.
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The caveat “phi is a morphism” is probably just to clarify that we’re talking about “all morphisms X -> Y [in a given category]” and not simply all functions or something.
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For more context, the derived functor of Hom(–, B) is called the Ext functor, and the exactness of that sequence (if the typo were fixed) is the statement of the universal coefficient theorem (for cohomology): https://en.wikipedia.org/wiki/Universal_coefficient_theorem The solution to this problem is the “Example: mod 2 cohomology of the real projective space” on that page. It’s (Z/2Z)[x] / <x^(n+1)> or 🍔[x]/<x^(n+1)>, i.e. the ring of polynomials of degree n or less with coefficients in 🍔 = Z/2Z, meaning coefficients of 0 or 1.
Okay I have some reading to do haha. Thanks for the explanation!
As a programmer (who also did quite some math) it never ceases to amaze me how often math just uses single character variable/function names that apparently have a specific meaning. For instance the P^(n)® thingy. Without knowing this specific notation, one might easily assume it meant something else like power sets. Even within the niche I’m more familiar with (machine learning) there was plenty of that stuff going around.
Then again, this meme has an incentive to make it harder, it wouldn’t be funny if it explained symbols.
Math builds up so much context that it’s hard to avoid the use of shorthand and reused names for things. Every math book and paper will start with definitions. So it’s not really on you for not recognizing it here
🍕(–, B) : C -> Set denotes the contravariant hom functor, normally written Hom(–, B). In this case, C is a category, and B is a fixed object in that category. The – can be replaced by either an object or morphism of C, and that defines a map from C to Set.
For any given object X in C, the hom-set Hom(X, C) is the set of morphisms X -> B in C. For a morphism f : X -> Y in C, the Set morphism Hom(f, B) : Hom(Y, B) -> Hom(X, B) is defined by sending each g : Y -> B to gf : X -> B. This is the mapping C -> Set defined by Hom(–, C), and it’s a (contravariant) functor because it respects composition: if h : X -> Y and f : Y -> Z then fh : X -> Z and Hom(fh, C) = Hom(h, C)Hom(f, C) sends g : Z -> B to gfh : X -> B.
–
P^(n)® AKA RP^n is the n-dimensional real projective space.
–
The caveat “phi is a morphism” is probably just to clarify that we’re talking about “all morphisms X -> Y [in a given category]” and not simply all functions or something.
–
For more context, the derived functor of Hom(–, B) is called the Ext functor, and the exactness of that sequence (if the typo were fixed) is the statement of the universal coefficient theorem (for cohomology): https://en.wikipedia.org/wiki/Universal_coefficient_theorem The solution to this problem is the “Example: mod 2 cohomology of the real projective space” on that page. It’s (Z/2Z)[x] / <x^(n+1)> or 🍔[x]/<x^(n+1)>, i.e. the ring of polynomials of degree n or less with coefficients in 🍔 = Z/2Z, meaning coefficients of 0 or 1.
Okay I have some reading to do haha. Thanks for the explanation!
As a programmer (who also did quite some math) it never ceases to amaze me how often math just uses single character variable/function names that apparently have a specific meaning. For instance the P^(n)® thingy. Without knowing this specific notation, one might easily assume it meant something else like power sets. Even within the niche I’m more familiar with (machine learning) there was plenty of that stuff going around.
Then again, this meme has an incentive to make it harder, it wouldn’t be funny if it explained symbols.
Math builds up so much context that it’s hard to avoid the use of shorthand and reused names for things. Every math book and paper will start with definitions. So it’s not really on you for not recognizing it here