User giorgio mossa - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T05:07:33Z http://mathoverflow.net/feeds/user/14969 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/64365/natural-transformations-as-categorical-homotopies/75686#75686 Answer by Giorgio Mossa for Natural transformations as categorical homotopies Giorgio Mossa 2011-09-17T17:11:38Z 2013-03-07T16:21:09Z <p>Following the previous indication of Professor Brown I want to add another possible way to see natural transformation which is a generalization of the previous definition.</p> <blockquote> <p>Given categories $\mathcal C$ and $\mathcal D$ and two functors between them $\mathcal F,\mathcal G \colon \mathcal C \to \mathcal D$ then a natural transformation $\tau$ can be defined as a functor $\tau \colon \mathcal C \to (\mathcal F \downarrow \mathcal G)$ which arrow components are the diagonal functions, sending each arrow $f \in \mathcal C(c,c')$, with $c,c' \in \mathcal C$ to $(f,f) \in (\mathcal F \downarrow \mathcal G)(\tau(c),\tau(c'))$.</p> </blockquote> <p><strong>Edit</strong>: I think the definition of natural transformation proposed by professor Brown probably can be even a more natural than the one proposed in the question. I think that more details are worthed. </p> <p>The key ingredient for that definition is the concept of arrow category of a given category $\mathbf D$: such category have morphism of $\mathbf D$ as objects and commutative square as morphisms. </p> <p>This category come equipped with two functors $\mathbf {source}, \mathbf{target} \colon \text{Arr}(\mathbf D) \to \mathbf D$ such that for each object (i.e. a morphisms of $\mathbf D$) $f \colon d \to d'$ we have $$\mathbf{source}(f)=d$$ $$\mathbf{target}(f)=d'$$ while for each $f \in \mathbf D(x,x')$, $g \in \mathbf D(y,y')$ and a morphism $\alpha \in \text{Arr}(\mathbf D)(f,g)$ (i.e. a quadruple $\langle f,g, \alpha_0,\alpha_1\rangle$ where $\alpha_0 \in \mathbf D(x,y)$ and $\alpha_1 \in \mathbf D(x',y')$ such that $\alpha_1 \circ f = g \circ \alpha_0$) we have $$\mathbf{source}(\alpha)=\alpha_0$$ $$\mathbf{target}(\alpha)=\alpha_1$$ it's easy to prove that these data give two functors (which gives to $\text{Arr}(\mathbf D)$ the structure of a graph internal to $\mathbf{Cat}$).</p> <p>Now let's take a look to this new definition of natural transformation:</p> <blockquote> <p>A natural transformation $\tau$ between two functors $F,G \colon \mathbf C \to \mathbf D$ is a functor $\tau \colon \mathbf C \to \text{Arr}(\mathbf D)$ such that $\mathbf{source} \circ \tau = F$ and $\mathbf{target}\circ \tau = G$.</p> </blockquote> <p>A functor of this kind associate to every object $c \in \mathbf C$ a morphism $\tau_c \colon F(c) \to G(c)$ in $\mathbf D$, while to every $f \in \mathbf C(c,c')$ it gives the commutative triangle expressing the equality $$\tau_{c'} \circ F(f)=\tau_{c'} \circ \mathbf {source}(\tau_f)=\mathbf {target}(\tau_f) \circ \tau_c = G(f) \circ \tau_c$$ certifying the naturality (in the ordinary sense) of the $\tau_c$. This definition reminds the notion of homotopy between maps $f,g \colon X \to Y$ as map of kind $X \to Y^I$ (i.e. an homotopy as a (continuous) family of path of $Y$). </p> <p>That's not all, indeed we can reiterate the construction of the arrow category obtaining what I think is called a cubical set $$\mathbf D \leftarrow \text{Arr}(\mathbf D) \leftarrow \text{Arr}^2(\mathbf D)\leftarrow \dots$$ where each arrow should be thought as the pair of functors $\mathbf{source}<em>{n+1},\mathbf{target}</em>{n+1} \colon \text{Arr}^{n+1}(\mathbf D) \to \text{Arr}^n (\mathbf D)$.</p> <p>In this way we can associate to each category a cubical set. There's also a natural way to associate to every functor a (degree 0) mapping of cubical sets.</p> <p>If we consider natural transformation as maps from a category to an arrow category then this correspondence associate to each natural transformation a degree 1 map between such cubical sets (by degree one I mean that the induced map send every object of $\text{Arr}^n(\mathbf C)$ in an object of $\text{Arr}^{n+1}(\mathbf D)$). I've found really beautiful this construction because it shows an analogy between categories-functors-natural transformation and complexes-map of complexes-complexes homotopies.</p> http://mathoverflow.net/questions/64365/natural-transformations-as-categorical-homotopies Natural transformations as categorical homotopies Giorgio Mossa 2011-05-09T09:42:35Z 2013-03-07T16:21:09Z <p>Every text book I've ever read about Category Theory gives the definition of natural transformation as a collection of morphisms which make the well known diagrams commute. There is another possible definition of natural transformation, which appears to be a categorification of homotopy:</p> <blockquote> <p>given two functors $\mathcal F,\mathcal G \colon \mathcal C \to \mathcal D$ a natural transformation is a functor $\varphi \colon \mathcal C \times 2 \to \mathcal D$, where $2$ is the arrow category $0 \to 1$, such that $\varphi(-,0)=\mathcal F$ and $\varphi(-,1)=\mathcal G$.</p> </blockquote> <p>My question is:</p> <blockquote> <p>why doesn't anybody use this definition of natural transformation which seems to be more "natural" (at least for me)? </p> </blockquote> <p>(Edit:) It seems that many people use this definition of natural transformation. This arises the following question: </p> <blockquote> <p>Is there any introductory <strong>textbook (or lecture)</strong> on category theory that introduces natural transformation in this "homotopical" way rather then the classical one? </p> </blockquote> <p>(Edit2:) Some days ago I've read a post in nlab about <a href="http://ncatlab.org/nlab/show/transfor" rel="nofollow">$k$-transfor</a>. In particular I have been interested by the discussion in the said post, because it seems to prove that the homotopical definition of natural transformation should be the right one (or at least a slight modification of it). On the other end this definition have always seemed to be the most natural one, because historically category theory develop in the context of algebraic topology, so now I've a new question:</p> <blockquote> <p>Does anyone know the logical process that took Mac Lane and Eilenberg to give their (classical) definition of natural transformation?</p> <p>Here I'm interested in the topological/algebraic motivation that move those great mathematicians to such definition rather the other one. </p> </blockquote> http://mathoverflow.net/questions/120100/in-your-opinion-what-are-the-relative-advantages-of-n-fold-categories-and-n-cate/120104#120104 Answer by Giorgio Mossa for In your opinion, what are the relative advantages of n-fold categories and n-categories? Giorgio Mossa 2013-01-28T12:11:09Z 2013-01-28T12:11:09Z <p>I believe is more a matter of tastes, personally I find easier and simpler n-fold categories than categories.</p> <p>For me n-fold categories are more natural and so are easier, for different reasons: for start one interesting thing is the various sources and targets of the composition are given just by $k-1$-cells (faces), where for $n$-categories sources are given by a $i$-cells for each $i &lt; k$, this gives an intuitive representation of $k$-cells as $k$-dimensional cubes, with orientation for each pair of opposite faces, and a representation of composition as pasting cubes along the faces coherently with these orientations. To do something similar with $n$-categories you should work with $k$-cells as $k$-globes and see compositions as a sort of pasting of globes which involves also deformations of such globes and so (at least by me) it's a little more difficult to figure.</p> <p>On the other end this cubical approach has proven to be more easier to write computations: consider the case of fundamental group in which to do computations it usually preferred to use maps from the cubical interval rather then maps from the spheres.</p> <p>Another point in favor of n-fold categories is that every n-categories can be seen as a n-fold category in which every cells have collapsed faces.</p> <p>I something else come to my mind I reserve the right to add something later. :)</p> http://mathoverflow.net/questions/119913/what-is-the-difference-between-a-function-and-a-morphism/119926#119926 Answer by Giorgio Mossa for What is the difference between a function and a morphism? Giorgio Mossa 2013-01-26T09:51:38Z 2013-01-26T09:51:38Z <p>Another interesting reason why categories cannot be identified always with categories having functions for morphisms is given in this <a href="http://www.tac.mta.ca/tac/reprints/articles/6/tr6abs.html" rel="nofollow">paper</a>, by Peter Freyd in which is proven that there are some categories which aren't concrete: i.e. which don't have any faithful functor from the category in $\mathbf{Set}$ (the category of sets and functions). Having a such functor is necessary condition to have functions as morphisms.</p> http://mathoverflow.net/questions/13358/category-theory-and-model-theory-as-natural-counterparts/106611#106611 Answer by Giorgio Mossa for Category theory and model theory as "natural" counterparts Giorgio Mossa 2012-09-07T15:22:57Z 2012-09-07T15:22:57Z <p>I'm not an expert in model theory anyway I'll try to answer your questions.</p> <p>From what I get your problem come from the fact that both model theory and category theory are related with the study of stuctured objects and morphisms between them. There are categories which aren't at all build up from structured objects and morphisms stucture-preserving, for instance monoids, groups and posets are categories too, and seeing this objects as categories is useful for some applications. Model theory instead deal exactly with models of a theory which are exactly stuctured objects and the stucture preserving morphisms, so it deals with categories of models of given theories (to be exact if I'm not mistaking, model theory also deal with theories' morphisms and derived morphisms between theories' models, but also this can be seen in terms objects and morphism). </p> <p>After this not too short introduction let's try to answer your questions:</p> <p><strong>Answer #1</strong>: I suppose that the textbook you are referring to were written in time when the deep connection between model theory and category theory weren't well known. Try to take a look to book about categorical logic.</p> <p><strong>Answer #2</strong>: As I said above categories can be viewed as models of a particular (first order) theory, by the way this is not really useful because of the size issues I mentioned above. By the way category theory via notions of categories (with enough structure), functors (preserving the said structure) and natural transformations offer a new way to define the notion of theory, model and model transformation. In this way it become possible to study the notion of model of a theory in any category, where classical model theory become simply the study the theory of models in $\mathbf{Set}$, the category of sets and functions.</p> <p><strong>Answer #3</strong>:I don't know if there's any satisfactory answer to this question, mostly because as I said category theory and model theory are really different theories which aims to study different objects (the first one deal with theories and models, the second with categories, functors and natural transformations, but also other objects if we consider higher category theory as category theory). Maybe it could be more interesting studying the relation between classical (i.e. set theoretic) model theory and categorical model theory, but I don't know enough to talk about this.</p> <p><strong>Answer #4</strong>:If by level of abstraction you mean if one can be consider as a special case of the other I guess the answer is yes and no: you can build a first order theory of categories, functors natural transformation but from another point of view model theory can be completely rephrased in categorical term. Seeing from this point of view the question seems to me very similar to the chicken or the egg causality dilemma, and I don't think it's really useful this point of view, I would never consider group theory just as the study of the models of the theory of groups. :)</p> <p>I hope this helps.</p> http://mathoverflow.net/questions/73772/concrete-example-of-infty-categories Concrete example of $\infty$-categories. Giorgio Mossa 2011-08-26T13:39:50Z 2012-08-05T10:34:49Z <p>I've seen many different notion of $\infty$-categories, actual I've seen the operadic-globular ones of Batanin and Leinster and the opetopic too and eventually I'll see the simplicial ones too. Although there are so many notion of $\infty$-category so far I've only seen the following examples:</p> <ul> <li><p>$\infty$-grupoids as fundamental groupoids topological spaces;</p></li> <li><p>$(\infty,1)$-categories, mostly via topological example and application in algebraic geometry (in particular in derived algebraic geometry);</p></li> <li><p>strict $(\infty,\infty)$-categories, and their $n$-dimensional versions, for instance the various categories of strict-$n$-categories (here I intend $n \in \omega+{\infty}$).</p></li> </ul> <blockquote> <p>There are other example of $\infty$-categories, especially from algebraic topology or algebraic geometry, but also mathematical physics and computer science and logic? In particular I wondering if there's a concrete example, well known, weak $(\infty,\infty)$-category.</p> </blockquote> <p>(Edit:) after the a discussion with Mr.Porter I think adding some specifications may help: </p> <blockquote> <p>I'm looking for models/presentations of $\infty$-weak-categories for which is possible to give a combinatorial description, in which is possible to make manipulations and explicit calculations, but also $\infty$-categories arising in practice in various mathematical context. </p> </blockquote> http://mathoverflow.net/questions/86016/alternative-characterization-of-homotopy-equivalence Alternative characterization of homotopy equivalence Giorgio Mossa 2012-01-18T17:15:20Z 2012-06-11T12:51:59Z <p>Using the formalism of model categories its possible define the concept of homotopy as done <a href="http://ncatlab.org/nlab/show/homotopy" rel="nofollow">here</a>.</p> <p>If we take as model category $\mathbf{Top}$ having homotopy-equivalence as weak-equivalence and fibration and co-fibration defined in the standard topological way, these type of homotopies are just homotopies as defined in basic courses of algebraic topology.</p> <p>From this point of view seems that weak equivalence are what really matter, so here's my question:</p> <blockquote> <p>Is there any way to characterize homotopy equivalence (in $\mathbf{Top}$) without using the concept of homotopy? </p> </blockquote> <p>I'm wondering if there's a way to discriminate homotopy equivalence <em>without using the concept of homotopy at all</em>, meaning that I'm looking for a criteria which enable to say that a certain continuous map $f \colon X \to Y$ is an homotopy equivalence without looking for a morphism $g \colon Y \to X$ and continuous maps $\mathcal F \colon X \times I \to X$ and $G \colon Y \times I \to Y$ which are indeed respectively homotopies of $g \circ f$ with $1_X$ and $f \circ g$ with $1_Y$.</p> http://mathoverflow.net/questions/90863/what-is-the-relationship-between-fopl-and-higher-order-logics/90898#90898 Answer by Giorgio Mossa for What is the relationship between FOPL and Higher Order Logics? Giorgio Mossa 2012-03-11T13:12:49Z 2012-03-11T13:12:49Z <p>As François G. Dorais pointed out you have to read carefully wikipedia's article. </p> <p>The main difference in expressiveness between first order logic and second order logic is given by the semantics. </p> <p>We can turn a second order language in a special kind of first order language simply considering <em>second order variables</em> as variables ranging over a different sort, or making no distinction between <em>first and second order variables</em> and using two unary predicates, one for first order objects (i.e. individuals), the other for second order ones (i.e. relations), to distinguish when a variable have to range over individuals or when it ranges over relations.<br> With these trick we can completely translate second order syntax in a first order one.</p> <p>The same trick cannot be applied in general for the second order semantics, in particular for the standard semantics. In standard semantics we impose that in every interpretation the second order variables range over the <em>power set</em> of the domain of the interpretation (i.e. the range of variation for first order variables). If we want to reduce second order semantics to first order ones with the trick above we cannot require such strict condition, in this case the domain of variation for second order variable could be every family of subset of the domain of the interpretation. These kind of things are allowed in Henkin semantics which are simply a rewriting of first order multisorted semantics for second order logics but which are also weaker semantics (less expressive) when compared with standard ones.</p> http://mathoverflow.net/questions/89307/higher-categories-in-logic Higher categories in logic Giorgio Mossa 2012-02-23T17:46:13Z 2012-02-24T11:47:07Z <p>I've read somewhere (probably in the nlab) that higher category theory has application in logic. By the way since now the only applications of higher category theory I've seen are in homotopy theory and mathematical physics, so I was wondering if anyone could give me some example of applications of higher categorical methods in logic (any reference would be appreciated).</p> http://mathoverflow.net/questions/84381/computations-in-infty-categories Computations in $\infty$-categories Giorgio Mossa 2011-12-27T12:34:12Z 2011-12-31T23:04:11Z <p>Direct to the point. Since now I've looked a lot of presentations of $\infty$-categories, but it seems that the only way to do explicit computations on these objects is via model categories. Is that so, or maybe there are other way of doing computations in $\infty$-categories, and if there such different ways what are they?</p> <p>Edit: I really want to thank all people that answered this question, unfortunately I can choose just one answer. Btw because here is yet midnight I wish everybody a happy new year.</p> http://mathoverflow.net/questions/83956/weak-algebraic-structures Weak algebraic structures Giorgio Mossa 2011-12-20T15:50:35Z 2011-12-21T12:09:27Z <p>The following question can be thought as a sequel of <a href="http://mathoverflow.net/questions/73772/concrete-example-of-infty-categories" rel="nofollow">this one</a>. </p> <p>Here I'm looking for a big list of example of <em>weak algebraic structures</em>: here weak means that the structure (i.e. operations) need not to satisfy equations but rather they must satisfy equations up to higher equivalence, the sort of conditions that can be expressed via commutative diagrams up to higher equivalence.</p> <p>In particular I'm interested in example of structures of this sort in which we can do explicit calculations that in theory can be implemented in a computer.</p> <p>It would be nice if every example come equipped with some reference where said example is presented.</p> http://mathoverflow.net/questions/69251/is-mac-lane-still-the-best-place-to-learn-category-theory/69266#69266 Answer by Giorgio Mossa for Is Mac Lane still the best place to learn category theory? Giorgio Mossa 2011-07-01T14:46:26Z 2011-11-17T15:52:01Z <p>I doubt that someone could learn higher category theory (and more in general higher dimensional algebra) without first studying a little of category theory, mostly because the definition given in such context use a lot of category theoretic machinery. About the textbook reference: MacLane's "Category theory for working mathematicians" may be a little outdated but I think it is still one of the most complete book of basic category theory second just to Borceux's books. Anyway there isn't a best book to learn basic category theory, any person could find a book better than another one, so I suggest you to take a look a some of these books, then choose which one is the best for you:</p> <p>S. MacLane: Category theory for working mathematicians (I've already said a lot about this)</p> <p>S. Awodey: Category theory (Peculiar because it has very low prerequisites and it's rich of examples too) </p> <p>J. Adamek,H. Herrlich, G. Strecker: Abstract and concrete category theory (freely avaible at at this site "http://katmat.math.uni-bremen.de/acc/acc.pdf", maybe the book with the greatest number of examples from topology and algebra)</p> <p>After you have read one of these book, you could also use Borceux's books and read some more advanced chapter of category theory which aren't discussed in the previous books.</p> <p>F. Borceux: Handbook of Categorical Algebra 1: Basic Category Theory</p> <p>F. Borceux: Handbook of Categorical Algebra 2: Categories and Structures</p> <p>F. Borceux: Handbook of Categorical Algebra 3: Categories of Sheaves</p> <p>For higher category theory I know just few reference: </p> <p>Leinster's "Higher Operads Higher Categories" (http://arxiv.org/abs/math/0305049), </p> <p>and </p> <p>Lurie's "Higher Topos Theory" (http://arxiv.org/abs/math/0608040) </p> <p>other good reference in higher category theory and higher dimensional algebra in general are Baez'<a href="http://math.ucr.edu/home/baez/TWF.html" rel="nofollow">This week's finds</a> and <a href="http://arxiv.org/find/all/1/AND+au%3A+Baez_John+ti%3A+AND+algebra+AND+Higher+dimensional/0/1/0/all/0/1" rel="nofollow">arxiv articles Higher dimensional algebra*</a>.</p> <p>Hope this may help.</p> http://mathoverflow.net/questions/41057/categories-first-or-categories-last-in-basic-algebra/80393#80393 Answer by Giorgio Mossa for Categories First Or Categories Last In Basic Algebra? Giorgio Mossa 2011-11-08T14:39:39Z 2011-11-09T18:57:49Z <p>A little preliminary: I'm an undergraduate student and I started to study category theory as self-taught at the beginning of second year of university, mostly because of my interest in logic and foundations. Since then I've enjoyed of this fact because knowing some category theory helped me to understand lots of concepts that I've learned more quickly then what I would have done without it, also category theory move me to study some branch of maths like algebraic topology and algebraic geometry. Now I would distinguish between "category theory" and "the language and instrument of category theory": while the first is an abstract and too specific branch of math, so not adequate to be considered in a undergraduate courses, the second is the very useful conceptual tool that should be taught also to undergraduate students. What I mean here is that (the language of) category theory shouldn't be teached in a specific course but it should be taught during the regular courses.</p> <p>I believe that some basic concepts like the ones of category and functor could be taught since first courses of algebra, that's because these concepts are not more abstract than those of groups-group homomorphism,ring-ring homomorphism, vector space-linear map which are taught in the first year's courses. Categories and functors can be easily shown to a young public respectively as graphs with structure (i.e. operations) and as graph morphisms preserving the structure. Many example can be given to those concepts which can be understood by undergraduates: the categories of graphs' points and graphs' paths, the category of sets and functions, the category of groups and group homomorphisms, vectorial spaces and linear maps, but also monoids, groups and poset as categories. In particular its very useful made these last example in first courses because they help in familiarizing with abstraction before mind is corrupted by concrete (I remember that after having done some basic algebra I found a lot of difficulties to understand why monoids should be categories with one object). Another good set of examples of category the are quite easy to understand and (in my personal opinion cool) are those of objects (which can be molecules, automaton's states, dynamic system states,...) and processes transforming one object into another. These examples are pretty cool because they open the way to application of category theory also to other science, besides giving really concrete examples of categories.</p> <p>Obviously categorical concepts should be introduced in a very gradual way, for instance its useless teaching natural transformation before having seen homotopies and groups' representation (or equivalently groups' actions), same apply for other more complex concepts: every thing need to be introduced at right time.</p> <p>Many would object that probably concepts should be presented every time when they are needed. To those people I would say that probably they right, anyway no-one have ever introduced to me abstract concepts like the ones of groups and rings with some motivation, same apply to topological spaces, the motivations for introducing these objects came late, when where introduced some results which gives us a more abstract framework in which some kind of problems tend to simplify and generalize. </p> <p>Last motivations of teaching category theory early is that many times seeing thing from an abstract point of view helps when we want to switch constructions from categories, where these constructions are build naturally, to other categories (it comes to my mind the example of homotopies of complexes in homological algebra) and also shows deep unity of lots of mathematical objects that maybe at first seem unrelated. </p> <p>Before to end I would also like to add some motivation to why <strong>not</strong> waiting to teach category theory in advanced courses: if you do so usually happens that these categorical concepts are presented in very fast way that make difficult to take familiarity with said concepts and that doesn't allow to deeply understand the meaning and usefulness of categorical results.</p> <p>One last comment: I don't know why but every time I think to those people which consider category theory too abstract and useless they remind me of what Kronecker said about Cantor, and this make me smile.</p> http://mathoverflow.net/questions/3242/canonical-examples-of-algebraic-structures/79277#79277 Answer by Giorgio Mossa for Canonical examples of algebraic structures Giorgio Mossa 2011-10-27T15:40:15Z 2011-10-27T15:40:15Z <p>I think you'll find out very different my example of algebraic structures.</p> <p>Monoid: the monoid of words over a finite alphabet.</p> <p>Group: the group of words over a finite alphabet.</p> <p>Ring: the polinomial ring.</p> <p>Module: $\mathbb K^n$ for some $\mathbb K$ field.</p> <p>Category: the path category over some graph $\mathbb G$.</p> <p>$R$-Algebra (for some ring $R$): $M_n(R)$, i.e. the ring of matrix over the ring $R$. </p> <p>These is exactly those algebraic structures that come in my mind when I think/try to prove fact about algebraic structures. Why? Because they're free object and are the terms models of their corresponding algebraic theories, so every other algebraic models derive from these by adding relations.</p> http://mathoverflow.net/questions/77951/what-are-n-poset What are $n$-poset? Giorgio Mossa 2011-10-12T19:41:24Z 2011-10-12T20:44:33Z <p>Yesterday I was wandering for the $n$-lab and I've found the definition of $n$-poset. Following this <a href="http://ncatlab.org/nlab/show/n-poset" rel="nofollow">post</a> it seems that a $n$-poset should be a $(n,n+1)$-category. Now an $(n,r)$-category should be a category such that every $k$-morphism is an equivalence for $k\geq r$ and every pair of parallel $k$-morphisms with $k \geq n$ are equivalent. Now here're my problems: I suppose that this objects should generalize in any some way the notion of poset to higher categorical structure, i.e. it should be a categorification of the notion of poset, but I don't get why this should be the case</p> <blockquote> <p>could anyone explain to me how $n$-posets generalize the notion of poset? </p> </blockquote> http://mathoverflow.net/questions/76083/categories-presented-with-arrows-only-no-objects-partial-monoids/76089#76089 Answer by Giorgio Mossa for Categories presented with Arrows only, no objects: partial monoids Giorgio Mossa 2011-09-21T21:50:38Z 2011-09-22T09:08:21Z <p>Of course you can define a (just-arrow) category $\mathcal C$ like a partial algebra which consist of:</p> <p>a set $\mathcal C$ (namely the set of arrows of your category), a set $D_\mathcal{C} \subseteq \mathcal C \times \mathcal C$ (the set of pair of composable arrows) and a map $\circ \colon D_\mathcal{C} \to \mathcal C$, which is the composition for this "category". In this structure we call identities all the elements $f \in \mathcal C$ such that for each $g,h \in \mathcal C$ with $(g,f),(f,h) \in D_\mathcal{C}$ we have $g\circ f=g$ and $f \circ h=h$. The composition have to satisfy the following axioms:</p> <p>*for each triple $h,g,f \in \mathcal C$ we have that these three statements are equivalent:</p> <p>$(g,f) \in D_\mathcal{C}$ and $(h,g\circ f) \in D_\mathcal{C}$ </p> <p>$(h,g) \in D_\mathcal{C}$ and $(h\circ g, f) \in D_\mathcal{C}$ </p> <p>$(h,g) \in D_\mathcal{C}$ and $(g,f) \in D_\mathcal{C}$</p> <p>and in this case the equality $h\circ(g \circ f)=(h \circ g) \circ f$ holds;</p> <p>*for each $f \in \mathcal C$ there are two arrows $g,h \in \mathcal C$ which are identities such that $(f,g), (h,f) \in D$ and $f \circ g=f=h \circ f$.</p> <p>With these data you have a concept of category just-arrow. With this definition of category a functor $F$ from the category $\mathcal C$ to the category $\mathcal D$ is just a function $F \colon \mathcal C \to \mathcal D$ between the sets of the arrows such that:</p> <ul> <li><p>for each pair $f,g \in \mathcal C$ if $(g,f) \in D_\mathcal{C}$ then $(\mathcal F(g),\mathcal F(f)) \in D_\mathcal{D}$ and $\mathcal F(g \circ f)= \mathcal F(g) \circ \mathcal F(f)$;</p></li> <li><p>for each identity $f \in \mathcal C$ also $\mathcal F(f)$ is an identity.</p></li> </ul> <p>The category of just-arrow categories and functors between them is proven to be equivalent to $\mathbf{Cat}$, the category of (ordinary) categories and functors between them.</p> http://mathoverflow.net/questions/75788/mathematics-needed-for-higher-dimensional-category-theory/75794#75794 Answer by Giorgio Mossa for Mathematics needed for higher dimensional category theory? Giorgio Mossa 2011-09-18T22:46:12Z 2011-09-20T10:59:17Z <p>As many other have just said you cannot think to study just some particular subjects ignoring some other areas, expecially if you want to do research. Most of math was born from the observation of some similar phenomena in many different areas: for instance the concept of category itself was born from the observation that in math we deal every time with collections of structures and morphisms preserving those structures, that led to the abstraction of category, similarly I strongly doubt that Grothendieck could invent the concept of (generalized) sheaf if first he hadn't known the many concrete sheaves that appear in topology, differential geometry and algebraic geometry, so it couldn't get to the concept of (Grothendieck's) topos, and without that I'm not so sure that Lawvere could get to the concept of elementary topos while doing his research in logic. This are just some example of as math have evolved thanks to interaction of different areas (for instance, as you can see in the example above, from interaction of geometry and logic). </p> <p>Just to answer to your comment about analysis there's a professor in Italy who studies higher dimensional category theory for his research in analysis, so analysis need higher category theory. </p> <p>Of course the best place where you can get a lot of intuition of higher category theory is algebraic topology where higher categories are used to model homotopy types for topological spaces, via $\infty$-groupoids, and directed space, via $(n,r)$-categories where $n,r \in \omega \cup {\infty}$ but you can find a lot of higher dimensional category theory in logic and computer science too, I've seen some application in calculability theory and model theory where (higher) category theory is used to model the semantic of theories, in particular type theory (if you're interested in application of higher categorical logic-model theory you can take a look to Makkai's work and also Mike Shulman's work on homotopy type theory). Also in mathematical physics there are a lot of higher category theory as John Baez's work prove.</p> <p>I suppose above you were referring to Cheng-Lauda "Illustrated guide book", that's a good book if you want to learn many approaches to $n$-categories, but in higher category theory there's a lot of more then just $(n,r)$-categories (like usually Mr.Shulman says), Leinster's "Higher operads, Higher categories" is more complete from this point of view because it presents a lot of stuff like generalized multicategories/operads or $fc$-multicategories. Anyway if you want some references on higher category theory you can find some <a href="http://mathoverflow.net/questions/69251/is-mac-lane-still-the-best-place-to-learn-category-theory/69266#69266" rel="nofollow">here</a>.</p> <p>Hope this may help you.</p> <p>(Edit: I've improved a little the answer now that I've found some other references.) </p> http://mathoverflow.net/questions/74425/a-question-on-the-sum-of-element-orders-of-a-finite-group/74426#74426 Answer by Giorgio Mossa for A question on the sum of element orders of a finite group Giorgio Mossa 2011-09-03T08:43:43Z 2011-09-03T20:02:12Z <p>Edit: I misunderstood the question, I'll try to fix here. I don't have the complete answer but I'll try to give a partial answer: let $G$ be a group of order $|G|$ and for each $d \mid |G|$ let $n_d$ indicate the number of elements of order $d$ in $G$; then if $|G|$ is even $|G| \nmid \sum_{d \mid |G|}n_d d$. Indeed we have that if $d$ is a odd divisor of $|G|$ (not equal to $1$) either $n_d=0$ or exists a odd prime numeber $p$ such that $p-1 \mid n_d$ and so $n_d$ is even, on the other hand if $d$ is even clearly $n_d d$ is also even and so $\sum_{1 \ne d \mid |G|} n_d d$ must be even. Thus $\sum_{d \mid |G|}n_d d$ is odd and so $|G| \nmid \sum_{d \mid |G|} n_d d$, because by hypothesis $|G|$ is even.</p> http://mathoverflow.net/questions/72490/why-are-operads-useful/72680#72680 Answer by Giorgio Mossa for Why are operads useful? Giorgio Mossa 2011-08-11T14:02:34Z 2011-08-11T14:02:34Z <p>I'm not an expert by the way I could give you an answer based on my personal experience in my study of category theory.</p> <blockquote> <p>Operads allow to make lots of constructions and to encode information of many mathematical object into an algebraic structure, this algebraic structure allows to work in a simpler way with the above mentioned objects.</p> </blockquote> <p>In pratical I think operads are similar to homotopy/homology groups, which encode homotopical information of topological spaces and enable to distinguish such spaces in a simple way, studying algebraic structures. This is useful because is more simple classifying groups rather then topological spaces. </p> <p>More in general I think the usefulness of all such structures derive by the fact that usually (concrete) algebraic structures are easier to work with, but I emphasise that these are just my thoughts.</p> http://mathoverflow.net/questions/66641/relation-between-monads-operads-and-algebraic-theories Relation between monads, operads and algebraic theories Giorgio Mossa 2011-06-01T09:26:18Z 2011-07-14T16:49:15Z <p>I've begun to interest in algebraic theories and their categorical models: in particular monads, generalized multicategories and operads, lawvere theories and their generalization. Is there any reference that treat systematically the relation between such models of theories, where model means a presentation of theory?</p> http://mathoverflow.net/questions/19152/why-is-a-topology-made-up-of-open-sets/68930#68930 Answer by Giorgio Mossa for Why is a topology made up of 'open' sets? Giorgio Mossa 2011-06-27T14:15:02Z 2011-06-27T14:15:02Z <p>I want to complete something said by Andrew Stacey above: like him I think that the only reason to motivate the use of the open sets it's because they are more easy to use. Topology is the study of property preserved by invertible continuous transformation (following the Erlangen program): this definition clearly need the notion of continuity, I've always thought of continuity as the relation of proximity of points, so the first thing to do topology is to define the notion of proximity and neighbourhood are most natural way to do so (at least for me). Anyway deal with neighbourhood is more complex than working with open set, for example the definition of topology with neighbourhood require five axioms while classical definition with open sets require just three axioms, so while it seems more natural the study of topology via neighbourhood it is more convenient dealing with open sets which allow to simplify proofing work.</p> <p>I hope this answer my help.</p> http://mathoverflow.net/questions/68235/what-is-higher-dimensional-algebra What is higher dimensional algebra? Giorgio Mossa 2011-06-19T21:12:11Z 2011-06-21T21:51:38Z <p>Could anyone explain what higher dimensional algebra is?</p> <p>I tried to look on the web but I couldn't find a satisfactory definition, the ones that I found are too vague. What I'm looking for is a good definition of higher dimensional algebra, what it deals with and some references about the subject.</p> http://mathoverflow.net/questions/67154/equivalence-in-infty-categories Equivalence in $\infty$-categories Giorgio Mossa 2011-06-07T16:14:18Z 2011-06-20T03:28:31Z <p>In every $n$-category (weak or strict) can be defined the concept of equivalence via a recursive definition: * an equivalence in a set ($0$-category) is just an identity; * for each $n \in \mathbb N$ an equivalence between two object (or $0$-cell), let say $a$ and $b$, in a $n+1$-category is just a $1$-cell $f \colon a \to b$ such that exist a $1$-cell $g \colon b \to a$ and two $2$-cells $\alpha \colon g \circ f \to 1_a$ and $\beta \colon f \circ g \to 1_b$ which are equivalence into the $n$-categories $\hom(g\circ f, 1_a)$ and $\hom(f \circ g, 1_b)$ respectively. There is a good formal definition of equivalence also for $\infty$-category?</p> http://mathoverflow.net/questions/66121/is-pa-consistent-do-we-know-it/66771#66771 Answer by Giorgio Mossa for Is PA consistent? do we know it? Giorgio Mossa 2011-06-02T21:16:41Z 2011-06-03T07:04:52Z <p>I'd been very interested in foundational questions for a long time so I think I can say something about it. To understand the question I think it is necessary make some comment: mathematical logic is the study of mathematical theories via mathematics itself. To make theorem about theories first you have to define what is a formula, a theory and a proof, so first you need a (meta-)theory in which you make your demonstration about your theory: usually this (meta-)theory is Zermelo Fraenkel set theory without infinite axiom. This means that every proof of a theorem about a theory is valid if the meta-theory is consistent, if that's not the case then we cannot say anything because in a inconsistent theory you can deduce everything (that if you use classical logic). This says that we cannot prove the consistency of a theory in an absolute sense but we can only reduce consistency of a theory to the consistency of another theory: so I think today is pointless wonder if PA is consistent, unless you don't believe in the existence of natural number which satisfy all of the Peano Axioms, such existence is the proof of consistence of PA, because it's well know (or I hope so) that a theory with a model is consistent. (In particular if you accept the consistence of ZFC the you can prove PA consistency in this theory). Just one more thing: if you don accept the existence of natural numbers which satisfy PA (and so of a universe which is a model of ZFC) then you cannot accept lots of mathematics that is derived from it. I hope this answer your question.</p> http://mathoverflow.net/questions/66313/monad-arising-from-operad Monad arising from operad Giorgio Mossa 2011-05-28T21:24:36Z 2011-05-30T09:18:56Z <p>It's known that from every operad arises a cartesian monad whose algebras are the algebras for the operad. Leinster proved that there are different operads from which arise the same monad, in this way he proved that operads cannot be identified with monads. Now I'm wondering: is it true that every cartesian monad arises from an operad, i.e. every such a monad is the monad associated to an operad?</p> <p>I little specification, what I mean here by operad is a generalized operad i.e. a $T$-operad for some cartesian monad $(T,\mu,\eta)$ in a cartesian category $\mathcal C$.</p> http://mathoverflow.net/questions/123353/the-semicat-of-morphisms-which-are-neither-right-nor-left-invertible Comment by Giorgio Mossa Giorgio Mossa 2013-03-01T20:25:11Z 2013-03-01T20:25:11Z Just a curiosity, how does a class of morphisms which are neither left nor right invertible be category? identities are always both right and left invertible. http://mathoverflow.net/questions/915/is-there-a-high-concept-explanation-for-why-characteristic-2-is-special Comment by Giorgio Mossa Giorgio Mossa 2012-12-10T11:41:56Z 2012-12-10T11:41:56Z For the combinatorial point of view I guess that the importance of 2 is linked to the fact that for every set $X$ there's the canonical representation of each subset of $X$ with an element of $2^X$ which by the way is the support of a $F_2$ vector space, the space $\prod_{x \in X} \mathbb Z/2 \mathbb Z$. About other anomalies related to characteristic 2, I wonder if they are due to the fact that in characteristic 2 inverse coincide with identity and so we don't have the (involutive) symmetry respect to 0. http://mathoverflow.net/questions/107266/is-homotopy-definable-by-categorical-means Comment by Giorgio Mossa Giorgio Mossa 2012-09-15T20:47:54Z 2012-09-15T20:47:54Z you're welcome :) http://mathoverflow.net/questions/107266/is-homotopy-definable-by-categorical-means Comment by Giorgio Mossa Giorgio Mossa 2012-09-15T17:07:15Z 2012-09-15T17:07:15Z @HansStricker Have you take a look to this: <a href="http://ncatlab.org/nlab/show/homotopy" rel="nofollow">ncatlab.org/nlab/show/homotopy</a> . http://mathoverflow.net/questions/13358/category-theory-and-model-theory-as-natural-counterparts/106626#106626 Comment by Giorgio Mossa Giorgio Mossa 2012-09-09T14:08:15Z 2012-09-09T14:08:15Z Really cool stuff, thanks. http://mathoverflow.net/questions/13358/category-theory-and-model-theory-as-natural-counterparts/106626#106626 Comment by Giorgio Mossa Giorgio Mossa 2012-09-08T16:49:11Z 2012-09-08T16:49:11Z @AndrejBauer I completly agree about the comparison between categorical logic and model theory, but why should category theory be compared with algebra? http://mathoverflow.net/questions/104896/interpretations-as-morphisms Comment by Giorgio Mossa Giorgio Mossa 2012-08-21T20:51:35Z 2012-08-21T20:51:35Z By the way I suppose things work in this way: we have two interpretations $\Delta,\Gamma \colon \mathbf W \to \mathbf V$, where $\mathbf W$ and $\mathbf V$ are respectively a $L$ and a $K$ family of structures. An homotopy $\chi$ associate to every $A \in \mathbf V$ an isomorphism $\chi_A \colon \Delta(A) \to \Gamma(A)$, like an homotopy $h$ between to continuous function $f,g \colon X \to Y$ associate to every $a \in X$ a path $h_a \colon f(a) \to f(b)$. Hope this seems like a possible clarification. :) http://mathoverflow.net/questions/104896/interpretations-as-morphisms Comment by Giorgio Mossa Giorgio Mossa 2012-08-21T20:41:45Z 2012-08-21T20:41:45Z @HansStricker I'm guessing that the problem of homotopies in the context of Hodge's book is that he doesn't state clearly what should it be the family of bijection induced by the formula/homotopy. http://mathoverflow.net/questions/101573/untyped-higher-category-theory Comment by Giorgio Mossa Giorgio Mossa 2012-07-07T15:01:23Z 2012-07-07T15:01:23Z By object above I mean a $0$-morphism/cell of the higher category. http://mathoverflow.net/questions/101573/untyped-higher-category-theory Comment by Giorgio Mossa Giorgio Mossa 2012-07-07T15:00:52Z 2012-07-07T15:00:52Z @MircoMannucci If what you're interested is just an higher category in which there's only one object then I think you're looking for higher monoidal categories. http://mathoverflow.net/questions/92355/what-is-a-self-dual-category/92358#92358 Comment by Giorgio Mossa Giorgio Mossa 2012-03-27T15:05:46Z 2012-03-27T15:05:46Z @MartinBrandenburg sorry, I haven't read correctly the answer. I've voted for delete this answer. Thanks for the correction. http://mathoverflow.net/questions/86774/defining-topological-spaces-with-the-notion-of-continuous-path Comment by Giorgio Mossa Giorgio Mossa 2012-01-27T15:38:37Z 2012-01-27T15:38:37Z @GuillaumeBrunerie you're asking if knowing what are the continuous functions of type $I^n \to X$ characterize the topology, am I right? http://mathoverflow.net/questions/86016/alternative-characterization-of-homotopy-equivalence Comment by Giorgio Mossa Giorgio Mossa 2012-01-19T12:41:04Z 2012-01-19T12:41:04Z @TomGoodwillie: yeah you're right. The point is, because homotopy equivalences, fibrations and cofibrations seem to be the only things you need to rebuild the whole homotopy theory, I was wondering if there is any why to present this model category structure of $\mathbf{Top}$ without using the concept of homotopy. http://mathoverflow.net/questions/86016/alternative-characterization-of-homotopy-equivalence/86021#86021 Comment by Giorgio Mossa Giorgio Mossa 2012-01-18T21:36:45Z 2012-01-18T21:36:45Z I've edited the question and added some specification, I hope I made myself more clear. Thanks. http://mathoverflow.net/questions/84381/computations-in-infty-categories Comment by Giorgio Mossa Giorgio Mossa 2011-12-28T23:09:58Z 2011-12-28T23:09:58Z @DylanWilson: I've never things about 1-category theory from this point of view. it's really clarifying from the higher dimensional point of view. Thanks a lot! :)