The third problem set is available here. The fourth problem set is available here.
Simplicial Objects in Algebraic Topology: By May, J. P. | eBay
The fifth problem set is available here. Two more good Coq tutorials are here and here. References for simplicial sets: Riehl's leisurely introduction to simplicial sets Friedman's illustrated introduction to simplicial sets May's Simplicial objects in algebraic topology which is still the best comprehensive reference References for homotopy theory and model categories: May's A concise course in algebraic topology Hatcher's Algebraic topology Dwyer and Spalinski's notes A good reference for base-change functors in the context of spaces : May and Sigurdsson's Parametrized homotopy theory References for the connection between dependent type theory and locally cartesian-closed categories: Seely's original paper Hofmann's fix via fibrations Curien's fix via relaxing the substitution rule Curien, Garner, and Hofmann's perspective on the fixes The best reference for the correspondence between cartesian-closed categories and the untyped lambda calculus: Scott and Lambek's book Introduction to higher order categorical logic.
References for lambda calculus and logic: Barendregt's "The lambda calculus: Its syntax and semantics" brief notes.
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A similar construction can be performed for every category C , to obtain the nerve NC of C. Here, NC [ n ] is the set of all functors from [ n ] to C , where we consider [ n ] as a category with objects 0,1, In particular, the 0-simplices are the objects of C and the 1-simplices are the morphisms of C. The degeneracy maps s i lengthen the sequence by inserting an identity morphism at position i.
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We can recover the poset S from the nerve NS and the category C from the nerve NC ; in this sense simplicial sets generalize posets and categories. Another important class of examples of simplicial sets is given by the singular set SY of a topological space Y. Here SY n consists of all the continuous maps from the standard topological n -simplex to Y.
The singular set is further explained below. The following isomorphism shows that a simplicial set X is a colimit of its simplices: . In this process the orientation of the simplices of X is lost. The geometric realization is functorial on sSet. It is significant that we use the category CGHaus of compactly-generated Hausdorff spaces, rather than the category Top of topological spaces, as the target category of geometric realization: like sSet and unlike Top , the category CGHaus is cartesian closed ; the categorical product is defined differently in the categories Top and CGHaus , and the one in CGHaus corresponds to the one in sSet via geometric realization.
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The singular set of a topological space Y is the simplicial set SY defined by. This definition is analogous to a standard idea in singular homology of "probing" a target topological space with standard topological n-simplices. Furthermore, the singular functor S is right adjoint to the geometric realization functor described above, i. Intuitively, this adjunction can be understood as follows: a continuous map from the geometric realization of X to a space Y is uniquely specified if we associate to every simplex of X a continuous map from the corresponding standard topological simplex to Y, in such a fashion that these maps are compatible with the way the simplices in X hang together.
In order to define a model structure on the category of simplicial sets, one has to define fibrations, cofibrations and weak equivalences.
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One can define fibrations to be Kan fibrations. A map of simplicial sets is defined to be a weak equivalence if its geometric realization is a weak equivalence of spaces.
A map of simplicial sets is defined to be a cofibration if it is a monomorphism of simplicial sets. It is a difficult theorem of Daniel Quillen that the category of simplicial sets with these classes of morphisms satisfies the axioms for a proper closed simplicial model category.
A key turning point of the theory is that the geometric realization of a Kan fibration is a Serre fibration of spaces. With the model structure in place, a homotopy theory of simplicial sets can be developed using standard homotopical algebra methods. Furthermore, the geometric realization and singular functors give a Quillen equivalence of closed model categories inducing an equivalence.
It is part of the general definition of a Quillen adjunction that the right adjoint functor in this case, the singular set functor carries fibrations resp. When C is the category of sets , we are just talking about the simplicial sets that were defined above. The intuitions are given in the book "Nonabelian Algebraic Topology However this book does not deal with the analogues of simplicial groups, so there is still lots to do in this area.
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