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

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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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