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## Hot answers tagged adjoint-functors

19 votes

### Understanding the tensor-hom adjunction intuitively

arctic tern's answer is very nice and complete. But less formally, the adjunction really says something quite simple: By the universal property of the tensor product, a linear map out of $U \otimes V$ ...
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16 votes
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Executive summary: the isomorphism is not particularly mysterious. It arises from the obvious choice of a trace from the group algebra of $G$ to the group algebra of $H$, given by the formula $$\sum_{... • 14.9k 15 votes ### Understanding the Beck-Chevalley condition \require{AMScd} When you do algebraic geometry, you can consider diagrams like$$ \begin{CD} X @>f>> A\\ @VgVV \\ B \end{CD} $$where X,A,B are spaces and g,f maps thereof. It is ... • 11.9k 15 votes Accepted ### Is there a notion of "schemeification" analogous to that of sheafification of a presheaf? There is an "affine schemification", a left adjoint to the inclusion of affine schemes into locally ringed spaces. Indeed, given a locally ringed space X, the left adjoint just sends X to \... • 332k 15 votes Accepted ### Why the forgetful functor from \mathbf{Ab} to \mathbf{Grp} does not admit a right adjoint? Hint: Left adjoints preserve colimits, so it suffices to prove that U does not preserve colimits. For an even bigger hint, hover over the box below. • 65.1k 15 votes Accepted ### What are some examples of self-adjoint functors? Is this an example? This is a relatively uncommon scenario, but here are a few more examples. As you suggest, (-)^\mathrm{op} : \mathbf{Cat} \to \mathbf{Cat} is self-adjoint. More generally, this should be true for ... • 5,728 13 votes ### Let C,D be categories and F:C\to D and G:D\to C be adjoint functors. Then F is fully faithful iff the unit is an isomorphism? F is fully faithful iff the map \text{Hom}(x, y) \to \text{Hom}(Fx, Fy) is an isomorphism. By adjunction, the map \text{Hom}(Fx, Fy) \to \text{Hom}(x, GFy) is always an isomorphism. Now use the ... • 421k 13 votes ### Is the tensor product of two projective sheaves projective? This fails in general. In fact, it fails for condensed abelian groups, which is a mayor annoyance. Projective condensed abelian groups M are in fact exactly the retracts of \mathbb Z[S] where S ... • 1,116 12 votes Accepted ### Non-monadic adjunction A classical example : the forgetful functor from topological spaces to sets. The left adjoint is the "discrete space" functor (sending a set X to the discrete space with underlying space X), and ... • 26.1k 11 votes ### Understanding the tensor-hom adjunction intuitively There is a de-linearized version of tensor-hom adjunction, called currying in computer science, which states there is a bijection \hom(U\times V,W)\cong \hom(U,\hom(V,W)) in the category \mathsf{... • 152k 11 votes Accepted ### Quick proof that free objects on sets of different cardinality are not isomorphic? Counterexample: The Jónsson–Tarski algebra. Let \mathbf K be the class of algebras \mathfrak A=(A,f,g,h) of signature (2,1,1) satisfying the identities g(f(x,y))=x,\ h(f(x,y))=y,\ f(g(z),h(z))=... • 78.7k 11 votes Accepted ### Intuition about coreflective subcategories. Your intuitive description of reflective subcategories doesn't distinguish left adjoints from right adjoints so you could apply it equally well to either. Anyway, I think the best way to get a handle ... • 421k 10 votes Accepted ### Composing functors with natural transformations Given functors$$\mathcal{A} \xrightarrow{F} \mathcal{B} \overset{G}{\underset{H}{\rightrightarrows}} \mathcal{C} \xrightarrow{K} \mathcal{D}$$and a natural transformation \alpha : G \to H, we can ... • 65.1k 10 votes Accepted ### Hartshorne Exercise II. 1.18 Yes: to prove two morphisms of sheaves are equal, it suffices to prove they induce the same maps on each stalk. Explicitly, let us suppose F and G are sheaves on a space X and a,b:F\to G are ... • 332k 10 votes ### Is “monoidal category enriched over itself” the same as “closed monoidal category”? The two concepts are distinct. In general it is true that every monoidal closed category is enriched over itself, and this fact is exploited for studying enriched presheaves. The converse does not ... • 18.2k 9 votes Accepted ### Adjoints between the category of sets and the category of left G-sets. The forgetful functor from G-sets to sets has adjoints on both sides. The left adjoint sends any set A to the product G\times A with the G-action defined by g(h,a)=(gh,a) (for all g,h\in G... • 72.6k 9 votes ### Let C,D be categories and F:C\to D and G:D\to C be adjoint functors. Then F is fully faithful iff the unit is an isomorphism? It's actually possible to prove something a bit more general : F is faithful if and only if every component of \eta  is a monomorphism. F is full if and only if every component of \eta is a ... • 20.9k 9 votes ### Show that a functor which preserves colimits has a right adjoint This is false in general without further hypotheses; see adjoint functor theorem. The reference to CWM, as you say, is only a reference for the uniqueness. The various adjoint functor theorems do ... • 421k 9 votes ### Commuting right adjoints implies commuting left adjoints. There are two results that piece together to give you your answer. Theorem 1. If \mathcal{C} \overset{F}{\underset{G}{\rightleftarrows}} \mathcal{D} \overset{K}{\underset{H}{\rightleftarrows}} \... • 65.1k 9 votes Accepted ### What exactly are the `size issues' preventing formation of presheaves being a left adjoint to some forgetful functor? The problem is that you cannot choose a domain and codomain for such a putative adjunction consistently and simultaneously. The statement that we have is that the category of presheaves on C is the ... • 52.6k 9 votes Accepted ### Right adjoint is fully faithful iff the counit is an isomorphism (without Yoneda) This result uses the naturalness of the map$$ \Phi \colon \mathrm{Hom}(A,GB) \xrightarrow\sim \mathrm{Hom}(FA,B).  In particular, if $f\colon A\to GB$ and $g\colon B\to B’$, then $\Phi(G(g)f)=g\Phi(... • 5,616 9 votes Accepted ### Explicit description of a left adjoint to the forgetful functor from Unif to Top It does not follow from the existence of a left adjoint$F\colon \mathsf{Top}\to \mathsf{Unif}$that every topological space has a canonical uniformity compatible with its topology. There's no reason ... • 77.8k 9 votes Accepted ### Bifunctoriality stronger than functoriality in each variable? Yes, bifunctoriality is a stronger condition than functoriality in each variable, and yes, the commutative diagram you write down need not commute in the second case. I don't actually know an example ... • 421k 9 votes ### Left adjoint to the forgetful functor from Hilbert spaces to topological spaces The left adjoint does not exist. Otherwise, also the forgetful functor$\mathbf{Hilb} \to \mathbf{Top} \to \mathbf{Set}$to the category of sets had a left adjoint (since left adjoints "compose&... 8 votes Accepted ### What can be said about adjunctions between groups (regarded as one-object categories)? Since groups as one-object categories are a special case of groupoids, that is, categories in which every morphism is an isomorphism, for any adjunction$F\dashv H\colon\mathcal G_1\to\mathcal G_2$... • 15.6k 8 votes Accepted ### Uniqueness of Left Adjoint The left adjoint is unique up to natural isomorphism. To see this, let$G : \mathcal{D} \to \mathcal{C}$be a functor and suppose$F,F' : \mathcal{C} \to \mathcal{D}$with are two left adjoints for$G$... • 65.1k 8 votes Accepted ### Is the free group functor an isomorphism? OK, so you can do that to convert a set to a group. Now take that group and apply the forgetful functor: what set do you get? In particular, if$S = \{a\}$, then$G$looks like the group$\Bbb Z$. ... • 94k 8 votes Accepted ### Does the forgetful functor$U: \mathbf{Gph} \to \mathbf{Set}$have a left adjoint? Whenever I say graph in this answer, I mean simple graph. We need to be a bit more precise here, and specify what the arrows are in$\mathbf{Gph}$. A natural choice would be maps$f: G \to G'\$ ...
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