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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 24th 2012
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   Author: Urs
   Format: MarkdownItex_[[tensor product of algebras]]_

   tensor product of algebras

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeSep 27th 2012
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   Author: TobyBartels
   Format: MarkdownItexTensor product is coproduct only for *commutative* algebras.

   Tensor product is coproduct only for commutative algebras.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeAug 20th 2020
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   Author: Urs
   Format: MarkdownItexAfter the statement that pushout = tensor product in $CRing$, i have added pointer to _[category of monoids -- pushouts -- of commutative monoids](https://ncatlab.org/nlab/show/category+of+monoids#PushoutOfCommutativeMonoids)_ <a href="https://ncatlab.org/nlab/revision/diff/tensor+product+of+algebras/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/tensor+product+of+algebras/6">v6</a>, <a href="https://ncatlab.org/nlab/show/tensor+product+of+algebras">current</a>

   After the statement that pushout = tensor product in , i have added pointer to category of monoids – pushouts – of commutative monoids

   diff, v6, current

4. • CommentRowNumber4.
   • CommentAuthorRodMcGuire
   • CommentTimeAug 23rd 2020
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   Author: RodMcGuire
   Format: MarkdownItexI'm not sure why Urs used this strange link syntax that doesn't work. I tried to fix it and make it more standard changing `See at _[pushouts of commutative monoids](category+of+monoids#PushoutOfCommutativeMonoids)_` to `See at _[[category of monoids#PushoutOfCommutativeMonoids|pushouts of commutative monoids]]_` <a href="https://ncatlab.org/nlab/revision/diff/tensor+product+of+algebras/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/tensor+product+of+algebras/7">v7</a>, <a href="https://ncatlab.org/nlab/show/tensor+product+of+algebras">current</a>

   I’m not sure why Urs used this strange link syntax that doesn’t work. I tried to fix it and make it more standard changing

   See at _[pushouts of commutative monoids](category+of+monoids#PushoutOfCommutativeMonoids)_

   to

   See at _[[category of monoids#PushoutOfCommutativeMonoids|pushouts of commutative monoids]]_

   diff, v7, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeAug 24th 2020
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   Author: Urs
   Format: MarkdownItexThanks for noticing that my link didn't work. I wonder why: I use the syntax all the time (maybe it dates from a time when the other syntax wasn't available yet), and it seems I didn't have a typo in it. Checking what happens when I paste my code into the _[[Sandbox]]_... Ah, there it works. (!?)

   Thanks for noticing that my link didn’t work. I wonder why: I use the syntax all the time (maybe it dates from a time when the other syntax wasn’t available yet), and it seems I didn’t have a typo in it.

   Checking what happens when I paste my code into the Sandbox…

   Ah, there it works. (!?)

6. • CommentRowNumber6.
   • CommentAuthorGeoffVooys
   • CommentTimeDec 10th 2024
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   Author: GeoffVooys
   Format: MarkdownItexI'm a little worried that the definition of $R \downarrow \mathbf{Rig}$ or $R \downarrow \mathbf{Ring}$ does not necessarily capture the correct category of $R$-algebras (in the sense that for a commutative object $A$ and an object $R$ with a map $f:A \to R$, the map determines an $A$-algebra structure if and only if it factors through the centre $Z(R)$ of $R$). Here's an explicit example of what can go wrong: if you let $A = \mathbb{C}$ and $\sigma:\mathbb{C} \to \mathbb{C}$ denote complex conjugation then the ring of twisted polynomials $\mathbb{C}[x,\sigma]$ is a ring with a map $\mathbb{C} \to \mathbb{C}[x,\sigma]$ but which does not factor through $Z(\mathbb{C}[x,\sigma])$ because $Z(\mathbb{C}[x,\sigma]) = \mathbb{R}[x^2]$. Alternatively and more explicitly, $i\cdot x = i\cdot x$ but $ x\cdot i= \overline{\imath}x = -i\cdot x$. This suggests that in this case $\mathbb{C} \downarrow \mathbf{Ring} \ne \mathbb{C}\mathbf{Alg}$, or at least not in the ``usual'' sense. Did I misinterpret the sense in which algebra is meant on this page? Certainly taking the coslice category of the commutative object over the base is a nice generalization.

   I’m a little worried that the definition of or does not necessarily capture the correct category of -algebras (in the sense that for a commutative object and an object with a map , the map determines an -algebra structure if and only if it factors through the centre of ).

   Here’s an explicit example of what can go wrong: if you let and denote complex conjugation then the ring of twisted polynomials is a ring with a map but which does not factor through because . Alternatively and more explicitly, but . This suggests that in this case , or at least not in the “usual” sense.

   Did I misinterpret the sense in which algebra is meant on this page? Certainly taking the coslice category of the commutative object over the base is a nice generalization.