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More Information: <a href="http://lussumo.com/docs">Documentation</a>, <a href="http://lussumo.com/community">Community Support</a>.</p></div> <div id="Content"><div id="NoticeCollector" class="Notices"><div class="Notice"><strong>Welcome to nForum</strong> <br />If you want to take part in these discussions either <a href="/people.php?ReturnUrl=http%3A%2F%2Fnforum.ncatlab.org%2Fdiscussion%2F2396%2Ffrobenius-reciprocity%2F">sign in now</a> (if you have an account), <a href="https://nforum.ncatlab.org/people/?PostBackAction=ApplyForm">apply for one now</a> (if you don't).</div></div><div class="ContentInfo Top"> <h1><a href="https://nforum.ncatlab.org/5/"><a href="https://nforum.ncatlab.org/18/">nLab</a> > </a> <a href="https://nforum.ncatlab.org/5/">Latest Changes</a>: Frobenius reciprocity</h1> <a href="#pgbottom">Bottom of Page</a> <div class="PageInfo"> <p>1 to 28 of 28</p> <ol class="PageList PageListEmpty"> <li> </li> </ol> </div> </div> <div id="ContentBody"> <script type="text/javascript"> //<![CDATA[ function toggle_source(id) { var mysrc = document.getElementById("CommentBody_" + id).firstChild; if (mysrc.className == "source") { if (mysrc.style.display == "none") { mysrc.style.display = "block"; } else { mysrc.style.display = "none"; } } } var commentIds = new Array(0); function hide_sources() { for (i = 0; i < commentIds.length; i++) { var myself = document.getElementById("Source" + commentIds[i]); var mycmt = document.getElementById("CommentBody_" + commentIds[i]); if (mycmt.firstChild.className != "source") { myself.style.display = "none"; } } } window.onload = hide_sources; //]]> </script> <ol id="Comments"><li id="Comment_20405"> <a id="Item_1"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>1.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/10/">zskoda</a></li> <li><span>CommentTime</span>Jan 12th 2011</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=20405#Comment_20405">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_20405"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/10/">zskoda</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Stub [[Frobenius reciprocity]].</code></div><div> <p>Stub <a href="https://ncatlab.org/nlab/show/Frobenius reciprocity">Frobenius reciprocity</a>.</p> </div> </div> </li><li id="Comment_21123" class="Alternate"> <a id="Item_2"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>2.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/31/">FinnLawler</a></li> <li><span>CommentTime</span>Jan 31st 2011</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=21123#Comment_21123">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_21123"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/31/">FinnLawler</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Added Lawvere's Frobenius condition for hyperdoctrines to [[Frobenius reciprocity]].</code></div><div> <p>Added Lawvere’s Frobenius condition for hyperdoctrines to <a href="https://ncatlab.org/nlab/show/Frobenius reciprocity">Frobenius reciprocity</a>.</p> </div> </div> </li><li id="Comment_21125"> <a id="Item_3"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>3.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/3/">Mike Shulman</a></li> <li><span>CommentTime</span>Jan 31st 2011</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=21125#Comment_21125">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_21125"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/3/">Mike Shulman</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Thanks. I added the topos-theory usage. And I'm pretty sure that it's about cartesian closedness of the *inverse* image functor, not the existential quantification, so I fixed that.</code></div><div> <p>Thanks. I added the topos-theory usage. And I’m pretty sure that it’s about cartesian closedness of the <em>inverse</em> image functor, not the existential quantification, so I fixed that.</p> </div> </div> </li><li id="Comment_21126" class="Alternate"> <a id="Item_4"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>4.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/4/">Urs</a></li> <li><span>CommentTime</span>Jan 31st 2011</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=21126#Comment_21126">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_21126"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/4/">Urs</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>It would be nice to have a comment on how the section "In representation theory" relates to that "In category theory" if it does, or else a warning that it does not. Hm, does it?</code></div><div> <p>It would be nice to have a comment on how the section “In representation theory” relates to that “In category theory” if it does, or else a warning that it does not. Hm, does it?</p> </div> </div> </li><li id="Comment_21147"> <a id="Item_5"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>5.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/31/">FinnLawler</a></li> <li><span>CommentTime</span>Feb 1st 2011</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=21147#Comment_21147">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_21147"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/31/">FinnLawler</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Mike wrote > I'm pretty sure that it's about cartesian closedness of the inverse image functor, not the existential quantification, so I fixed that. Argh, yes, you're quite right. I don't know how I managed to write that. I don't know any representation theory, but from scanning the Wikipedia page on induced representations it looks as though Frobenius reciprocity in that context means the adjunction $i_! \dashv i^*$ coming from an inclusion $i \colon H \hookrightarrow G$ of a subgroup, in the indexed category $G \mapsto k \text{-} Vect^G$. But I won't add this until we hear from someone who knows what they're talking about...</code></div><div> <p>Mike wrote</p> <blockquote> <p>I’m pretty sure that it’s about cartesian closedness of the inverse image functor, not the existential quantification, so I fixed that.</p> </blockquote> <p>Argh, yes, you’re quite right. I don’t know how I managed to write that.</p> <p>I don’t know any representation theory, but from scanning the Wikipedia page on induced representations it looks as though Frobenius reciprocity in that context means the adjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>i</mi> <mo>!</mo></msub><mo>&dashv;</mo><msup><mi>i</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">i_! \dashv i^*</annotation></semantics></math> coming from an inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo lspace="0.11111em">&colon;</mo><mi>H</mi><mo>&hookrightarrow;</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">i \colon H \hookrightarrow G</annotation></semantics></math> of a subgroup, in the indexed category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>&map;</mo><mi>k</mi><mtext>-</mtext><msup><mi>Vect</mi> <mi>G</mi></msup></mrow><annotation encoding="application/x-tex">G \mapsto k \text{-} Vect^G</annotation></semantics></math>. But I won’t add this until we hear from someone who knows what they’re talking about…</p> </div> </div> </li><li id="Comment_21521" class="Alternate"> <a id="Item_6"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>6.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/31/">FinnLawler</a></li> <li><span>CommentTime</span>Feb 22nd 2011</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=21521#Comment_21521">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_21521"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/31/">FinnLawler</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Some minor edits to [[Frobenius reciprocity]]. What I meant by my previous comment (I don't think it was clear) is that it seems that in representation theory the term 'Frobenius reciprocity' refers to the _existence_ of the induction-restriction adjunctions; but in category theory (especially with hyperdoctrines) it refers to a _property_ of the analogous adjunctions. So I don't know why Lawvere chose the term.</code></div><div> <p>Some minor edits to <a href="https://ncatlab.org/nlab/show/Frobenius reciprocity">Frobenius reciprocity</a>.</p> <p>What I meant by my previous comment (I don’t think it was clear) is that it seems that in representation theory the term ’Frobenius reciprocity’ refers to the <em>existence</em> of the induction-restriction adjunctions; but in category theory (especially with hyperdoctrines) it refers to a <em>property</em> of the analogous adjunctions. So I don’t know why Lawvere chose the term.</p> </div> </div> </li><li id="Comment_21523"> <a id="Item_7"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>7.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/6/">Tim_Porter</a></li> <li><span>CommentTime</span>Feb 22nd 2011</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=21523#Comment_21523">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_21523"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/6/">Tim_Porter</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Slightly tangentially, I have been playing around with pullback/pushforward and 'transfer' as discussed by Turaev in his HQFT book. If one has a group homomorphism $f: G\to H$ you can pullback crossed $H$-algebras to crossed $G$-algebras, and provided $Ker f$ is finite, you can push them forward. If $f$ is a cofinite inclusion you can also do an induction or transfer process that is left adjoint to pullback. (This is the analogue of the restriction/induction adjoint pair of representation theory, but because it is not simply representations in Vect, the usual Frobenius reciprocity situation does not hold and the pushforward and induction processes do not coincide. Has anyone seen anything like this before?</code></div><div> <p>Slightly tangentially, I have been playing around with pullback/pushforward and ’transfer’ as discussed by Turaev in his HQFT book. If one has a group homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mi>G</mi><mo>&rightarrow;</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">f: G\to H</annotation></semantics></math> you can pullback crossed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>-algebras to crossed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-algebras, and provided <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Ker</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">Ker f</annotation></semantics></math> is finite, you can push them forward. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a cofinite inclusion you can also do an induction or transfer process that is left adjoint to pullback. (This is the analogue of the restriction/induction adjoint pair of representation theory, but because it is not simply representations in Vect, the usual Frobenius reciprocity situation does not hold and the pushforward and induction processes do not coincide. Has anyone seen anything like this before?</p> </div> </div> </li><li id="Comment_21527" class="Alternate"> <a id="Item_8"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>8.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/3/">Mike Shulman</a></li> <li><span>CommentTime</span>Feb 22nd 2011</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=21527#Comment_21527">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_21527"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/3/">Mike Shulman</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Is Lawvere's the original usage of the term in category theory? I wonder if this would be a good question for the categories email list. I would really like to know why that condition is called "Frobenius".</code></div><div> <p>Is Lawvere’s the original usage of the term in category theory?</p> <p>I wonder if this would be a good question for the categories email list. I would really like to know why that condition is called “Frobenius”.</p> </div> </div> </li><li id="Comment_21539"> <a id="Item_9"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>9.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/31/">FinnLawler</a></li> <li><span>CommentTime</span>Feb 23rd 2011</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=21539#Comment_21539">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_21539"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/31/">FinnLawler</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>I had assumed, though for no compelling reason, that the term was Lawvere's. He introduced hyperdoctrines in _Adjointness in foundations_, I think, and in _Equality in hyperdoctrines_ ([here](http://books.google.ie/books?id=3Lyvsc694T4C&pg=PA1&lpg=PA1&dq=lawvere+%22equality+in+hyperdoctrines%22&source=bl&ots=7OCKiQAuSm&sig=8KyAEaANR0GQ43WD5MFCpPR8g44&hl=en&ei=1x1BS6yRJ87RjAfMwoCtDQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CAoQ6AEwAA#v=onepage&q&f=false)), not long afterwards, he uses the term 'Frobenius reciprocity' without giving any source. Anything else I could say would be a guess, though. I too would like to know what the reasoning behind the name was.</code></div><div> <p>I had assumed, though for no compelling reason, that the term was Lawvere’s. He introduced hyperdoctrines in <em>Adjointness in foundations</em>, I think, and in <em>Equality in hyperdoctrines</em> (<a href="http://books.google.ie/books?id=3Lyvsc694T4C&pg=PA1&lpg=PA1&dq=lawvere+%22equality+in+hyperdoctrines%22&source=bl&ots=7OCKiQAuSm&sig=8KyAEaANR0GQ43WD5MFCpPR8g44&hl=en&ei=1x1BS6yRJ87RjAfMwoCtDQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CAoQ6AEwAA#v=onepage&q&f=false">here</a>), not long afterwards, he uses the term ’Frobenius reciprocity’ without giving any source. Anything else I could say would be a guess, though. I too would like to know what the reasoning behind the name was.</p> </div> </div> </li><li id="Comment_25083" class="Alternate"> <a id="Item_10"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>10.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/24/">Todd_Trimble</a></li> <li><span>CommentTime</span>Aug 11th 2011</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=25083#Comment_25083">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_25083"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/24/">Todd_Trimble</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Mike Shulman has kindly uploaded a file I sent him, consisting of handwritten string diagram calculations for the proof of the second proposition (bottom of the page) of [[Frobenius reciprocity]]. I don't know what the history is behind either use of 'Frobenius' (either as in Frobenius algebra or Frobenius reciprocity). But I have removed some discussion that had appeared on that page: > The relationship between the two usages is not clear. In fact, since the category-theoretic usage is a special case of being a [[Hopf adjunction]], it seems as though it might have been misnamed, since [[Frobenius algebras]] and [[Hopf algebras]] are similar, but different. > The word "Frobenius" is also sometimes used in category theory to denote a condition which is in some way analogous to the characteristic property of a [[Frobenius algebra]]. because the two propositions taken together indicate to me a reasonable relationship between the two usages. Whether such a relationship was in Lawvere's mind when he wrote Adjointness in Foundations, I have no idea. I'd like to know what he was thinking.</code></div><div> <p>Mike Shulman has kindly uploaded a file I sent him, consisting of handwritten string diagram calculations for the proof of the second proposition (bottom of the page) of <a href="https://ncatlab.org/nlab/show/Frobenius reciprocity">Frobenius reciprocity</a>.</p> <p>I don’t know what the history is behind either use of ’Frobenius’ (either as in Frobenius algebra or Frobenius reciprocity). But I have removed some discussion that had appeared on that page:</p> <blockquote> <p>The relationship between the two usages is not clear. In fact, since the category-theoretic usage is a special case of being a <a href="https://ncatlab.org/nlab/show/Hopf adjunction">Hopf adjunction</a>, it seems as though it might have been misnamed, since <a href="https://ncatlab.org/nlab/show/Frobenius algebras">Frobenius algebras</a> and <a href="https://ncatlab.org/nlab/show/Hopf algebras">Hopf algebras</a> are similar, but different. The word “Frobenius” is also sometimes used in category theory to denote a condition which is in some way analogous to the characteristic property of a <a href="https://ncatlab.org/nlab/show/Frobenius algebra">Frobenius algebra</a>.</p> </blockquote> <p>because the two propositions taken together indicate to me a reasonable relationship between the two usages. Whether such a relationship was in Lawvere’s mind when he wrote Adjointness in Foundations, I have no idea. I’d like to know what he was thinking.</p> </div> </div> </li><li id="Comment_25092"> <a id="Item_11"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>11.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/3/">Mike Shulman</a></li> <li><span>CommentTime</span>Aug 12th 2011</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=25092#Comment_25092">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_25092"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/3/">Mike Shulman</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Thanks! If Lawvere did know of this relationship at some level (which seems conceivable), it might finally actually explain the use of "Frobenius" for $f_!(C\times f^*B)\cong (f_!C)\times B$ as originating from Frobenius algebras (and having nothing to do with the representation-theoretic "Frobenius reciprocity").</code></div><div> <p>Thanks! If Lawvere did know of this relationship at some level (which seems conceivable), it might finally actually explain the use of “Frobenius” for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>f</mi> <mo>!</mo></msub><mo stretchy="false">(</mo><mi>C</mi><mo>×</mo><msup><mi>f</mi> <mo>*</mo></msup><mi>B</mi><mo stretchy="false">)</mo><mo>&cong;</mo><mo stretchy="false">(</mo><msub><mi>f</mi> <mo>!</mo></msub><mi>C</mi><mo stretchy="false">)</mo><mo>×</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f_!(C\times f^*B)\cong (f_!C)\times B</annotation></semantics></math> as originating from Frobenius algebras (and having nothing to do with the representation-theoretic “Frobenius reciprocity”).</p> </div> </div> </li><li id="Comment_25093" class="Alternate"> <a id="Item_12"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>12.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/192/">hilbertthm90</a></li> <li><span>CommentTime</span>Aug 12th 2011</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=25093#Comment_25093">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_25093"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/192/">hilbertthm90</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>When I see $f_{!}(C\times f^*B)\simeq (f_{!}C)\times B$ the first thing that jumps into my mind is something along the lines of the Grothendieck-Riemann-Roch theorem. This type of thing keeps appearing for me while thinking about Fourier-Mukai transforms on different cohomology theories.</code></div><div> <p>When I see <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>f</mi> <mo>!</mo></msub><mo stretchy="false">(</mo><mi>C</mi><mo>×</mo><msup><mi>f</mi> <mo>*</mo></msup><mi>B</mi><mo stretchy="false">)</mo><mo>&simeq;</mo><mo stretchy="false">(</mo><msub><mi>f</mi> <mo>!</mo></msub><mi>C</mi><mo stretchy="false">)</mo><mo>×</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f_{!}(C\times f^*B)\simeq (f_{!}C)\times B</annotation></semantics></math> the first thing that jumps into my mind is something along the lines of the Grothendieck-Riemann-Roch theorem. This type of thing keeps appearing for me while thinking about Fourier-Mukai transforms on different cohomology theories.</p> </div> </div> </li><li id="Comment_25097"> <a id="Item_13"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>13.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/20/">David_Corfield</a></li> <li><span>CommentTime</span>Aug 12th 2011</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=25097#Comment_25097">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_25097"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/20/">David_Corfield</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>There's a MathOverflow [question](http://mathoverflow.net/questions/18799/ubiquity-of-the-push-pull-formula) about formulas of that type.</code></div><div> <p>There’s a MathOverflow <a href="http://mathoverflow.net/questions/18799/ubiquity-of-the-push-pull-formula">question</a> about formulas of that type.</p> </div> </div> </li><li id="Comment_25105" class="Alternate"> <a id="Item_14"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>14.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/10/">zskoda</a></li> <li><span>CommentTime</span>Aug 12th 2011</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=25105#Comment_25105">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_25105"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/10/">zskoda</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>As far as Frobenius terminology, may I add to the confusion by bringing up the question of relation of the notions of quasi-Frobenius and pseudo-Frobenius rings (both notions are e.g. in Carl Faith's 2-volume Algebra, mainly vol. 2).</code></div><div> <p>As far as Frobenius terminology, may I add to the confusion by bringing up the question of relation of the notions of quasi-Frobenius and pseudo-Frobenius rings (both notions are e.g. in Carl Faith’s 2-volume Algebra, mainly vol. 2).</p> </div> </div> </li><li id="Comment_25113"> <a id="Item_15"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>15.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/24/">Todd_Trimble</a></li> <li><span>CommentTime</span>Aug 12th 2011</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=25113#Comment_25113">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_25113"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/24/">Todd_Trimble</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Zoran, I don't have that book and I'm nowhere near a university library. Can you tell us what those notions are exactly? Do they have much to do with Frobenius algebras?</code></div><div> <p>Zoran, I don’t have that book and I’m nowhere near a university library. Can you tell us what those notions are exactly? Do they have much to do with Frobenius algebras?</p> </div> </div> </li><li id="Comment_43892" class="Alternate"> <a id="Item_16"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>16.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/4/">Urs</a></li> <li><span>CommentTime</span>Dec 9th 2013</li><li><em>(edited Dec 9th 2013)</em></li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=43892#Comment_43892">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_43892"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/4/">Urs</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Let me come back to the (old) question discussed above on what exactly the relation between the representation theoretic and the hyperdoctrinal use of "Frobenius reciprocity" is and how it is related to Fourier-Mukai transforms. So it is true that most texts on representation theory say that Frobenius reciprocity means just the existence of the adjunction $(f_! \dashv f^\ast) = (Ind \dashv Res)$. An exception is [the PlanetMath page](http://planetmath.org/frobeniusreciprocity) which claims without further ado that this is equivalent to the projection formula $$ Ind(Res(W) \otimes V) \simeq W \otimes Ind(V) \,. $$ Indeed what is true is that with an adjunction $(f_! \dashv f^\ast)$ between closed monoidal categories given, then this projection formula is equivalent to $f^\ast$ being a strong closed functor. This is one of the basic statements highlighted in May's [Isomorphisms between left and right adjoints](http://www.tac.mta.ca/tac/volumes/11/4/11-04abs.html), some of which is extracted a bit at _[[Wirthmüller context]]_. And that then also clarifies the relation to Fourier-Mukai-type transforms mentioned in #12 above, via some discussion as in _[Abstract integral transforms](https://dl.dropboxusercontent.com/u/12630719/integral.pdf)_ I've added some brief cross-links to these entries now, accordingly, and in particular added some comments to _[[Frobenius reciprocity]]_. But I don't have time today to do this justice.</code></div><div> <p>Let me come back to the (old) question discussed above on what exactly the relation between the representation theoretic and the hyperdoctrinal use of “Frobenius reciprocity” is and how it is related to Fourier-Mukai transforms.</p> <p>So it is true that most texts on representation theory say that Frobenius reciprocity means just the existence of the adjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mo>!</mo></msub><mo>&dashv;</mo><msup><mi>f</mi> <mo>&ast;</mo></msup><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>Ind</mi><mo>&dashv;</mo><mi>Res</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f_! \dashv f^\ast) = (Ind \dashv Res)</annotation></semantics></math>. An exception is <a href="http://planetmath.org/frobeniusreciprocity">the PlanetMath page</a> which claims without further ado that this is equivalent to the projection formula</p> <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>Ind</mi><mo stretchy="false">(</mo><mi>Res</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">)</mo><mo>&otimes;</mo><mi>V</mi><mo stretchy="false">)</mo><mo>&simeq;</mo><mi>W</mi><mo>&otimes;</mo><mi>Ind</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mspace width="0.16667em"/><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ind(Res(W) \otimes V) \simeq W \otimes Ind(V) \,. </annotation></semantics></math> <p>Indeed what is true is that with an adjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mo>!</mo></msub><mo>&dashv;</mo><msup><mi>f</mi> <mo>&ast;</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f_! \dashv f^\ast)</annotation></semantics></math> between closed monoidal categories given, then this projection formula is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>f</mi> <mo>&ast;</mo></msup></mrow><annotation encoding="application/x-tex">f^\ast</annotation></semantics></math> being a strong closed functor. This is one of the basic statements highlighted in May’s <a href="http://www.tac.mta.ca/tac/volumes/11/4/11-04abs.html">Isomorphisms between left and right adjoints</a>, some of which is extracted a bit at <em><a href="https://ncatlab.org/nlab/show/Wirthmüller context">Wirthmüller context</a></em>.</p> <p>And that then also clarifies the relation to Fourier-Mukai-type transforms mentioned in #12 above, via some discussion as in <em><a href="https://dl.dropboxusercontent.com/u/12630719/integral.pdf">Abstract integral transforms</a></em></p> <p>I’ve added some brief cross-links to these entries now, accordingly, and in particular added some comments to <em><a href="https://ncatlab.org/nlab/show/Frobenius reciprocity">Frobenius reciprocity</a></em>. But I don’t have time today to do this justice.</p> </div> </div> </li><li id="Comment_54596"> <a id="Item_17"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>17.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/1141/">Noam_Zeilberger</a></li> <li><span>CommentTime</span>Sep 12th 2015</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=54596#Comment_54596">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_54596"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/1141/">Noam_Zeilberger</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Just a small comment: the term itself was introduced by Lawvere in the "Equality in hyperdoctrines" paper, not yet in "Adjointness in foundations". I've updated the reference accordingly.</code></div><div> <p>Just a small comment: the term itself was introduced by Lawvere in the “Equality in hyperdoctrines” paper, not yet in “Adjointness in foundations”. I’ve updated the reference accordingly.</p> </div> </div> </li><li id="Comment_86480" class="Alternate"> <a id="Item_18"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>18.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/17/">John Baez</a></li> <li><span>CommentTime</span>Aug 27th 2020</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=86480#Comment_86480">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_86480"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/17/">John Baez</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Eliminated "sometimes Frobenious" - this spelling is rare enough (isn't it just a misspelling?) that I don't think we need bother people with it in the first sentence (and perhaps propagate it). I'm doing some other small typo fixes on this page. <a href="https://ncatlab.org/nlab/revision/diff/Frobenius+reciprocity/40">diff</a>, <a href="https://ncatlab.org/nlab/revision/Frobenius+reciprocity/40">v40</a>, <a href="https://ncatlab.org/nlab/show/Frobenius+reciprocity">current</a></code></div><div> <p>Eliminated “sometimes Frobenious” - this spelling is rare enough (isn’t it just a misspelling?) that I don’t think we need bother people with it in the first sentence (and perhaps propagate it).</p> <p>I’m doing some other small typo fixes on this page.</p> <p><a href="https://ncatlab.org/nlab/revision/diff/Frobenius+reciprocity/40">diff</a>, <a href="https://ncatlab.org/nlab/revision/Frobenius+reciprocity/40">v40</a>, <a href="https://ncatlab.org/nlab/show/Frobenius+reciprocity">current</a></p> </div> </div> </li><li id="Comment_86482"> <a id="Item_19"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>19.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/17/">John Baez</a></li> <li><span>CommentTime</span>Aug 27th 2020</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=86482#Comment_86482">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_86482"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/17/">John Baez</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>I improved the proof of Proposition 1.1. In the original proof $\pi$ was not explained. A commutative square was drawn without clarifying that one of the sides was, at least in the absence of some other definition of $\pi$, being defined as the composite of the other three. Furthermore hom-tensor adjointness and $f_!$-$f^*$ adjointness were being deployed in a rapid and inexplicit way. I think it's worth having a proof that's really easy to follow. <a href="https://ncatlab.org/nlab/revision/diff/Frobenius+reciprocity/40">diff</a>, <a href="https://ncatlab.org/nlab/revision/Frobenius+reciprocity/40">v40</a>, <a href="https://ncatlab.org/nlab/show/Frobenius+reciprocity">current</a></code></div><div> <p>I improved the proof of Proposition 1.1. In the original proof <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math> was not explained. A commutative square was drawn without clarifying that one of the sides was, at least in the absence of some other definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>, being defined as the composite of the other three. Furthermore hom-tensor adjointness and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>f</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">f_!</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^*</annotation></semantics></math> adjointness were being deployed in a rapid and inexplicit way. I think it’s worth having a proof that’s really easy to follow.</p> <p><a href="https://ncatlab.org/nlab/revision/diff/Frobenius+reciprocity/40">diff</a>, <a href="https://ncatlab.org/nlab/revision/Frobenius+reciprocity/40">v40</a>, <a href="https://ncatlab.org/nlab/show/Frobenius+reciprocity">current</a></p> </div> </div> </li><li id="Comment_86500" class="Alternate"> <a id="Item_20"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>20.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/17/">John Baez</a></li> <li><span>CommentTime</span>Aug 28th 2020</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=86500#Comment_86500">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_86500"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/17/">John Baez</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>I added a second basic result relating strong closed functors to the projection formula. I moved these argument to a separate section from the discussion of Wirthmueller contexts, which is actually more technical. <a href="https://ncatlab.org/nlab/revision/diff/Frobenius+reciprocity/46">diff</a>, <a href="https://ncatlab.org/nlab/revision/Frobenius+reciprocity/46">v46</a>, <a href="https://ncatlab.org/nlab/show/Frobenius+reciprocity">current</a></code></div><div> <p>I added a second basic result relating strong closed functors to the projection formula. I moved these argument to a separate section from the discussion of Wirthmueller contexts, which is actually more technical.</p> <p><a href="https://ncatlab.org/nlab/revision/diff/Frobenius+reciprocity/46">diff</a>, <a href="https://ncatlab.org/nlab/revision/Frobenius+reciprocity/46">v46</a>, <a href="https://ncatlab.org/nlab/show/Frobenius+reciprocity">current</a></p> </div> </div> </li><li id="Comment_86510"> <a id="Item_21"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>21.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/17/">John Baez</a></li> <li><span>CommentTime</span>Aug 28th 2020</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=86510#Comment_86510">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_86510"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/17/">John Baez</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>I compressed the facts about closed monoidal functors and the projection formula into one super-general proposition, and added some context to help people understand it. <a href="https://ncatlab.org/nlab/revision/diff/Frobenius+reciprocity/47">diff</a>, <a href="https://ncatlab.org/nlab/revision/Frobenius+reciprocity/47">v47</a>, <a href="https://ncatlab.org/nlab/show/Frobenius+reciprocity">current</a></code></div><div> <p>I compressed the facts about closed monoidal functors and the projection formula into one super-general proposition, and added some context to help people understand it.</p> <p><a href="https://ncatlab.org/nlab/revision/diff/Frobenius+reciprocity/47">diff</a>, <a href="https://ncatlab.org/nlab/revision/Frobenius+reciprocity/47">v47</a>, <a href="https://ncatlab.org/nlab/show/Frobenius+reciprocity">current</a></p> </div> </div> </li><li id="Comment_100229" class="Alternate"> <a id="Item_22"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>22.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/4/">Urs</a></li> <li><span>CommentTime</span>Jun 21st 2022</li><li><em>(edited Jun 21st 2022)</em></li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=100229#Comment_100229">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_100229"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/4/">Urs</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>checking whether I can point the reader to the section *[In cartesian categories](https://ncatlab.org/nlab/show/Frobenius+reciprocity#InCategoryTheory)* for discussion of Frobenius reciprocity for base change between slices of finitely complete categories, I found I could not. Have now added brief mentioning of the relevant pasting law argument ([here](https://ncatlab.org/nlab/show/Frobenius+reciprocity#InCategoryTheory#PastingLawForLexCategories)) but I am not claiming that there would not be a more thorough edit needed to do justice to this section. <a href="https://ncatlab.org/nlab/revision/diff/Frobenius+reciprocity/52">diff</a>, <a href="https://ncatlab.org/nlab/revision/Frobenius+reciprocity/52">v52</a>, <a href="https://ncatlab.org/nlab/show/Frobenius+reciprocity">current</a></code></div><div> <p>checking whether I can point the reader to the section <em><a href="https://ncatlab.org/nlab/show/Frobenius+reciprocity#InCategoryTheory">In cartesian categories</a></em> for discussion of Frobenius reciprocity for base change between slices of finitely complete categories, I found I could not.</p> <p>Have now added brief mentioning of the relevant pasting law argument (<a href="https://ncatlab.org/nlab/show/Frobenius+reciprocity#InCategoryTheory#PastingLawForLexCategories">here</a>) but I am not claiming that there would not be a more thorough edit needed to do justice to this section.</p> <p><a href="https://ncatlab.org/nlab/revision/diff/Frobenius+reciprocity/52">diff</a>, <a href="https://ncatlab.org/nlab/revision/Frobenius+reciprocity/52">v52</a>, <a href="https://ncatlab.org/nlab/show/Frobenius+reciprocity">current</a></p> </div> </div> </li><li id="Comment_109639"> <a id="Item_23"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>23.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/4/">Urs</a></li> <li><span>CommentTime</span>May 19th 2023</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=109639#Comment_109639">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_109639"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/4/">Urs</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>added pointer to: * [[Robert A. G. Seely]], p. 511 of: *Hyperdoctrines, Natural Deduction and the Beck Condition*, Zeitschr. f. math Logik und Grundlagen d. Math. **29** (1983) 505-542 &lbrack;[pdf](https://www.math.mcgill.ca/seely/ZML/ZML.PDF)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/Frobenius+reciprocity/55">diff</a>, <a href="https://ncatlab.org/nlab/revision/Frobenius+reciprocity/55">v55</a>, <a href="https://ncatlab.org/nlab/show/Frobenius+reciprocity">current</a></code></div><div> <p>added pointer to:</p> <ul> <li><a href="https://ncatlab.org/nlab/show/Robert A. G. Seely">Robert A. G. Seely</a>, p. 511 of: <em>Hyperdoctrines, Natural Deduction and the Beck Condition</em>, Zeitschr. f. math Logik und Grundlagen d. Math. <strong>29</strong> (1983) 505-542 [<a href="https://www.math.mcgill.ca/seely/ZML/ZML.PDF">pdf</a>]</li> </ul> <p><a href="https://ncatlab.org/nlab/revision/diff/Frobenius+reciprocity/55">diff</a>, <a href="https://ncatlab.org/nlab/revision/Frobenius+reciprocity/55">v55</a>, <a href="https://ncatlab.org/nlab/show/Frobenius+reciprocity">current</a></p> </div> </div> </li><li id="Comment_109641" class="Alternate"> <a id="Item_24"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>24.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/4/">Urs</a></li> <li><span>CommentTime</span>May 19th 2023</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=109641#Comment_109641">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_109641"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/4/">Urs</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>and this one: * [[Duško Pavlović]], p. 164 in: *Maps II: Chasing Diagrams in Categorical Proof Theory*, Logic Journal of the IGPL, **4** 2 (1996) 159–194 &lbrack;[doi:10.1093/jigpal/4.2.159](https://doi.org/10.1093/jigpal/4.2.159), [pdf](http://www.isg.rhul.ac.uk/dusko/papers/1996-mapsII-IGPL.pdf)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/Frobenius+reciprocity/55">diff</a>, <a href="https://ncatlab.org/nlab/revision/Frobenius+reciprocity/55">v55</a>, <a href="https://ncatlab.org/nlab/show/Frobenius+reciprocity">current</a></code></div><div> <p>and this one:</p> <ul> <li><a href="https://ncatlab.org/nlab/show/Duško Pavlović">Duško Pavlović</a>, p. 164 in: <em>Maps II: Chasing Diagrams in Categorical Proof Theory</em>, Logic Journal of the IGPL, <strong>4</strong> 2 (1996) 159–194 [<a href="https://doi.org/10.1093/jigpal/4.2.159">doi:10.1093/jigpal/4.2.159</a>, <a href="http://www.isg.rhul.ac.uk/dusko/papers/1996-mapsII-IGPL.pdf">pdf</a>]</li> </ul> <p><a href="https://ncatlab.org/nlab/revision/diff/Frobenius+reciprocity/55">diff</a>, <a href="https://ncatlab.org/nlab/revision/Frobenius+reciprocity/55">v55</a>, <a href="https://ncatlab.org/nlab/show/Frobenius+reciprocity">current</a></p> </div> </div> </li><li id="Comment_110871"> <a id="Item_25"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>25.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/1691/">nLab edit announcer</a></li> <li><span>CommentTime</span>Jun 23rd 2023</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=110871#Comment_110871">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_110871"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/1691/">nLab edit announcer</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Fix small typo Josselin Poiret <a href="https://ncatlab.org/nlab/revision/diff/Frobenius+reciprocity/56">diff</a>, <a href="https://ncatlab.org/nlab/revision/Frobenius+reciprocity/56">v56</a>, <a href="https://ncatlab.org/nlab/show/Frobenius+reciprocity">current</a></code></div><div> <p>Fix small typo</p> <p>Josselin Poiret</p> <p><a href="https://ncatlab.org/nlab/revision/diff/Frobenius+reciprocity/56">diff</a>, <a href="https://ncatlab.org/nlab/revision/Frobenius+reciprocity/56">v56</a>, <a href="https://ncatlab.org/nlab/show/Frobenius+reciprocity">current</a></p> </div> </div> </li><li id="Comment_119553" class="Alternate"> <a id="Item_26"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>26.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/356/">Dmitri Pavlov</a></li> <li><span>CommentTime</span>Oct 23rd 2024</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=119553#Comment_119553">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_119553"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/356/">Dmitri Pavlov</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Fixed the pullback in the diagram for Frobenius reciprocity for cartesian categories. <a href="https://ncatlab.org/nlab/revision/diff/Frobenius+reciprocity/57">diff</a>, <a href="https://ncatlab.org/nlab/revision/Frobenius+reciprocity/57">v57</a>, <a href="https://ncatlab.org/nlab/show/Frobenius+reciprocity">current</a></code></div><div> <p>Fixed the pullback in the diagram for Frobenius reciprocity for cartesian categories.</p> <p><a href="https://ncatlab.org/nlab/revision/diff/Frobenius+reciprocity/57">diff</a>, <a href="https://ncatlab.org/nlab/revision/Frobenius+reciprocity/57">v57</a>, <a href="https://ncatlab.org/nlab/show/Frobenius+reciprocity">current</a></p> </div> </div> </li><li id="Comment_120283"> <a id="Item_27"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>27.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/3667/">Evan Cavallo</a></li> <li><span>CommentTime</span>Dec 30th 2024</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=120283#Comment_120283">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_120283"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/3667/">Evan Cavallo</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Add a section on the Frobenius condition for weak factorization systems (which I plan to mention at [[cubical-type model category]]). Please advise if there is a better place to put this. <a href="https://ncatlab.org/nlab/revision/diff/Frobenius+reciprocity/59">diff</a>, <a href="https://ncatlab.org/nlab/revision/Frobenius+reciprocity/59">v59</a>, <a href="https://ncatlab.org/nlab/show/Frobenius+reciprocity">current</a></code></div><div> <p>Add a section on the Frobenius condition for weak factorization systems (which I plan to mention at <a href="https://ncatlab.org/nlab/show/cubical-type model category">cubical-type model category</a>). Please advise if there is a better place to put this.</p> <p><a href="https://ncatlab.org/nlab/revision/diff/Frobenius+reciprocity/59">diff</a>, <a href="https://ncatlab.org/nlab/revision/Frobenius+reciprocity/59">v59</a>, <a href="https://ncatlab.org/nlab/show/Frobenius+reciprocity">current</a></p> </div> </div> </li><li id="Comment_120963" class="Alternate"> <a id="Item_28"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>28.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/4/">Urs</a></li> <li><span>CommentTime</span>2 days ago</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2396/frobenius-reciprocity/?Focus=120963#Comment_120963">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_120963"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/4/">Urs</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>This entry had not a single reference on Frobenius reciprocity in representation theory. Now it has one: * Katerina Hristova: *Frobenius Reciprocity for Topological Groups*, Communications in Algebra **47** 5 (2019) &lbrack;[doi:10.1080/00927872.2018.1529773](https://doi.org/10.1080/00927872.2018.1529773), [arXiv:1801.00871](https://arxiv.org/abs/1801.00871)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/Frobenius+reciprocity/61">diff</a>, <a href="https://ncatlab.org/nlab/revision/Frobenius+reciprocity/61">v61</a>, <a href="https://ncatlab.org/nlab/show/Frobenius+reciprocity">current</a></code></div><div> <p>This entry had not a single reference on Frobenius reciprocity in representation theory. Now it has one:</p> <ul> <li>Katerina Hristova: <em>Frobenius Reciprocity for Topological Groups</em>, Communications in Algebra <strong>47</strong> 5 (2019) [<a href="https://doi.org/10.1080/00927872.2018.1529773">doi:10.1080/00927872.2018.1529773</a>, <a href="https://arxiv.org/abs/1801.00871">arXiv:1801.00871</a>]</li> </ul> <p><a href="https://ncatlab.org/nlab/revision/diff/Frobenius+reciprocity/61">diff</a>, <a href="https://ncatlab.org/nlab/revision/Frobenius+reciprocity/61">v61</a>, <a href="https://ncatlab.org/nlab/show/Frobenius+reciprocity">current</a></p> </div> </div> </li></ol> </div><div class="ContentInfo Middle"> <div class="PageInfo"> <p>1 to 28 of 28</p> <ol class="PageList PageListEmpty"> <li> </li> </ol> </div> </div></div> <a id="pgbottom" > </a> </div> </div></body> </html>