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  1. Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in almost all areas of mathematics.
    en.wikipedia.org/wiki/Category_theory
    en.wikipedia.org/wiki/Category_theory
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    In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called ' weak equivalences ', ' fibrations ' and ' cofibrations ' satisfying certain axioms relating them. These abstract from the category of topological spaces or of chain complexes ( derived category theory).
    en.wikipedia.org/wiki/Model_category
    Nevertheless, this is not at all what we will do here. To make a loose analogy, categorical model theory relates to studying definable sets of a fixed structure in roughly the same way that thermodynamics relates to quantum physics: category theory looks at an entire class of structures, together with morphisms between them.
    Sections 7 and 8 describe in detail two basic examples of model categories, namely the category Top of topological spaces and the category ChR of nonnegative chain complexes of modules over a ring R. The homotopy theory of Top is of course fa-miliar, and it turns out that the homotopy theory of ChR is what is usually called homological algebra.
    math.jhu.edu/~eriehl/616/DwyerSpalinski.pdf
    Part of model theory studies the definable subsets of such structures: sets that can be expressed from the relations and functions using simple set operations such as union, intersection, complement, and projection. This approach has had quite a bit of success (one of the easiest outcomes to describe is the theory of o- minimality [vdD98, PW06]).
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    Model category - Wikipedia

    In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstract from the category of topological spaces or of chain … See more

    Model categories can provide a natural setting for homotopy theory: the category of topological spaces is a model category, with the homotopy corresponding to the usual theory. Similarly, objects that are thought of as … See more

    Every closed model category has a terminal object by completeness and an initial object by cocompleteness, since these objects are the limit and colimit, respectively, of the … See more

    Cofibrations can be characterized as the maps which have the left lifting property with respect to acyclic fibrations, and acyclic cofibrations are characterized as the maps which have the left lifting property with respect to fibrations. Similarly, fibrations … See more

    The definition given initially by Quillen was that of a closed model category, the assumptions of which seemed strong at the time, motivating … See more

    Topological spaces
    The category of topological spaces, Top, admits a standard model category structure with the usual (Serre) fibrations and with weak equivalences as weak homotopy equivalences. The cofibrations are not the usual … See more

    The homotopy category of a model category C is the localization of C with respect to the class of weak equivalences. This definition of … See more

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  4. Category Theory - Stanford Encyclopedia of Philosophy

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