plane geometry - Does the intersection of two moving curves sweep out a continuous line?

ct.class idea – This is just not a tensor: tensoring abelian teams over teams Answer

Hello pricey customer to our community We will proffer you an answer to this query ct.class idea – This is just not a tensor: tensoring abelian teams over teams ,and the respond will breathe typical via documented info sources, We welcome you and proffer you fresh questions and solutions, Many customer are questioning in regards to the respond to this query.

ct.class idea – This is just not a tensor: tensoring abelian teams over teams

$ newcommand { Cats} { mathsf {Cats}} newcommand { MonCats} { mathsf {MonCats}} newcommand { BrMonCats} { mathsf {BrMonCats}} newcommand { SymMonCats} { mathsf { SymMonCats}} newcommand { CMon} { mathsf {CMon}} newcommand { Mon} { mathsf {Mon}} newcommand { Z} { mathbb {Z}} newcommand { Ab} { mathsf {Ab}} newcommand { Grp} { mathsf {Grp}} newcommand { Sets} { mathsf {Sets}} newcommand { eHom} { mathbf {Hom}} newcommand { C} { mathcal {C}} newcommand { V} { mathcal {V}} newcommand { Obj} { mathrm {Obj}} $Permit $ C $ breathe a $ V $-enriched class, let $ V and Obj ( V) $, and let $ A in Obj ( C) $.

  • the Tensor of $ V $ with $ A $ (too known as the come from $ V $ with $ A $) is the objective, if it exists $ V level A $ from $ C $ in order that we a. to have $ V $-natural isomorphism
    $$ eHom_ C (V odot A, -) cong eHom_ V (V, eHom_ C (A, -)).
  • Dual that Cotensor of $ V $ with $ A $ (too known as the energy of $ V $ with $ A $) is the objective, if it exists $ V pitchfork A $ from $ C $ in order that we a. to have $ V $-natural isomorphism
    $$ eHom_ C (-, V Pitchfork A) cong eHom_ V (V, eHom_ C (-, A)).

Aside from that, $ C $ is known as $ V $-Co / Tensored if it has all co / tensors.

An instance of that is any co / full class $ mathcal {C} $, whose $ Sentences $-Co / tensors are given by
commence {align *} X odot A & cong coprod_ {x in X} A, X Pitchfork A & cong prod_ {x in X} A. aim {align *}
Another instance is the class $ Ab $:

  • $ Ab $ is enriched about itself: given $ A, B in Obj ( Ab) $, we now have an Abelian group $ eHom_ Ab (A, B) $ whose product $ (f, g) mapsto f * g $ is obtained by point-wise multiplication, i.e. by definition $[f*g](a) = f (a) g (a) $. This relies upon crucially on the commutativity of $ B $that ensures that $ f * g $ is once more a morphism of teams:
    commence {align *}
    [f*g](ab) & = f (ab) g (ab) & = f (a) coloration {crimson} {f (b)} coloration {blue} {g (a)} g (b) & = f (a) coloration {blue} {g (a)} coloration {crimson} {f (b)} g (b) & = [f*g](a)[f*g](B). aim {align *}
  • $ Ab $ is tensored over itself through the tensor product of Abelian teams $ (A, B) mapsto A otimes_ mathbb {Z} B $;
  • $ Ab $ is cotensored through the interior $ eHom $ talked about above, $ (A, B) mapsto eHom_ Ab (A, B) $.

Now, $ Ab $ is just not enriched $ Grp $, since there isn’t any significant tensor product for the latter; nonetheless, it’s “faux co / tensored” about it since we now have isomorphisms
commence {align *} eHom_ Ab (G ^ mathrm {ab} otimes_ ZA, B) & cong eHom_ Grp (G, eHom_ Ab (A, B)), eHom_ Ab (A, eHom_ Grp (G, B)) & cong eHom_ Grp (G, eHom_ Ab (A, B)), aim {align *}
so $ G ” odot” A = G ^ mathrm {ab} otimes_ ZA $ and $ G “ Pitchfork“ A = eHom_ Grp (G, B) $.

This status is just not unique to $ Grp $ and $ Ab $: it too happens with $ Mon $ and $ CMon $, with $ BrMonCats $ and $ SymMonCats $, and appears extra common to pairs of the figure. to happen $ ( Mon ( C), CMon ( C)) $ to the $ C $ a symmetric monoid class.


A second associated level is you could get the $ Sentences $-Co / tensoring of $ Grp $ together with the unmindful functor $ | {-} | colon Grp to Sets $ to $ Sentences $ Define “half-tensor products” $ triangleleft $ and $ triangleright $, given by
commence {align *} G triangleleft H & = | H | odot G, & cong coprod_ {h in H} G, G triangleright H & = | G | odot H, & cong coprod_ {g in G} H. aim {align *}
As talked about right here, $ G triangle left H $ is the free group on symbols $ a otimes b $ Quotient via the left distributivity relations $ (a + b) otimes c sim a otimes c + b otimes c $, and comparable for $ triangleright $.

Because of this final level, whereas monoids in $ ( Ab, otimes_ Z, Z) $ are rings which might be “monoids” in $ ( Grp, triangleleft,?) $ ought to breathe Almost ringswrestle with not essentially commutative addition and solely the left distributive regulation. The drawback, nonetheless, is that $ triangleleft $ there isn’t any $ Grp $ a monoid class construction: it appears at greatest a variant of the conception of a “neglectful monoid class” on it.

As with the disloyal co / tensors above, this kind of tensor product seems to emerge in lots of different contexts, together with $ MonCats $ with $ Cats $-Tensors or possibly $ BrMonCats $ with “wrong” $ MonCats $-Tensors “.


This is a quite peculiar status: we now have these very pure “faux-co / tensors”, however the habitual class theoretic phrases do not fairly acquire them. About Zulip, Reid Barton steered grouping the classes $ Sentences $, $ Grp $, $ Ab $, $ Ab $, $ ldots $ ($ = ( Grp _ { mathbb {E} _n} ( portions)) _ {n in mathbb {N}} $) In a “$ mathbb {N} $-graded monoidal category “, however right here too the models and associates appear to breathe problematic …

So — in a nutshell — what precisely is occurring right here?

  • How ought to we get these “faux co / tensors” from. behold? $ My _ { mathbb {E} _ {n}} ( C) $ from $ Mon _ { mathbb {E} _ { leq n-1}} ( C) $?
  • What precisely are the “false mono-idal structures” $ triangleleft $ and $ triangleright $ At $ Grp $whose monoids are conjectural to revive near-rings?

we are going to proffer you the answer to ct.class idea – This is just not a tensor: tensoring abelian teams over teams query through our community which brings all of the solutions from a number of dependable sources.