Definition df-termc 50055

Description: Definition of the proper class (termcnex 50158) of terminal categories, or final categories, i.e., categories with exactly one object and exactly one morphism, the latter of which is an identity morphism (termcid 50068). These are exactly the thin categories with a singleton base set. Example 3.3(4.c) of [Adamek] p. 24.

As the name indicates, TermCat is the class of all terminal objects in the category of small categories (termcterm3 50097). TermCat is also the class of categories to which all categories have exactly one functor (dftermc2 50102). See also dftermc3 50113 where TermCat is defined as categories with exactly one disjointified arrow.

Unlike https://ncatlab.org/nlab/show/terminal+category 50113, we reserve the term "trivial category" for (SetCat‘1_o), justified by setc1oterm 50073.

Followed directly from the definition, terminal categories are thin (termcthin 50059). The opposite category of a terminal category is "almost" itself (oppctermco 50087). Any category 𝐶 is isomorphic to the category of functors from a terminal category to the category 𝐶 (diagcic 50122).

Having defined the terminal category, we can then use it to define the universal property of initial (dfinito4 50083) and terminal objects (dftermo4 50084). The universal properties provide an alternate proof of initoeu1 18035, termoeu1 18042, initoeu2 18040, and termoeu2 49820. Since terminal categories are terminal objects, all terminal categories are mutually isomorphic (termcciso 50098).

The dual concept is the initial category, or the empty category (Example 7.2(3) of [Adamek] p. 101). See 0catg 17711, 0thincg 50040, func0g 49671, 0funcg 49667, and initc 49673.

(Contributed by Zhi Wang, 16-Oct-2025.)

Assertion

Ref	Expression
df-termc	⊢ TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

Distinct variable group:   𝑥,𝑐

Detailed syntax breakdown of Definition df-termc

Step	Hyp	Ref	Expression
1		ctermc 50054	. 2 class TermCat
2		vc	. . . . . . 7 setvar 𝑐
3	2	cv 1558	. . . . . 6 class 𝑐
4		cbs 17236	. . . . . 6 class Base
5	3, 4	cfv 6516	. . . . 5 class (Base‘𝑐)
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1558	. . . . . 6 class 𝑥
8	7	csn 4579	. . . . 5 class {𝑥}
9	5, 8	wceq 1559	. . . 4 wff (Base‘𝑐) = {𝑥}
10	9, 6	wex 1798	. . 3 wff ∃𝑥(Base‘𝑐) = {𝑥}
11		cthinc 49999	. . 3 class ThinCat
12	10, 2, 11	crab 3413	. 2 class {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
13	1, 12	wceq 1559	1 wff TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

This definition is referenced by:  istermc  50056