Eric Auld November 25, 2019 Enriched Category Theory The notion of a locally small category is so skeletal as to be mostly taxonomic. More structure is needed to prove nonvacuous theorems: for instance, the existence of limits, and knowing that certain functors preserve limits, or structures on the hom sets, which this note deals with. The presence of an identity morphism in each EndA paq, which we can think of as kind of a choice of basis for each element a P A , already gives us some vestige of an algebraic structure: EndA paq is a monoid. Namely, it is a set S with an associative multiplication map S ˆ S Ñ S and distinguished element e P S which satisfies s ¨ e “ e ¨ s “ s for all s. The set of endomorphisms of an abelian group A are again a monoid, in the same way. We would somehow like to express the extra structure that function composition is bilinear with respect to addition of functions. This means that composition is a map EndAb A b End A Ñ End A. This clues us into the fact that defining a monoid object in a category shouldn't always necessarily be using the cartesian product to model the multiplication. What would be the analog in Cat, if I do use the cartesian product? I'd need a category C , an associative multiplication map C ˆ C Ñ C , and an identity object in C . It turns out that this is pretty much the kind of C I need to create a monoid object inside of C : I just need to allow the unitality and the associativity to be a natural isomorphism, instead of on the nose. (Mostly for philosophical reasons...how objects are defined in Set, for instance.) Monoidal Categories A monoidal category consists of a category V, a bifunctor ´ b ´ : V ˆ V Ñ V, a special object I P V, and three natural isomorphisms α; λ, ρ. pV;W;Xq V bpW bXq V IbV V V bI α – λ – ρ – pV;W;Xq pV bW qbX V V V V These must be coherent in the following ways α V b pW b pX b Y qq pV b W q b pX b Y q V b pI b W q pV b Iq b W 1bλ ρb1 V b ppW b Xq b Y q ppV b W q b Xq b Y V b W λ pV b pW b Xqq b Y I b I I ρ A strict monoidal category is when all the isomorphisms are required to be identity mor- phisms. 1 Example: From Mac Lane's text, examples of (generally weak) monoidal categories: Example: Any category with finite products can be given a cartesian monoidal structure with unit as terminal element. Examples are pSet; ˆ; t˚uq, pAb; `; 0q, pSet˚; ˆ; t˚uq. Example: Any category with finite coproducts can be given a cocartesian monoidal structure with unit as initial element. Examples are pSet; \; ?q, pCRing; bZ; Zq (or more generally pRCAlg; bR;Rq), pSet˚; _; t˚uq Example: The category 2 is monoidal (strict, in fact). We think of it as false Ñ true, and the product ^ being conjunction (\and"), and the unit being true. Example: pSet˚; ^; t˚; ˚uq is monoidal. Recall S ^ t˚; ˚u – S. I would say the genesis of a monoidal category is defining some operation, and picking out how we would like it to interact with whatever structure is there already (e.g. I want an operation which is bilinear with respect to the addition that is already there). Of course, the whole thing is kind of circular: a strict monoidal category is a monoid object in the monoidal category pCat; ˆ; 1q. Monoidal categories allow us to define monoid objects inside them... A (lax) monoidal functor pV; b;Iq ÝÑpF U; d;Jq is a normal functor V Ñ U, along with data pV;W q F pV qdF pW q J ÝÑ F pIq pV;W q F pV bW q Note that these are not required to be isomorphisms, and note that they kind of seem to go the wrong way at first. (They sort of \undistribute" F , or take me not from \where I landed" F pIq to \where I should be" J, but the other way.) This is the direction that gives \image of monoid object is monoid object". They are required to satisfy unitality and associativity F pW q d J F pW q d F pIq ρ pV;W;Xq F pV qdF pW qdF pXq (two ways of \unrolling" U – “ F pV qdF pW qdF pXqq ρV F pV;W;Xq F pV bW bXq F pW q – F pW b Iq (sim left unital) 2 Remark: A lax monoidal functor takes a monoid object to a monoid object, as follows. If F : pV; b;Iq Ñ pU; d;Jq is lax monoidal, and pV; µ, eq is a monoid object in V, then we get F µ a map F pV q d F pV q Ñ F pV b V q ÝÝÑ F pV q. Finish Example: The forgetful functor Ab Ñ Set is lax monoidal pAb; b; Zq Ñ pSet; ˆ; t˚uq. To give the data of a lax monoidal functor, I need to choose a map t˚u Ñ |Z|, and a natural transformation pA;Bq |A|ˆ|B| pA;Bq |AbB| 1 We choose to take pa; bq ÞÑ a b b, and t˚u ÝÑ Z. |A| ˆ t˚u |A| ˆ |Z| – Note that it takes a ring (monoid object in Ab to its multiplicative monoid. |A| – |A bZ Z| Notice that this sort of tells us why we have the maps going the way it does...how would we come up with a natural map |A b B| Ñ |A| ˆ |B|? Example: More generally, it turns out that the \forgetful" functor VpI; ´q : V Ñ Set is lax monoidal from pV; b;Iq Ñ pSet; ˆ; t˚uq. To show this I need natural maps VpI; vq ˆ VpI; wq Ñ VpI; v b wq and t˚u Ñ VpI;Iq. For the former, I can do fbg pf; gq ÞÑ I ÝÑ„ I b I ÝÝÝÑ v b w: Example: Let R be a commutative ring. Then the free functor pSet; \; ?q Ñ pRMod; `; 0q is strong monoidal. (Flesh this out) Why does R have to be commutative here? Example: Similarly, the free functor pSet; ˆ; t˚uq Ñ pRMod; b;Rq is strong monoidal when R is commutative. (I guess it takes a monoid to an R algebra?) Example: In homological algebra, we get a natural map H‚pC‚q b H‚pD‚q Ñ H‚pC‚ b D‚q, the algebraic homology cross product. Note on the left hand side we have a tensor product of graded modules (not chain complexes), so just HjC‚ b HkD‚. This map makes homology into a lax monoidal i j`k“i functor pCh‚pRModq; b;Rq Ñ pGràpRModà ; b;Rq. (The Kunneth theorem says that when R is a PID, the homology cross product is injective, and has cokernel some stuff involving Tor.) A strong monoidal functor is one where the maps are required to be isomorphisms. A strict monoidal functor is where they are required to be identities. 3 Closed Monoidal Categories Let V be monoidal such that for each w P V, the functor v ÞÑ v b w has a right adjoint Vpw; ´q. ´bw Vpv b w; xq – Vpv; Vpw; xqq V K V Vpw,´q Claim: Vp´; ´q is the action on objects of a bifunctor VˆV Ñ V, and there is an isomorphism of trifunctors Vp´ b ´; ´q – Vp´; Vp´; ´qq: f f ˚“Vpf;xq Proof: Given w ÝÑ w1, I want to define Vpw; xq ÐÝÝÝÝÝÝ Vpw1; xq by Vpv b w; xq – Vpv; Vpw; wqq p1bfq˚ 1 1 Vpv b w ; xq – Vpv; Vpw ; xqq Let A be any category. ' 1 1 Lemma: To define a map a ÝÑ a , it is sufficient to define set maps A px; aq Ñ A px; a q for all x, natural in x. (Yoneda embedding full...just take x “ a.) Remark: Of course it is necessary also...choosing ' gives us those natural set maps. Remark: In particular the map of hom objects Vpw1; xq ÝÑ Vpw; xq is defined by its effect on generalized points Vpv; Vpw1; xqq ÝÑ Vpv; Vpw; xqq. This assignment of arrows is functorial, because p1 b fq˚ is. (I can first precompose with f, then g, or precompose with gf.) Therefore I have defined the isomorphism Vp´ b ´; ´q – Vp´; Vp´; ´qq in a way that is natural in w. I have assumed naturality in v and x (adjunction), so I'm done. The unit for this adjunction looks like v ÞÑ Vpw; v b wq, and the counit (\evaluation") looks like Vpw; vq b w Ñ v. As an example of the unit, think about the identity map A b B Ñ A b B. It should in- duce a map A Ñ rB; A b Bs which just says \tensor with whichever element you chose". Enriched Categories Given a monoidal category V, a category C enriched over V (AKA a V-category) is 4 • A collection of objects c P C • For each c; c1 P C , a \hom object" C pc; c1q in V • Composition morphisms C pa; bq b C pb; cq ÝÑ C pa; cq • \Identities" I ÝÑ C pc; cq (generalized point in C pc; cq) These are supposed to satisfy the following coherence axioms: ´1 I b C pb; cq λ C pb; cq C pa; bq b C pb; cq b C pc; dq ˝b1 1b˝ i b1 a C pa; cq b C pc; dq C pa; bq b C pb; dq ˝ ˝ C pa; aq b C pb; cq ˝ C pb; cq C pa; dq (and similarly for right unit) (unitality) (associativity) Note that any V category C has a easily formed opposite V category C op.