Zhen Lin
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May 18 |
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Drect limit of sequences To be clear, filtered colimits do commute with finite limits in a Grothendieck category – but this is basically part of the definition! For an example of a Grothendieck category that is not locally finitely presentable, one just has to look for a topos that is not locally finitely presentable. |
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May 18 |
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Drect limit of sequences @DavidWhite Filtered colimits commute with finite limits in any finitely accessible category (hence Paul's emphasis on finitary algebraic theory). In particular it will be true in a locally finitely presentable abelian category, but Grothendieck categories need not be locally finitely presentable. Moreover it is not true in general that filtered colimits commmute with finite limits: take $\mathbf{Set}^\mathrm{op}$, for example. |
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May 18 |
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Directed colimits of maps in a combinatorial model category Directed colimits are indexed by directed posets, filtered colimits are indexed by filtered categories. |
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May 15 |
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On the large cardinals foundations of categories Yes, that much is easy. Actually I prefer not to use any universes at all and just work in NBG for such simple scenarios: because, say, locally small categories are actual objects in the universe of discourse, I can then form the metacategory of locally small categories at the syntactic level and do what I need there. My belief is that, so long as nothing sophisticated is being done, iterating the ZFC-to-NBG construction $\omega$ times should be enough to guarantee the existence of any definable categories one might like to use. The difficulty is when one wants to do universal constructions... |
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May 15 |
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On the large cardinals foundations of categories Yes, that definition is much closer to that of a Grothendieck universe. |
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May 15 |
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On the large cardinals foundations of categories @Asaf: My point is that sometimes one is doing much more than mere bookkeeping. It should go without saying that for the first 2 years that I studied category theory I was in the anti-universe camp because I thought it was all just a matter of being careful with various things, but I have now seen that this just isn't feasible in some areas. I'm afraid there's nothing more I can say if you don't know the category theory involved! |
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May 14 |
answered | On the large cardinals foundations of categories |
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May 14 |
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On the large cardinals foundations of categories Perhaps, if all you are doing is really just bookkeeping. But for other purposes (e.g. the one considered in my preprint) it is absolutely crucial that no new functions be added. |
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May 14 |
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On the large cardinals foundations of categories It appears to me that when one is doing sophisticated things with category theory, one must be prepared to climb up the universe hierarchy. For instance, the quasicategory of small quasicategories in a universe is going to be a simplicial set in the next universe, so we would have to juggle the Joyal model structure on large simplicial sets with the Joyal model structure on small simplicial sets. It seems crucial that not only the same theorems be proved about all things (universe polymorphism) but also that we get nice embeddings of each thing into their respective enlargement. |
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May 14 |
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On the large cardinals foundations of categories A universe in the sense of Grothendieck et al. satisfies a second-order replacement axiom. How do we guarantee this using the fibred topos language? |
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May 14 |
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On the large cardinals foundations of categories @Asaf A chain of end-extensions does sound better, although I am unfamiliar with the technical definition of that. |
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May 14 |
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On the large cardinals foundations of categories As a practising categorist, let me emphasise how mind-bending it is to work in an arbitrary chain of models: when I think about expanding the universe, I don't expect cardinals to suddenly collapse together, or for sets to gain new subsets, or any of the other things that set theorists are used to! Rather, my basic minimum criterion is that hom-sets are preserved, and that implies there are no new functions and no new subsets. |
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May 8 |
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Is this square a push-out square? See Fernando Muro's answer. For my answer, you only need an additive category and the existence of kernels and cokernels. |
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May 8 |
accepted | Is this square a push-out square? |
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May 3 |
answered | Is this square a push-out square? |
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Apr 26 |
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Infinity-categories vs Kan complexes Your definition is that of an inner Kan complex, also known as a quasicategory, at least when $X$ is a simplicial set. |
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Apr 24 |
answered | “Wrong” strictification of symmetric monoidal categories |
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Apr 24 |
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Can Inequivalent Topologies Have Same Sheaves/Cohomology? The short answer is, Grothendieck topologies are uniquely determined by the subcategory of sheaves. This is not so for pretopologies. |
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Apr 23 |
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Finitely Generated Commutative Z-algebra. This is Theorem 4.19 in Eisenbud. |
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Apr 23 |
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General Theory of Left-Exact Localization? Special cases of reflective localisation in locally presentable categories are studied in § 1.C of [Adámek and Rosický, _Locally presentable and accessible categories_] and, of course, the theory of left exact localisations of presheaf toposes is just the theory of Grothendieck topologies. |
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Apr 23 |
answered | Does this kind of endofunctor ever have an initial algebra? |
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Apr 22 |
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Internal Day convolution A locally cartesian closed category with finite colimits believes itself to be cocomplete (in the internal sense, of course). Caution: such categories may suffer from delusions; $\textbf{FinSet}$ is one example. |
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Apr 20 |
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Does a nonlinear additive function on R imply a Hamel basis of R? Previously posted on MSE: math.stackexchange.com/questions/366010 |
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Apr 19 |
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Needless axiom for Grothendieck topologies? You can make do with much less than a Grothendieck pretopology: all you need to have a good notion of sheaf (i.e. one that generates a topos) is a coverage. See here: ncatlab.org/nlab/show/coverage |
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Apr 12 |
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Need M combinatorial for existence of injective model structure on $M^G$? @Mike At first I thought "logical approach" meant "straightforward", but having skimmed the paper now, I see you mean "logical" in the literal sense! A very remarkable paper. |
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Apr 12 |
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The category theory of $(\infty, 1)$-categories @UrsSchreiber Thanks. Good to know that it's true in at least one pair of models! |
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Apr 12 |
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The category theory of $(\infty, 1)$-categories @DavidWhite I know about the Quillen equivalences, but how does that tell me what is happening to category-theoretic constructions inside the $(\infty, 1)$-categories? |
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Apr 11 |
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Does there exist a topology for a set X which is compact and Hausdorff? Crossposted at MSE: math.stackexchange.com/questions/358583/… |
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Apr 11 |
asked | The category theory of $(\infty, 1)$-categories |
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Apr 5 |
awarded | ● Enlightened |
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Apr 5 |
awarded | ● Nice Answer |
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Apr 4 |
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Are there non-categorical notions in topos theory? minor clarification |
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Apr 4 |
accepted | Are there non-categorical notions in topos theory? |
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Apr 4 |
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Are there non-categorical notions in topos theory? @Simon I have added a paragraph to address your comments. The answer is yes. |
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Apr 4 |
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Are there non-categorical notions in topos theory? added 583 characters in body |
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Apr 4 |
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Are there non-categorical notions in topos theory? Yes, but how is this any different from needing to know which object is $\mathbb{Z}$ in $\textbf{Ab}$ in order to recover the underlying set of an abelian group? |
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Apr 4 |
answered | Are there non-categorical notions in topos theory? |
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Apr 3 |
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question of topos and site Indeed, but that is almost by definition. |
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Apr 1 |
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A “mother of all groups”? What kind of structures have “mother of all”s? Perhaps you're thinking of a generalised Fraïssé limit: en.wikipedia.org/wiki/Age_%28model_theory%29 |
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Mar 28 |
accepted | question of topos and site |
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Mar 28 |
answered | question of topos and site |
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Mar 25 |
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Consistency of the concept of the collection of all collection Yes, the Russell class being a set and the universal class being a set are equiconsistent in ZF (in the silly sense of being equally inconsistent). But I wouldn't go as far as saying that the two classes are equal. By your own admission, the Russell class is empty; but the universal class is very much non-empty. |
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Mar 24 |
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Consistency of the concept of the collection of all collection Universal sets are inconsistent with strong forms of Cantor's theorem, yes. But the strong form of Cantor's theorem is known to be false in NFU, as Noah has stated. The point is that it is very hard for an entity to be inconsistent on its own – and anyone who actually read Russell's argument would know that the paradoxical set is not the universal set. |
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Mar 24 |
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Consistency of the concept of the collection of all collection The set of all sets is not inconsistent. It would be more accurate to say that the set $\{ x : x \notin x \}$ is inconsistent – and there is nothing about the universal set that says that the set $\{ x : x \notin x \}$ exists. |
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Mar 18 |
accepted | When does the direct image functor nicely push past the power/exists functor? |
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Mar 17 |
answered | When does the direct image functor nicely push past the power/exists functor? |
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Feb 24 |
accepted | abelian group objects category |
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Feb 23 |
answered | abelian group objects category |
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Feb 8 |
awarded | ● Fanatic |
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Feb 2 |
accepted | Category and the axiom of choice |
