# Tagged Questions

Categories and functors, universal properties, algebras and algebraic theories, topoi, enriched and internal categories, structured categories (abelian, monoidal, etc), higher categories.

Any colimit can be represented as a functor $F$ left adjoint to a particular diagonal functor $\Delta: C \rightarrow C^J$. The unit of this adjunction is the natural transformation $\eta_K: K \... 1answer 269 views ### Jets in synthetic differential geometry As I understand it from Kostecki's notes, the$k$-jet$j^k f$of a function$f: R^n \to R^m$should be the map $$f^{D_k(n)} : {(R^n)}^{D_k(n)} \to (R^m)^{D_k(n)},$$ where $$D_k(n) = \{(x_1, \ldots, ... 0answers 193 views ### A definition of the homotopy colimit of a coherent diagram Suppose I am given a homotopy coherent diagram of spaces of shape I (This is a simplicial functor F:\mathfrak{C}[I] \to Top, where \mathfrak{C} is the standard cofibrant replacement functor in ... 2answers 215 views ### Question about Enriched Categories and Functors How would one describe the process of enriching a category C over some monoidal category D? Is there some functor between them that adds structure to the hom-sets? 1answer 94 views ### A question on 2-bundles In this paper, the authors +John Baez and +Urs Schreiber defined (page 15) "transition functions" for a special kind of 2-bundles (those whose the base space is a ordinary smooth space augmented to a ... 0answers 81 views ### Does a left Kan lift of a homset functor Hom(*, - ): C -> {Set} through the forgetful functor {M-Set} -> {Set} exist? Let M be a monoid and let \mathsf{U} : M\mathsf{Set} \to \mathsf{Set} denote the functor which forgets the action on an M-set. Given a category C with a distinguished object p as baspoint, ... 0answers 177 views ### Algebraic K theory, Karoubi completion and splitting Suppose \mathcal{C}\subset \mathcal{C}' is a pair of pre-triangulated smooth DG categories over a characteristic-zero basefield (say, \mathbb{C}), such that \mathcal{C} is faithfully embedded in ... 1answer 841 views ### Variant of Conceptual Completeness Let \mathcal{C} and \mathcal{D} be pretopoi, and let f: \mathcal{C} \rightarrow \mathcal{D} be a pretopos functor (that is, a functor which preserves finite coproducts, finite limits, and ... 0answers 90 views ### What are universal abstract \sigma-algebras on \sigma-frames? Originally asked on MSE. In this paper, the authors make the following definitions: An (abstract) \sigma-algebra is a boolean algebra with countable joins. A \sigma-frame is a bounded lattice ... 0answers 59 views ### Determining finite abelian groups among algebraic theories by counting Let T be a Lawvere theory (algebraic theory) which contains the theory T_{\mathrm{grp}} of groups as a subtheory (so that T has, in general, more equations that T_\mathrm{grp}). Suppose that,... 9answers 3k views ### What is… A Grossone? Y. Sergeyev developed a positional system for representing infinite numbers using a basic unit called a "grossone", as well as what he calls an "infinity computer". The mathematical value of this ... 1answer 236 views ### Reference for t-structures on stable model categories What kind of definitions of t-structures on stable model categories have been investigated in the literature? Of course, one can always define a t-structure on a stable model category as a t-... 1answer 140 views ### Additive functors to abelian groups: “additional structure” and functors induced by “additive pseudo-functors”; references? Let \mathcal{A} be a small additive category. Consider the category PreSh(\mathcal{A}) of all additive functors from \mathcal{A}^{op} into abelian groups; note that this category is abelian and ... 0answers 224 views ### Do differential objects form triangulated categories? Let \mathcal{A} be a (fixed) additive category. To a differential object (A,a) for \mathcal{A} (so, a:A\to A and a^2=0) one may associate an \mathcal{A}-complex \dots \to A\stackrel{a}{\... 0answers 286 views ### What is an Elementary “Homotopy, Model” Topos? Context: Def (Rezk): A (Grothendieck) homotopy topos is a homotopy left exact Bousfield localization of the model category of simplicial presheaves sPsh(C) on a small simplicial category C. Thm (... 0answers 136 views ### Test categories applied to Dold-Kan correspondence? Let's see how this goes, this might be a bit rushed, if you spot any mistakes feel free to correct them. A test category X is a category that can be used in place of the simplex category \Delta to ... 1answer 166 views ### Is the modularisation of a unitary fusion category always unitary? Suppose \mathcal{C} is a unitary ribbon fusion category. Also assume that its symmetric centre has trivial twist and trivial pivotal structure, i.e. is tannakian. Thus, the Müger/Bruguières ... 1answer 245 views ### Reference request: Morita bicategory I have two closely related questions: Who first recognized that Morita equivalence was equivalence in the bicategory of Rings, Bimodules, and Intertwiners? I've heard this bicategory called the "... 1answer 237 views ### What kinds of limits does localization of commutative rings reflect? Localization of commutative rings is a left exact left adjoint, so it behaves nicely with plenty of things. Local-to-global principles are also abundant in commutative algebra, and I thought some of ... 1answer 108 views ### Category enriched over a monoidal 2-category Consider a monoidal 2-category (or bicategory) B. For example, B could by the 2-category (finite sets, finite correspondences, isomorphisms of correspondences) with monoidal structure given by ... 2answers 1k views ### “Why” are monadicity and descent related? This question is probably too vague for experts, but I really don't know how to avoid it. I've read in several places that under mild conditions, a morphism is an effective descent morphism iff the ... 0answers 142 views ### Which nice subcategories of \mathsf{Top} are locally cartesian closed? For a class \mathcal{J} of topological spaces, let \mathsf{Top}_\mathcal{J} denote the category of \mathcal{J}-generated spaces, i.e. those spaces X such that U\subseteq X is open iff f^{-1}... 2answers 443 views ### What categorical property of monoidal categories picks out the ones with duals? Recall that a monoidal category \mathcal C is rigid if every object X\in \mathcal C has both left and right duals, i.e. objects X^l and X^r with maps X^l \otimes X \to \mathbf 1 \to X \otimes ... 1answer 839 views ### Topology of categories, very basic facts surrounding Quillen's Higher Algebraic K-Theory I In his paper Higher Algebraic K-Theory I (see [here][1]), Quillen introduces a topological space BC, called the classifying space of C, and tries to relate its topology to the categorical ... 1answer 169 views ### Faithful exact functors to tensor categories Let P be a "nice" k-linear abelian tensor category (e.g. A tannakian category or a fusion category over a field k) and F: M\to P an additive k-linear exact and faithful functor. I want to ... 0answers 230 views ### Many-sorted nominal sets as sheaves The category of nominal sets can also be presented as the Schanuel topos, the category of sheaves on \mathbb{I}^\mathsf{op} under the atomic topology, where \mathbb{I} is the category of finite ... 1answer 601 views ### Is there an ∞-categorical interpretation of the Quillen S⁻¹S construction? The Quillen S⁻¹S construction (not to be confused with the Quillen Q-construction or the Quillen plus-construction), as defined by Grayson in Higher algebraic K-theory: II (page 219), takes as an ... 0answers 206 views ### Defining Inertia Stack Let X be a topos and F: \zeta \rightarrow X a stack on X. Now in the paper http://arxiv.org/pdf/math/0411337v2.pdf ( Definitions 2.1.1.5 and 2.1.1.1) of Lieblich, he describes the inertia stack ... 1answer 255 views ### What is known about the large cardinal strength of Shelah's categoricity conjecture? Shelah's categoricity conjecture states that for every Abstract Elementary Class \mathcal{K} there is a cardinal \mu depending only on \operatorname{LS}(K) (i.e. the Löwenheim–Skolem number of \... 1answer 380 views ### P.G.Goerss, J.F.Jardine, “Simplicial Homotopy Theory” prerequisites I know that such questions may be better suited for math.stackexchange, but I believe that that the topic of simplicial homotopy theory is advanced enough for mathoverflow. Besides, I know that there ... 0answers 120 views ### Topos Theory, internal Heyting Algebra Given a topos \mathcal{E} with subobject classifier \Omega. If we denote by N\Omega the former of all local operators on \Omega, that is, Lawvere–Tierney topologies of \mathcal{E}, it is ... 1answer 133 views ### What is an example of a colimit-dense generator which is not dense? An object G of a category \mathcal{C} is a dense generator if every object X is the colimit of the canonical diagram of copies of G mapping to X. (This canonical diagram is indexed by the ... 1answer 192 views ### Left adjoint of pullback In Awodey's Category Theory, p233 of 2nd ed. (or p205 of 1st ed.), he states: Indeed, the UMP of pullbacks essentially states that composition along any function α is left adjoint to pullback ... 1answer 529 views ### Categorical or simplicial introduction to modern homotopy theory I've heard some people saying that it would be easier to study homotopy theory first through simplicial sets. First, i thought that these people were not being serious. But it caught me wondering... ... 0answers 203 views ### \mathcal{M}(\mathcal{D}_X) and \mathcal{M}^r(\mathcal{D}_X) have natural tensor category structures? Write \mathcal{M}^\ell(\mathcal{D}_{X/S}) = \mathcal{M}(\mathcal{D}_{X/S}) for the category of left \mathcal{D}-modules over X and \mathcal{M}^r(\mathcal{D}_{X/S}) for the category of right \... 0answers 146 views ### Coinduction in a presheaf topos? Fix a presheaf topos \mathcal{E}\triangleq \mathbf{Set}^{\mathcal{C}^\mathsf{op}}. Then, for any object/presheaf X:\mathcal{E}, I can define the presheaf of relations on X as \wp(X)\triangleq\... 2answers 307 views ### In a fibration, where does the generic object live? In Bart Jacobs. Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics 141. North Holland, 1999. isbn: 9780444508539, the author writes, p. 326: Sometimes, for ... 0answers 191 views ### Operadic Lift of Lurie's Relative Tensor Product In Section 4.4 of his book Higher Algebra, Lurie introduces, for a monoid object A of a monoidal quasicategory C, and right and left A-modules M,N, the relative tensor product M\otimes_AN. ... 0answers 100 views ### Finitely presented algebras with isomorphic semilattices of congruences Let \mathbb{T} be a finitary algebraic theory. For each \mathbb{T}-algebra A, let Q (A) be the join semilattice of finitely generated congruences on A. There is an evident pushforward ... 1answer 180 views ### Exterior derivative as only (up to multiple) natural operator \Lambda ^kT^\ast \rightsquigarrow \Lambda ^{k+1}T^\ast In Kolar, Michor, & Slovak's book Natural Operations in Differential Geometry, it is proved the exterior derivative is universal in the following sense. Proposition 25.4. For k>0 all natural ... 2answers 340 views ### Natural operators in differential geometry - why are they natural? I'm reading bits and pieces of Kolar, Michor, & Slovak's Natural Operations in differential Geometry, and I'm having "doubt" about some of the definitions. All I'm trying to do is sheafify some of ... 1answer 273 views ### Model independent proof of colimit formula for left Kan extensions I am interested in finding a proof of the colimit formula for left Kan extensions (\infty,1)-categories which does not rely on a chosen model. (By a model inpendent proof I mean a proof which uses ... 2answers 180 views ### Equivalence of natural transformations Let \mathcal{C} be a small category and \mathrm{Cat} be the 2-category of small categories. Let F,G : \mathcal{C} \to \mathrm{Cat} be two functors and \theta : F \to G be a natural ... 1answer 178 views ### Definition of Left Operadic Kan Extension for \infty-operads In Lurie's book Higher Algebra, he makes the following definition: Definition 3.1.2.2: Let M^\otimes\to N(Fin_\ast)\times\Delta^1 be a correspondence from an \infty-operad A^\otimes to another ... 1answer 319 views ### Does the category of categories-mod-natural-isomorphism have any nonobvious autoequivalences? The simpler question is to study the 2-groupoid \mathrm{Aut}(\mathsf{Cat}) of all autoequivalences of the category of (small) categories (i.e. the groupoid of equivalences of categories \mathsf{Cat}... 0answers 65 views ### Bi-adjointness and isomorphisms of (co)limits Without getting into too many (presumably irrelevant) details, I have a functor F:A \to B between reasonably mellow categories so that for each object b \in B_0 the universal morphism$$g_b: \... 1answer 204 views ### What is the right notion of generalized element of a category? I've been working out how the internal language of a category C extends to taking the category itself as a type. The most obvious way to interpret$X : \mathbf{C}$is, of course, that$X$is an ... 0answers 121 views ### Topologized category of bounded chain complex I am reading Segal's paper 'categories and cohomology theories' [1], but there is one claim (in the last example in sec.2) I don't quite understand: Let$\mathcal{C}$be the category of bounded chain ... 1answer 732 views ### Lemma 2 from Beilinson's “Coherent Sheaves on$\mathbb{P}^n$and Problems of Linear Algebra”, intuition? This is a followup to here. Consider Lemma 2 from Beilinson's paper "Coherent Sheaves on$\mathbb{P}^n$and Problems of Linear Algebra", as follows. Lemma 2. For any pair$i$,$j$such that$0 \...
In Lurie's book "Higher Topos Theory", a class $\mathsf{W}$ of morphisms in a category $\mathcal{A}$ is called perfect if Every isomorphism belongs to $\mathsf{W}$. it satisfies "2 out of 3 ...