F. Borceux. Handbook of Categorical Algebra 2: Categories and Structures. Number 51 in Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... (TmX ) 11) The morphisms X,x are completely determined by their restrictions c X,x,x : Ta(X, a(mX (X) x) TTX, x X) and with respect to these composition morphisms the commutativity conditions (10) and (11) simply become generalizations of the axioms for a V category (see [15] [4]) x)# a(e X (x) x) Ta(e TX (x) e X (x) # a(x, x) a(x, x)# c e TX (x) e X (x) x 11 a(x, x) Ta( TeX ) x) # # # (Tu) # # # TeX (x) x,x Ta(m TX (X) mX (X) # a(mX (X) x) a(mX (m TX (X) x) x) # a(x, x) T (Ta(X, a(x, x) a(mX (X) x) Ta(TmX (X) c m TX ....

F. Borceux, Handbook of Categorical Algebra 2: Categories and Structures (Cambridge University Press, Cambridge 1994).

....) for varying n and i form a set of finitely presentable strong generators. So the category of # spaces if locally finitely presentable. The main property of locally presentable categories for our purpose is that every object is presentable, hence small with respect to the whole category (see [Bor, proposition 5210] I injectives, I cofibrations and regular I cofibrations. Given a cocomplete category C and a set I of maps, we denote . by I inj the subcategory of C consisting of maps which have the right lifting property with respect to the maps in I.MapsinI inj are referred to as I ....

....model category. Remark B3. Note that the smallness condition (1) of the lifting lemma is automatically satisfied if the model category C is cofibrantly generated and the category D is locally presentable (see App. A) because then every object of D is small with respect to the whole category [Bor, proposition 5210] Proof. We use the numbering of the model category axioms as given in [BF,11] Limits and colimits exist in D by assumption. Model category axioms CM2 (saturation) and CM3 (closure properties under retracts) are clear. One half of CM4 (lifting properties) holds by definition ....

F. Borceux. Handbook of categorical algebra 2: categories and structures. Encyclopedia of Mathematics and its Applications 51 (Cambridge University Press, 1994).

....in linear extensions of algebraic theories which generalize the well known principle for modules (see for example Proposition III.2.10 of [1] 1. Preliminaries on algebraic theories. We recall the notion of algebraic theory in the sense of Lawvere. Good reference for algebraic theories is [4] The algebraic theory, or simply theory, formalizes the concept of an algebraic object as a set together with n ary operations for various n 2 N and equational relations. 1.1. Definition. A theory is a category T with the following properties: T has a set of objects equal to the set of natural ....

....T alg Sets; X 7 X(1) has a left adjoint, the free T algebra functor F T : Sets T alg. We let n T (or simply n ) denote the free T algebra generated by f1; Delta Delta Delta ; ng. The category T op is equivalent to the category of finitely generated free T algebras. 1. 4. Algebraic functors. A morphism of theories OE : T 1 T induces an adjoint functor pair analogous to scalar restrictions and extensions: if X is a T algebra, i.e. a product preserving functor X : T Sets, then the composite XOE : T 1 Sets is again product preserving, so it defines an T 1 ....

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F. Borceux, Handbook of Categorical Algebra 2: Categories and Structures, Encyclopedia of Mathematics and its Applications 51, Cambridge University Press (1994).

.... notion of weighted colimit (or indexed colimit) BK75] and makes it possible to continue Lawvere s approach to the theory of metric spaces [Law73] by applying further insights from enriched category theory, in particular various results on weighted colimits (and their dual, weighted limits) [Kel82, Bor94]. As a consequence, many of the recently proposed definitions of generalized metric limit turn out to be theorems in enriched category theory. Furthermore, many other types of limits , such as the least upper bound of a directed subset or the limit of a Cauchy net, are expressible as weighted ....

....The enriched categorical definitions of weighted limit and colimit [BK75] are given for the special case of [0; 1] categories, that is, generalized metric spaces. Most definitions and facts of the present section are instances of general enriched categorical versions of them, see [Kel82] or [Bor94]. For all facts, elementary proofs can be given as well, some of which have been included here. In the next section, we shall show that limits of Cauchy sequences are weighted (co)limits. Let D and X be generalized metric spaces, and let f : D X; g : D [0; 1] be non expansive functions. An ....

F. Borceux. Handbook of categorical algebra 2: categories and structures, volume 51 of Encyclopedia of mathematics and its applications. Cambridge University Press, 1994.

....of models of the sketch. Note that (co)limits are finite (co)limits and that 1 (co)limits are countable (co)limits. However, filtered colimits are the usual filtered colimits, which are equivalent for many purposes to directed colimits, but strictly generalize them. The exposition in [4, 10, 11] gives the general case. Object C of category C is said to be presentable if the representable functor C(C; Gamma) C Sets preserves filtered colimits. The category A is said to be QUERY DATA DUALITY 11 accessible if A has filtered colimits and if there exists a small subcategory C ....

.... filtered colimits. The category A is said to be QUERY DATA DUALITY 11 accessible if A has filtered colimits and if there exists a small subcategory C op of A in which every object is presentable such that every object of A is a filtered colimit of a filtered diagram in C op , [4, 20, 15, 16, 11]. A functor is said to be flat if it is a filtered colimit of representable functors, 19, 20, 10] Every accessible category is, up to equivalence, presented as a category of flat functors from the opposite of the small subcategory of presentables to Sets, 20, 11] Therefore, each object ....

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Francis Borceaux, Handbook of Categorical Algebra 2: Categories and Structures, Cambridge University Press, 1994.

.... and to which we will often refer) Although this work involves enriched categories, no knowledge of them is either assumed or even necessary for a basic understanding of what follows; 2 a gentler introduction into enriched category theory than the somewhat demanding standard text [Kel82] is [Bor94, Chapter 6]. This article is an extract of the author s forthcoming thesis [Lu96] Without referring to it explicitly in the following, the thesis will present the material of the present article, some of which can only be adumbrated due to length limitations, in far more detail. I would like to thank Don ....

Francis Borceux. Handbook of Categorical Algebra 2: Categories and Structures. Number 51 in Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1994.

.... to which we will often refer) Although this work implicitly involves enriched categories, no knowledge of them is either assumed or even necessary for a basic understanding of what follows; a gentler introduction into enriched category theory than the somewhat demanding standard text [Kel82] is [Bor94, Chapter 6]. We would like to thank Don Sannella and Stefan Kahrs for many stimulating discussions. Get well soon Alan. Glory, Glory to the Hibees 2 Universal Algebra and Monads Definition 1 (Monad) A monad T = hT ; j; i on a category C is given by an endofunctor T : C C, called the action, and two ....

Francis Borceux. Handbook of Categorical Algebra 2: Categories and Structures. Number 51 in Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1994.

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F. Borceux. Handbook of Categorical Algebra 2: Categories and Structures. Number 51 in Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1994.

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Francis Borceux, Handbook of Categorical Algebra 2: Categories and Structures (1994). Cambridge University Press.

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F. Borceux, Handbook of Categorical Algebra 2: Categories and Structures, Encyclopedia of Mathematics and its Applications, Vol. 51, Cambridge University Press, Cambridge, 1994.

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Francis Borceux. Handbook of Categorical Algebra 2: Categories and Structures, volume 51 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1994.

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