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## 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?

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