Dari Algebraic Topology ke Aljabar

Leonhard Euler (1707–1783), mathématicien et p...
Leonhard Euler (1707–1783), mathématicien et physicien suisse (Photo credit: Wikipedia)

Dari Algebraic Topology ke Aljabar adalah judul slide presentasi saya dalam seminar KK Aljabar di ITB pertengahan bulan Mei lalu. Klik link nya untuk mengunduh slide.

Pada bulan Mei lalu, saya diminta untuk mengisi seminar di KK Aljabar ITB mengenai Algebraic Topology. Yang terpikir untuk saya presentasikan adalah cerita bagaimana Aljabar diaplikasikan di Topologi (Algebraic Topology) dan pada akhirnya hasilnya diaplikasikan kembali ke Aljabar (Homological Algebra). Rangkuman cerita yang menemani slidenya kira-kira sebagai berikut:

Pada awalnya topologi adalah studi terhadap bentuk (yang tidak terikat koordinat) dan agak berbeda dari topologi yang umum dipelajari saat ini (himpunan buka, fungsi kontinu dst). Studi topologi diawali oleh ketertarikan Leonhard Euler terhadap Graf dan Platonic Solids. Euler menemukan bahwa ada suatu “invarian” atau suatu rumus yang selalu dipenuhi oleh platonic solids dan graf yaitu #titik – #rusuk + #sisi = 2. Hal ini meyakinkan Euler bahwa terdapat sesuatu pola pada bentuk-bentuk ini yang tidak tergantung pilihan koordinat, inilah ide dasar dari Topologi.

Karena bentuk-bentuk ini tidak lagi bergantung koordinat. Dua buah bentuk yang “dianggap sama” bisa terlihat sangat berbeda, salah satu contoh terkenalnya adalah anekdot Donat dan Mug. Hal ini berujung kepada pertanyaan, kapankah dua bentuk topologi dapat dikatakan sama (homotopic) atau berbeda? Ternyata, membuktikan dua bentuk sama atau berbeda tidak mudah, oleh karena itu dicari dan didefinisikan invarian-invarian yang dapat membantu membedakan bentuk. Jika dua bentuk punya invarian yang sama, maka kedua bentuk itu tidak sama.

Ada berbagai jenis invarian yang dipelajari di Algebraic Topology, antara lain grup homotopy, grup homology (komutatif) dan ring cohomology. Tiga invarian ini juga terdefinisi untuk morfisma antara bentuk topologi. Ketiga invarian inilah yang pada tahun 40an menginspirasi lahirnya Category theory. Di slide tersebut diberikan salah satu contoh cohomology yaitu De Rham cohomology dan satu contoh homology yaitu simplicial homology. Aljabar banyak sekali memiliki aplikasi pada singular homology, salah satu bentuk homology yang susah dihitung tetapi sangat ampuh untuk theorem proving. Di akhir slide, diceritakan juga motivasi lahirnya Derived Category.

Metode-metode yang diaplikasikan di Algebraic Topology seperti pengambilan homology, category theory dan derived category ternyata sangat ampuh diaplikasikan kembali di Aljabar. Alasan pasti mengapa metode-metode ini (dikenal sebagai Homological Algebra) bisa seampuh itu masih belum diketahui. Pendek kata, Algebraic Topology akan sangat berguna dipelajari oleh para Algebraist baik dari sisi kepentingan sejarah maupun dari segi aplikasi dan kontribusi kepada ilmu Aljabar sendiri.

What Universal Property means to me?

Universal property of productFew weeks ago, I read a book that define Ring Localization by using Universal Property (or Universal Mapping Problem). This is not the first time I see definition of Localization of a Ring. However when it is defined by using Universal Property, it seems a lot less intuitive than what I experienced.

This is also not the first time I see some mathematical objects are defined by using Universal Property. Among the first definitions I see, there are Projective and Injective modules, Tensor product, Flat modules, product, pullback, pushout etc. I have encountered some of these definitions before and nearly all of them (if not all) defined in ways that, in my opinion, more intuitive.

This case both on Localization example and other examples makes me wonder why we even bother to define those objects via Universal Property. Speaking from Category Theory point of view, this approach (using Universal diagram etc) is of course the one most suitable and gives more generality. However, I think only being the most suitable is not enough reasons to do this — as my own view is that it is okay to use harder approach if this aid understanding.

I therefore tried to make sense of this approach by looking for another justifications. The first visit to wikipedia resulted in:

Before giving a formal definition of universal properties, we offer some motivation for studying such constructions.

  • The concrete details of a given construction may be messy, but if the construction satisfies a universal property, one can forget all those details: all there is to know about the construct is already contained in the universal property. Proofs often become short and elegant if the universal property is used rather than the concrete details. For example, the tensor algebra of a vector space is slightly painful to actually construct, but using its universal property makes it much easier to deal with.
  • Universal properties define objects uniquely up to isomorphism. Therefore, one strategy to prove that two objects are isomorphic is to show that they satisfy the same universal property.
  • Universal constructions are functorial in nature: if one can carry out the construction for every object in a category C then one obtains a functor on C. Furthermore, this functor is a right or left adjoint to the functor U used in the definition of the universal property.[1]
  • Universal properties occur everywhere in mathematics. By understanding their abstract properties, one obtains information about all these constructions and can avoid repeating the same analysis for each individual instance.

Also, I would like to add another remark to the second bullet. If we know one easy representation of the object, by using the isomorphic argument, we can say something like this: “Let x be an element of … (an object defined by Universal Property) then, x can be written as … (the easy representation)”.

By then, I was almost satisfied with wikipedia’s explanation.But then I discussed with two friends, and one of them argues that “by defining an object by using its Universal Property,  we practically shift our attention from the object itself to object in diagram (another object or morphism)”. He gave example that he found in a paper by saying “When the tensor product of modules A and B are 0? Apparently, it only happens if all bilinear morphism from Cartesian product of A and B are 0 map”. Now that’s a revelation, by using Universal Property definition of tensor product we shift our focus out of A and B to their bilinear morphism .. very neat 😀

Category Theory seems to get brighter 🙂