EMILY RIEHL. X. This data is subject to the following axiom: to any pair X f Y g Z there must be some specified composite map = Z

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1 THE ALGEBRA AND GEOMETRY OF -CATEGORIES EMILY RIEHL Abstract. -cateories are a sopisticated tool or te study o matematical structures wit ier omotopical inormation. Tis note, directed at te Friends o Harvard Matematics, introduces tis notion rom irst principles.. Te pilosopy o cateory teory Te undamental pilosopy o cateory teory is tat anytin one would want to know about a matematical object is determined by te maps (a.k.a. unctions or transormations, sometimes simply arrows ) to or rom it. For instance, unctions rom a sinleton set to a set classiy its elements; unctions rom to te set {, } identiy subsets. Linear maps rom R to a real vector space V correspond bijectively to its vectors. More reined analysis can be used to decode te entire vector space structure. Continuous unctions rom te sinleton space to a topoloical space T identiy its points. Maps rom T to te Sierpinski space classiy open sets. And so ort. Matematical objects o a ixed type assemble into a cateory. A cateory consists o objects, Y, Z,... and maps Y, Y Z,... includin speciied identities. Tis data is subject to te ollowin axiom: to any pair Y Z tere must be some speciied composite map 6666 Y Z and urtermore tis composition law must be bot associative and unital. Te irst o tese conditions says tat a sinle arrow is te speciied composite () and () wenever tese compositions are deined. Te latter says tat composition wit an identity as no eect. ()() 6666 Y Y Y 5555 Y For instance, tere is a cateory o sets and unctions; o vector spaces and linear transormations; and o topoloical spaces and continuous maps, to name just a ew. One mit say tat te objects o a cateory are te nouns and te maps te verbs in te lanuae appropriate to te matematical teory o Date: May 8, 22.

2 2 EMILY RIEHL interest [Maz7]. Te title o tis beautiul essay asks Wen is one tin equal to some oter tin? a question wic we now address. Deinition. A map Y is an isomorpism i tere exists a map Y suc tat te displayed composites are identities Y 6666 Y Y Y Te term derives rom te Greek: iso same morpic sape. Tis notion was called an equivalence in te oundational paper [EM45] in wic cateories are introduced as a ormalism wit wic to describe natural comparisons between parallel matematical constructions. An easy arument sows tat te map is also an isomorpism and urtermore uniquely determined. We say two objects are isomorpic i tere exists an isomorpism between tem. Te ollowin lemma says tat i two objects are isomorpic, ten tey are identical rom te vantae point o our pilosopy. Lemma. I and Y are isomorpic ten tere is a bijection between te collections o maps {Z } {Z Y } Proo. Composition wit te map Y deines a unction rom te collection o maps {Z } to te collection {Z Y }; composition wit Y deines a unction in te oter direction. To sow tat tese unctions deine a bijection, we prove tat tey are inverses. For tis, suppose we are iven a map Z. Its imae under te composite unction is () () by te associativity and unitality axioms. Te irst major teorem in cateory teory is tat te converse olds as well. It is not possible to overstate te importance o tis result wic says tat a matematical object is determined up to isomorpism by its universal property or, equivalently, by te (set-valued) unctor tat it represents. Lemma (Yoneda lemma). I tere exists a natural bijection {Z } {Z Y } or all objects Z in te cateory, ten and Y are canonically isomorpic. 2. A motivatin example rom omotopy teory Let us now brin tis discussion into te realm o omotopy teory. Fix a topoloical space T or instance a surace, peraps a torus, on wic, we mit imaine, lives a very small bu.

3 THE ALGEBRA AND GEOMETRY OF -CATEGORIES 3 Returnin to te pilosopy introduced above, we mit try to investiate te space T by considerin continuous maps rom simple eometric objects into T. In particular, we mit coose maps tat describe possible trajectories or te bu, wose positions are represented by points in T and wose meanderins are described by means o pats, i.e., continuous unctions rom te interval [, ] to T. Tese considerations naturally suest a cateory wose objects are points and wose morpisms are pats. Pats tat start and end at a common point can be composed by travelin twice as ast. Te composition o a pat rom position x to position y wit a pat rom position y to position z is te pat were te bu traverses over te course o te interval [, 2 ] and traverses over te course o te interval [ 2, ]. But tis composition law ails to be associative: a bu travelin alon te pat () spends al its time on and a quarter eac alon and wile a bu travelin alon () spends al its time alon and only a quarter eac on and. Wile tese pats aren t identical, tey are omotopic. Te omotopy takes te orm o a continuous unction rom te product [, ] [, ] to T as depicted by te ollowin scematic picture. [ [ [ ] Te construction suested above does yield a cateory Π T provided we instead deine maps to be omotopy classes o pats. Incidentally, te reason we suested tat T mit be a surace, suc as a torus, wit non-zero enus is so tat tis cateory will ave more tan one map between any two iven points. Assumin te space T is pat connected, te cateory Π T is equivalent (as a cateory) to te undamental roup o T, one o te most celebrated invariants in alebraic topoloy [Lur8]. Noneteless, tis construction is somewat unsatisyin, because te omotopy classes lose te inormation provided by te explicit omotopies. Te orettin involved wit tis truncation was necessary because cateories, as classically deined, are only -dimensional. Te ier omotopical inormation o te space T instead oranizes into an -cateory. 3. Simplicial sets An -cateory is a particular sort o simplicial set. A simplicial set consists o sets,, 2,... o simplices in varyin dimension wic we mit visualize

4 4 EMILY RIEHL in te ollowin manner x x x x x 2 x x 3333 x x 2 x Te data o a simplicial set also consists o unctions 2 3 tat speciy wic lower-dimensional simplices are aces o ier-dimensional simplices and wic ier-dimensional simplices represent deenerate copies o lowerdimensional simplices. Tese unctions satisy certain relations tat are evident rom teir eometric description. Example. To any cateory C tere is an associated simplicial set N C wit -simplices te objects o C, -simplices te maps, 2-simplices composable pairs o maps, 3- simplices composable triples, and so on. In tis simplicial set, a simplex is uniquely determined by its spine, te sequence o edes connectin te t vertex to te st to te 2nd to te 3rd and so on. Tis simplicial set is used to deine te classiyin space o a roup. Example. To any topoloical space T tere is an associated simplicial set ST wit -simplices te points o T, -simplices te pats, 2-simplices continuous unctions rom te topoloical 2-simplex into T, 3-simplices continuous unctions rom te topoloical 3-simplex to T, and so on. Tis simplicial set is used to deine te omoloy o a topoloical space. Simplicial sets temselves orm a cateory. Indeed, returnin to te pilosopy expressed above, a simplicial set is entirely described by maps to it. In particular, te n-simplices o correspond bijectively to maps n wose domain is te standard n-simplex. Here n is te simplicial set wit a sinle non-deenerate n- simplex, toeter wit its aces and teir deeneracies. Tere is a sub simplicial set Λ n k n or eac k n called te (n, k)-orn wic is te simplicial set ormed by te trowin away te non-deenerate n-simplex and its k-t (n )-dimensional ace. For example, te (2, )-orn is te simplicial set depicted below 2 2 A map Λ 2 speciies a pair o -simplices in so tat te taret vertex o one is te source vertex o te oter. 4. -cateories We are now prepared to deine an -cateory.

5 8888 THE ALGEBRA AND GEOMETRY OF -CATEGORIES 5 Deinition. An -cateory is a simplicial set so tat or eac n 2, k n any (n, k)-orn in extends to an n-simplex. Λ n k n We ave already seen two classes o examples. Example. For any cateory C, N C is an -cateory. Eac (2, )-orn admits an extension to a 2-simplex because composable maps ave a speciied composite. Eac (3, )- and (3, 2)-orn admits an extension because tis composition law is associative. Indeed, eac (n, k)-orn admits a unique extension to an n-simplex because te simplices in N C are uniquely caracterized by teir spine, wic is visible in te orn. Example. For any topoloical space T, ST is an -cateory. A map Λ n k ST corresponds to a continuous unction rom a topoloical realization o te (n, k)- orn into te space T. Tis topoloical orn includes into te topoloical n-simplex and tis inclusion is a deormation retract, meanin tere is a continuous retraction o te n-simplex onto te (n, k)-orn. Tis retraction may be used to deine te desired extension. Indeed, note tat tis arument works equally well or (n, )- and (n, n)-orns in ST. In eneral, we tink o an -cateory as a weak cateory wit maps in eac dimension. Te -simplices are te objects and te -simplices are te maps. For any composable pair o maps x y, y z, te extension condition in te case n 2 uarantees tat tere exists a 2-simplex x y z wic we tink o as a omotopy witnessin tat is a composite o and. Te 3-dimensional extension condition implies, amon oter tins, tat any two composites are temselves omotopic. Te 4-dimensional extension condition implies, amon oter tins, tat any two parallel omotopies ave a ier omotopy comparin tem, and so on. Tere is a sense in wic tese (ier) omotopies may temselves be composed; wit respect to tis composition law, all (ier) omotopies are invertible. One way to express tis property is to say tat between any two vertices in an -cateory tere is a topoloical space o maps, tou tis space is only well-deined up to omotopy type. 5. Equivalences in -cateories Finally, we return to a question considered at te beinnin: Wat sould it mean or two objects o an -cateory to be equivalent? A quick deinition is tat objects o an -cateory are equivalent just wen tey are isomorpic in te associated omotopy cateory. Tis leads to te ollowin concrete deinition.

6 6 EMILY RIEHL Deinition. A -simplex x y in a -cateory is an equivalence i tere exists a -simplex y x and 2-simplices y x x x y y x y Part o te maic o -cateories is visible in te ollowin result, wic says tat equivalences in an -cateory are automatically ininite-dimensional. Let J be te simplicial set wit two -simplices, ; two non-deenerate -simplices, ; and indeed two non-deenerate simplices in eac dimension continuin in te pattern depicted Teorem. Equivalences in an -cateory correspond to maps J. More precisely, a -simplex in is an equivalence i and only i it can be extended to a map wose domain is J. Te proo is reasonably elementary tou it requires some clever combinatorics. Te reader mit enjoy provin a special case: tat te data described above can be extended to dimension tree. Reerences [Lur8] J. Lurie, Wat is an -cateory?, Notices o te AMS, 55 (8), September, 28. [Maz7] B. Mazur, Wen is one tin equal to some oter tin?, In memory o Saunders MacLane, ttp:// mazur/expos.tml, June 2, 27. [EM45] S. Eilenber and S. MacLane, General teory o natural equivalences, Trans. Amer. Mat. Soc., 58, 945, Dept. o Matematics, Harvard University, Oxord Street, Cambride, MA 238 address: eriel@mat.arvard.edu