In Bousfield and Kan's book"Homotopy Limits, Completions and Localizations",they define homotopy direct limit for system of pointed simplicial sets(Ch XII S2 2.1 P327), while they define homotopy inverse limit for system of simplicial sets without pointed condition(P 295), why do they insist this condition for direct limit?
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You can define homotopy limits and colimits in pointed as well as unpointed spaces. It so happens that the two notions of homotopy limit coincide, basically because the forgetful functor from pointed to unpointed spaces is a right Quillen adjoint. That is, the homotopy limit of a diagram of pointed spaces is the same whether taken in the pointed or unpointed category. On the other hand, pointed and unpointed homotopy colimits are not equivalent, and it is important to be clear which one you are using.
To see why this is plausible, think of products and coproducts. The categorical product is the same in the pointed and unpointed categories. But the pointed coproduct is the wedge sum while the unpointed coproduct is disjoint union.
More generally, the pointed homotopy colimit is the quotient of the unpointed homotopy colimit by the classifying space of the indexing category.
answered Jan 31, 2021 at 10:30
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$\begingroup$ Thank you for your answer ! Is there any reference book about definitions in both pointed and unpointed cases? $\endgroup$ Commented Jan 31, 2021 at 12:12
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2$\begingroup$ Try the book "Cubical homotopy theory" by Munson and Volic. It has a section about homotopy colimits where both versions are discussed. In particular, see Remark 8.2.14 $\endgroup$ Commented Jan 31, 2021 at 12:35